space travel math

Space Math @ NASA

This site contains over 700 math problems, for all ages, relating to a variety of astronomical topics. Sort problems by astronomy topic or math course. Check out these problems related to the Sun: https://spacemath.gsfc.nasa.gov/sun.html

Contrary to everyday experience on Earth, the most efficient route through space may not be a straight line. Some mathematicians and NASA engineers have learned in recent years that take best advantage of gravity, and save fuel in the process, it may be necessary to make bizarre loops through space.

Take for example the Genesis spacecraft. In 2004, it started on its way home after having spent two years collecting solar particles in orbit around a Lagrange point , a point between Earth and the Sun where the gravity of both bodies is balanced.

But Genesis did not come straight home. It took a long curvy path, going past the Earth to make an extended million-mile loop (around another Lagrange point) before coming back to Earth. Amazingly, this seeming impractical path actually saves fuel by making use of gravity in the Earth-Sun system.

The design of interplanetary space probes can now be done by taking advantage of the competing gravitational tugs of the different planets and their moons, which create a vast network of passageways by which a spacecraft can travel over large distances while expending very little energy. Without recent advances in mathematics and computation, these fuel-saving, mission-enabling paths through space could not be found.

Click the links to the right to learn more about the gravitational passageways which form this interplanetary transport network.

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3 things you should know about how mathematics is used for space exploration, how honeybees are masters of geometry, and the uniqueness of a shuffled deck of cards..

Michelle Campbell Mekarski, PhD Canada Science and Technology Museum

Cassandra Marion, PhD Canada Aviation and Space Museum

Meet Cassandra Marion, Renée-Claude Goulet, and Michelle Campbell Mekarski.

They are Ingenium’s science advisors, providing expert scientific advice on key subjects relating to the Canada Aviation and Space Museum, the Canada Agriculture and Food Museum, and the Canada Science and Technology Museum.

In this colourful monthly blog series, Ingenium’s science advisors offer up quirky nuggets related to their areas of expertise. For the September edition, they explain how mathematics is an indispensable tool for space exploration, how honeybees are masters of geometry, and about the unbelievable uniqueness of a shuffled deck of cards.

Consider the exacting calculations required to launch the Lunar Module ascent stage from the Moon’s surface and rendezvous with the orbiting Apollo 11 command module from which this photo was taken, some 384,000 kms from Earth.

Math: an indispensable tool for space exploration

Mathematics are essential for the observation, exploration and understanding of space. From basic arithmetic, calculus, algebra, geometry, and statistics and probabilities.  It is interwoven into the fields of aerospace engineering, astronomy and planetary science. Without it, we would not have achieved spaceflight, we wouldn’t have satellite radio or GPS, we wouldn’t have landed on the Moon or flown a helicopter on Mars, or have ever seen high resolution images of Pluto. Here we review just a few of the countless ways we use math for space exploration.

To put a spacecraft in space, you first need a basic understanding of how our solar system operates. I’m referring to equations that describe gravity, Newton’s laws of motion, Kepler’s laws of planetary motion (how planets move in elliptical orbits around the Sun), and so on, that explain the forces acting on and motions of celestial bodies. These concepts are collectively known as “orbital mechanics,” which we have discovered or learned, through hundreds of years of observation, theory and manipulated equations.

Planning a mission to space requires math from simple unit conversions to budget management, to calculating astronomical distances, to modelling a spacecraft’s trajectory. To get to orbit, a rocket must launch off the surface of a rotating planet, which itself is also moving through space along its orbit around the Sun, all at different velocities.  To get to the Moon, we add more variables and must predict the position of the Moon relative to the Earth when the spacecraft will arrive.

To launch a rocket requires several types of calculations starting with how much thrust – the force working opposite gravity and drag in this case – you’ll need to lift the weight of the rocket’s payload from the Earth’s surface, through its atmosphere and accelerate it to orbital velocity (7.8 km/s). A commonly used rocket equation enables us to do that by calculating the delta-v (Δv), the difference in velocity that is proportional to the thrust per unit mass and the burn time of the rocket’s engines. We must also determine the specific amount of propellant needed to fire the engines for the proper duration. Furthermore, these values are needed for each rocket stage as most modern rockets consist of several rockets stacked one atop another which progressively separate at pre-determined altitudes to reduce the total mass of the spacecraft. Once in orbit, computer modeling is needed to track the spacecraft’s position and speed, and navigate through space.

Instruments onboard spacecraft and telescopes on the Earth’s surface are used for a range of communications, navigation and remote sensing (e.g., cameras, spectrometers), and require math to create, operate and/or interpret returned data. With these, we have determined and measured characteristics and features of planetary bodies millions of kilometres away.  Math is an exceptional and indispensable tool for understanding and exploring our solar system and beyond.

Space Math problems for classrooms @ NASA

By Cassandra Marion

Cells on a comb are exactly the right size to fit a honeybee. They serve as storage for pollen and honey, and as nursery for new bees.

Honeybees: masters of geometry

Honeybees are amazing engineers. Their innate ability to build complex wax honeycomb structures, with symmetrical, repeating hexagonal patterns has long puzzled bee observers and mathematicians alike. Why the hexagon, over any other shape? And how do bees know what to build? It appears nature figured out long before we did that the hexagon is simply the most efficient shape to build with.   From twelve to twenty days old, bees have a special gland in their stomach allowing them to convert the sugars they eat into wax. The wax comes out of their abdomen in flakes, which they then chew and apply to the structure in construction. Producing wax takes a lot of energy; a bee needs to eat 8 grams of honey for each gram of wax she makes. So every flake is precious, representing the work of the entire hive, and must be used most efficiently.    Honeybees use honeycombs as storage and nursery, so it’s logical they would also want to store the most in the least amount of space. A circular cell would hold the most, but when placed in a grid, each cell needs its own wall, and there is dead space between the circles – not a great use of a precious material. Bees have three basic shape options for tiling a surface with 100 percent coverage and no overlap: equilateral triangles, squares, or hexagons. These shapes are special because each of their sides can be shared with a neighbour, creating what is called a tessellation.   So why the hexagon, and not the triangle or the square? Because it reduces the amount of material needed for construction, while maximizing the storage capacity! This explanation is known as the “Honeycomb Conjecture,” which is thought to have been proposed some 2000 years ago, and at many other points in history. The idea is that hexagons are the most efficient way to pack shapes on a surface, so that each cell of the same size contains the maximum surface area, with the least perimeter.  It took until 1999 for this to be proven mathematically, by an American mathematician named Thomas C. Hales.     Another great reason for the hexagon? Research has shown that due to the repeating 120-degree angles in each corner of the cells, the forces in the structure are in balance. This makes the honeycomb structure very strong, stable, and resistant to crushing, allowing bees to build the cell walls thinner than a sheet of paper!   Thanks to mathematics, we know that bees build structures that maximize space and stability, while minimizing building material. This knowledge can come in handy as we innovate to find new solutions in building materials, architecture, materials, and even in electronics. Today, bees continue to be a popular subject of study, and they surely have many other lessons to teach us! 

By Renée-Claude Goulet

Un paquet de cartes mélangé tenu en éventail dans une main.

The unbelievable uniqueness of a shuffled deck

Find a standard deck of cards, shuffle it, and look at the order. 

You are almost certainly holding an arrangement of cards that has never existed in history. In fact, it’s almost certain that if humans continue to use playing cards for millions of years, no one will shuffle a deck of cards into that exact order ever again! 

How is that possible? Just how many ways are there to arrange a deck of cards?

Let’s start small. Imagine we have a deck with only two cards. We shuffle it, and lay out the cards one at a time. We have two possible options for the first card. If we lay that card down, we are left with one card to place second. So, in a two card deck, there are two possible ways to shuffle the deck: card A followed by card B or, card B followed by card A. 

What about a three card deck? We have three possible options for the first card. If we lay the first card down, that leaves two possible options for the second card. If we place the second card next to the first, we are left with one card, which goes last. If we write out all the possible combinations of cards, they are: ABC, ACB, BAC, BCA, CAB, or CBA. So, there are six possible ways to shuffle our deck of three cards (three times as many ways as in the two card deck). 

In a four card deck, we would have four options for the first card, which leaves three options for the second card, two options for the third, and the last card by default is placed fourth. In a four card deck, there are 24 possible ways to shuffle the deck (four times as many as in the three card deck). 

Noticing a pattern? We can calculate the number of possible combinations of any size deck of cards using a simple formula called a factorial . Start with the number of cards in your deck and multiply it by consecutively smaller whole numbers until you get to one. 

In a 2 card deck: 2 x 1 = 2 

In a 3 card deck: 3 x 2 x 1 = 6 

In a 4 card deck: 4 x 3 x 2 x 1 = 24

In a 5 card deck: 5 x 4 x 3 x 2 x 1 = 120

And so on. 

A deck of 10 cards has more than 3 million combinations. 

A deck of 20 cards has about two and a half quintillion combination (that’s 18 zeros). 

A standard 52 card deck has 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 combinations (that’s 68 digits long!). That is an incomprehensibly large number. It’s FAR larger than the age of the universe if we counted in milliseconds (the universe is 13.8 billion years old). It’s more combinations than there are atoms in our entire galaxy (which contains more than 100,000,000,000 stars ).  

Whenever you shuffle a deck of cards, you’re holding something that will (almost certainly) only exist this one time, ever. It’s not magic. It’s math. 

By Michelle Campbell Mekarski

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Renée-Claude is the Science Advisor at the Canada Agriculture and Food Museum, and an Ontario Certified Teacher. Through her background in biology, education and many years of experience creating and delivering programs and exhibits at the museum, she has developed an expertise in communicating key issues related to the science and innovation behind production of food, fibre and fuel, to a wide range of audiences.

As the Science Advisor at the Canada Science and Technology Museum, Michelle’s goal is to bridge the gap between the scientific community and the public — specializing in making science and technology engaging, accessible, and fun. Michelle earned a PhD in evolutionary biology and paleontology and has many years of experience developing and delivering science outreach activities. When away from her job at the museum, she can be found teaching at the University of Ottawa or Carleton University, digging for fossils, or relaxing by the water.

Cassandra is the Science Advisor for the Canada Aviation and Space Museum. She has a PhD in geology and planetary science and exploration. Her research is focused on meteorite impact craters in the Canadian Arctic. She has more than a decade of experience in education and public outreach, developing and delivering science programming. She is dedicated to sharing her passion for Earth and planetary sciences with communities near and far, and improving science literacy in Canada.

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Space Travel Calculator

Table of contents

Ever since the dawn of civilization, the idea of space travel has fascinated humans! Haven't we all looked up into the night sky and dreamed about space?

With the successful return of the first all-civilian crew of SpaceX's Inspiration4 mission after orbiting the Earth for three days, the dream of space travel looks more and more realistic now.

While traveling deep into space is still something out of science fiction movies like Star Trek and Star Wars, the tremendous progress made by private space companies so far seems very promising. Someday, space travel (or even interstellar travel) might be accessible to everyone!

It's never too early to start planning for a trip of a lifetime (or several lifetimes). You can also plan your own space trip and celebrate World Space Week in your own special way!

This space travel calculator is a comprehensive tool that allows you to estimate many essential parameters in theoretical interstellar space travel . Have you ever wondered how fast we can travel in space, how much time it will take to get to the nearest star or galaxy, or how much fuel it requires? In the following article, using a relativistic rocket equation, we'll try to answer questions like "Is interstellar travel possible?" , and "Can humans travel at the speed of light?"

Explore the world of light-speed travel of (hopefully) future spaceships with our relativistic space travel calculator!

If you're interested in astrophysics, check out our other calculators. Find out the speed required to leave the surface of any planet with the escape velocity calculator or estimate the parameters of the orbital motion of planets using the orbital velocity calculator .

One small step for man, one giant leap for humanity

Although human beings have been dreaming about space travel forever, the first landmark in the history of space travel is Russia's launch of Sputnik 2 into space in November 1957. The spacecraft carried the first earthling, the Russian dog Laika , into space.

Four years later, on 12 April 1961, Soviet cosmonaut Yuri A. Gagarin became the first human in space when his spacecraft, the Vostok 1, completed one orbit of Earth.

The first American astronaut to enter space was Alan Shepard (May 1961). During the Apollo 11 mission in July 1969, Neil Armstrong and Buzz Aldrin became the first men to land on the moon. Between 1969 and 1972, a total of 12 astronauts walked the moon, marking one of the most outstanding achievements for NASA.

In recent decades, space travel technology has seen some incredible advancements. Especially with the advent of private space companies like SpaceX, Virgin Galactic, and Blue Origin, the dream of space tourism is looking more and more realistic for everyone!

However, when it comes to including women, we are yet to make great strides. So far, 566 people have traveled to space. Only 65 of them were women .

Although the first woman in space, a Soviet astronaut Valentina Tereshkova , who orbited Earth 48 times, went into orbit in June 1963. It was only in October 2019 that the first all-female spacewalk was completed by NASA astronauts Jessica Meir and Christina Koch.

Women's access to space is still far from equal, but there are signs of progress, like NASA planning to land the first woman and first person of color on the moon by 2024 with its Artemis missions. World Space Week is also celebrating the achievements and contributions of women in space this year!

In the following sections, we will explore the feasibility of space travel and its associated challenges.

How fast can we travel in space? Is interstellar travel possible?

Interstellar space is a rather empty place. Its temperature is not much more than the coldest possible temperature, i.e., an absolute zero. It equals about 3 kelvins – minus 270 °C or minus 455 °F. You can't find air there, and therefore there is no drag or friction. On the one hand, humans can't survive in such a hostile place without expensive equipment like a spacesuit or a spaceship, but on the other hand, we can make use of space conditions and its emptiness.

The main advantage of future spaceships is that, since they are moving through a vacuum, they can theoretically accelerate to infinite speeds! However, this is only possible in the classical world of relatively low speeds, where Newtonian physics can be applied. Even if it's true, let's imagine, just for a moment, that we live in a world where any speed is allowed. How long will it take to visit the Andromeda Galaxy, the nearest galaxy to the Milky Way?

We will begin our intergalactic travel with a constant acceleration of 1 g (9.81 m/s² or 32.17 ft/s²) because it ensures that the crew experiences the same comfortable gravitational field as the one on Earth. By using this space travel calculator in Newton's universe mode, you can find out that you need about 2200 years to arrive at the nearest galaxy! And, if you want to stop there, you need an additional 1000 years . Nobody lives for 3000 years! Is intergalactic travel impossible for us, then? Luckily, we have good news. We live in a world of relativistic effects, where unusual phenomena readily occur.

Can humans travel at the speed of light? – relativistic space travel

In the previous example, where we traveled to Andromeda Galaxy, the maximum velocity was almost 3000 times greater than the speed of light c = 299,792,458 m/s , or about c = 3 × 10 8 m/s using scientific notation.

However, as velocity increases, relativistic effects start to play an essential role. According to special relativity proposed by Albert Einstein, nothing can exceed the speed of light. How can it help us with interstellar space travel? Doesn't it mean we will travel at a much lower speed? Yes, it does, but there are also a few new relativistic phenomena, including time dilation and length contraction, to name a few. The former is crucial in relativistic space travel.

Time dilation is a difference of time measured by two observers, one being in motion and the second at rest (relative to each other). It is something we are not used to on Earth. Clocks in a moving spaceship tick slower than the same clocks on Earth ! Time passing in a moving spaceship T T T and equivalent time observed on Earth t t t are related by the following formula:

where γ \gamma γ is the Lorentz factor that comprises the speed of the spaceship v v v and the speed of light c c c :

where β = v / c \beta = v/c β = v / c .

For example, if γ = 10 \gamma = 10 γ = 10 ( v = 0.995 c v = 0.995c v = 0.995 c ), then every second passing on Earth corresponds to ten seconds passing in the spaceship. Inside the spacecraft, events take place 90 percent slower; the difference can be even greater for higher velocities. Note that both observers can be in motion, too. In that case, to calculate the relative relativistic velocity, you can use our velocity addition calculator .

Let's go back to our example again, but this time we're in Einstein's universe of relativistic effects trying to reach Andromeda. The time needed to get there, measured by the crew of the spaceship, equals only 15 years ! Well, this is still a long time, but it is more achievable in a practical sense. If you would like to stop at the destination, you should start decelerating halfway through. In this situation, the time passed in the spaceship will be extended by about 13 additional years .

Unfortunately, this is only a one-way journey. You can, of course, go back to Earth, but nothing will be the same. During your interstellar space travel to the Andromeda Galaxy, about 2,500,000 years have passed on Earth. It would be a completely different planet, and nobody could foresee the fate of our civilization.

A similar problem was considered in the first Planet of the Apes movie, where astronauts crash-landed back on Earth. While these astronauts had only aged by 18 months, 2000 years had passed on Earth (sorry for the spoilers, but the film is over 50 years old at this point, you should have seen it by now). How about you? Would you be able to leave everything you know and love about our galaxy forever and begin a life of space exploration?

Space travel calculator – relativistic rocket equation

Now that you know whether interstellar travel is possible and how fast we can travel in space, it's time for some formulas. In this section, you can find the "classical" and relativistic rocket equations that are included in the relativistic space travel calculator.

There could be four combinations since we want to estimate how long it takes to arrive at the destination point at full speed as well as arrive at the destination point and stop. Every set contains distance, time passing on Earth and in the spaceship (only relativity approach), expected maximum velocity and corresponding kinetic energy (on the additional parameters section), and the required fuel mass (see Intergalactic travel — fuel problem section for more information). The notation is:

• a a a — Spaceship acceleration (by default 1   g 1\rm\, g 1 g ). We assume it is positive a > 0 a > 0 a > 0 (at least until halfway) and constant.
• m m m — Spaceship mass. It is required to calculate kinetic energy (and fuel).
• d d d — Distance to the destination. Note that you can select it from the list or type in any other distance to the desired object.
• T T T — Time that passed in a spaceship, or, in other words, how much the crew has aged.
• t t t — Time that passed in a resting frame of reference, e.g., on Earth.
• v v v — Maximum velocity reached by the spaceship.
• K E \rm KE KE — Maximum kinetic energy reached by the spaceship.

The relativistic space travel calculator is dedicated to very long journeys, interstellar or even intergalactic, in which we can neglect the influence of the gravitational field, e.g., from Earth. We didn't include our closest celestial bodies, like the Moon or Mars, in the destination list because it would be pointless. For them, we need different equations that also take into consideration gravitational force.

Newton's universe — arrive at the destination at full speed

It's the simplest case because here, T T T equals t t t for any speed. To calculate the distance covered at constant acceleration during a certain time, you can use the following classical formula:

Since acceleration is constant, and we assume that the initial velocity equals zero, you can estimate the maximum velocity using this equation:

and the corresponding kinetic energy:

Newton's universe — arrive at the destination and stop

In this situation, we accelerate to the halfway point, reach maximum velocity, and then decelerate to stop at the destination point. Distance covered during the same time is, as you may expect, smaller than before:

Acceleration remains positive until we're halfway there (then it is negative – deceleration), so the maximum velocity is:

and the kinetic energy equation is the same as the previous one.

Einstein's universe — arrive at the destination at full speed

The relativistic rocket equation has to consider the effects of light-speed travel. These are not only speed limitations and time dilation but also how every length becomes shorter for a moving observer, which is a phenomenon of special relativity called length contraction. If l l l is the proper length observed in the rest frame and L L L is the length observed by a crew in a spaceship, then:

What does it mean? If a spaceship moves with the velocity of v = 0.995 c v = 0.995c v = 0.995 c , then γ = 10 \gamma = 10 γ = 10 , and the length observed by a moving object is ten times smaller than the real length. For example, the distance to the Andromeda Galaxy equals about 2,520,000 light years with Earth as the frame of reference. For a spaceship moving with v = 0.995 c v = 0.995c v = 0.995 c , it will be "only" 252,200 light years away. That's a 90 percent decrease or a 164 percent difference!

Now you probably understand why special relativity allows us to intergalactic travel. Below you can find the relativistic rocket equation for the case in which you want to arrive at the destination point at full speed (without stopping). You can find its derivation in the book by Messrs Misner, Thorne ( Co-Winner of the 2017 Nobel Prize in Physics ) and Wheller titled Gravitation , section §6.2. Hyperbolic motion. More accessible formulas are in the mathematical physicist John Baez's article The Relativistic Rocket :

• Time passed on Earth:
• Time passed in the spaceship:
• Maximum velocity:
• Relativistic kinetic energy remains the same:

The symbols sh ⁡ \sh sh , ch ⁡ \ch ch , and th ⁡ \th th are, respectively, sine, cosine, and tangent hyperbolic functions, which are analogs of the ordinary trigonometric functions. In turn, sh ⁡ − 1 \sh^{-1} sh − 1 and ch ⁡ − 1 \ch^{-1} ch − 1 are the inverse hyperbolic functions that can be expressed with natural logarithms and square roots, according to the article Inverse hyperbolic functions on Wikipedia.

Einstein's universe – arrive at destination point and stop

Most websites with relativistic rocket equations consider only arriving at the desired place at full speed. If you want to stop there, you should start decelerating at the halfway point. Below, you can find a set of equations estimating interstellar space travel parameters in the situation when you want to stop at the destination point :

Intergalactic travel – fuel problem

So, after all of these considerations, can humans travel at the speed of light, or at least at a speed close to it? Jet-rocket engines need a lot of fuel per unit of weight of the rocket. You can use our rocket equation calculator to see how much fuel you need to obtain a certain velocity (e.g., with an effective exhaust velocity of 4500 m/s).

Hopefully, future spaceships will be able to produce energy from matter-antimatter annihilation. This process releases energy from two particles that have mass (e.g., electron and positron) into photons. These photons may then be shot out at the back of the spaceship and accelerate the spaceship due to the conservation of momentum. If you want to know how much energy is contained in matter, check out our E = mc² calculator , which is about the famous Albert Einstein equation.

Now that you know the maximum amount of energy you can acquire from matter, it's time to estimate how much of it you need for intergalactic travel. Appropriate formulas are derived from the conservation of momentum and energy principles. For the relativistic case:

where e x e^x e x is an exponential function, and for classical case:

Remember that it assumes 100% efficiency! One of the promising future spaceships' power sources is the fusion of hydrogen into helium, which provides energy of 0.008 mc² . As you can see, in this reaction, efficiency equals only 0.8%.

Let's check whether the fuel mass amount is reasonable for sending a mass of 1 kg to the nearest galaxy. With a space travel calculator, you can find out that, even with 100% efficiency, you would need 5,200 tons of fuel to send only 1 kilogram of your spaceship . That's a lot!

So can humans travel at the speed of light? Right now, it seems impossible, but technology is still developing. For example, a photonic laser thruster is a good candidate since it doesn't require any matter to work, only photons. Infinity and beyond is actually within our reach!

How do I calculate the travel time to other planets?

To calculate the time it takes to travel to a specific star or galaxy using the space travel calculator, follow these steps:

• Choose the acceleration : the default mode is 1 g (gravitational field similar to Earth's).
• Enter the spaceship mass , excluding fuel.
• Select the destination : pick the star, planet, or galaxy you want to travel to from the dropdown menu.
• The distance between the Earth and your chosen stars will automatically appear. You can also input the distance in light-years directly if you select the Custom distance option in the previous dropdown.
• Define the aim : select whether you aim to " Arrive at destination and stop " or “ Arrive at destination at full speed ”.
• Pick the calculation mode : opt for either " Einstein's universe " mode for relativistic effects or " Newton's universe " for simpler calculations.
• Time passed in spaceship : estimated time experienced by the crew during the journey. (" Einstein's universe " mode)
• Time passed on Earth : estimated time elapsed on Earth during the trip. (" Einstein's universe " mode)
• Time passed : depends on the frame of reference, e.g., on Earth. (" Newton's universe " mode)
• Required fuel mass : estimated fuel quantity needed for the journey.
• Maximum velocity : maximum speed achieved by the spaceship.

How long does it take to get to space?

It takes about 8.5 minutes for a space shuttle or spacecraft to reach Earth's orbit, i.e., the limit of space where the Earth's atmosphere ends. This dividing line between the Earth's atmosphere and space is called the Kármán line . It happens so quickly because the shuttle goes from zero to around 17,500 miles per hour in those 8.5 minutes .

How fast does the space station travel?

The International Space Station travels at an average speed of 28,000 km/h or 17,500 mph . In a single day, the ISS can make several complete revolutions as it circumnavigates the globe in just 90 minutes . Placed in orbit at an altitude of 350 km , the station is visible to the naked eye, looking like a dot crossing the sky due to its very bright solar panels.

How do I reach the speed of light?

To reach the speed of light, you would have to overcome several obstacles, including:

Mass limit : traveling at the speed of light would mean traveling at 299,792,458 meters per second. But, thanks to Einstein's theory of relativity, we know that an object with non-zero mass cannot reach this speed.

Energy : accelerating to the speed of light would require infinite energy.

Effects of relativity : from the outside, time would slow down, and you would shrink.

Why can't sound travel in space?

Sound can’t travel in space because it is a mechanical wave that requires a medium to propagate — this medium can be solid, liquid, or gas. In space, there is no matter, or at least not enough for sound to propagate. The density of matter in space is of the order 1 particle per cubic centimeter . While on Earth , it's much denser at around 10 20 particles per cubic centimeter .

Dreaming of traveling into space? 🌌 Plan your interstellar travel (even to a Star Trek destination) using this calculator 👨‍🚀! Estimate how fast you can reach your destination and how much fuel you would need 🚀

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ol{padding-top:0px;}.css-4okk7a ul:not(:first-child),.css-4okk7a ol:not(:first-child){padding-top:4px;} Spaceship and destination 👩‍🚀👨‍🚀

Spaceship acceleration

Spaceship mass

Mass of spaceship excluding fuel.

Destination

Select a destination from the list or type in distance by hand.

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Do you want to stop at destination point? If yes, the spaceship will start decelerating once it reaches the halfway point.

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You can compare Einstein's special relativity with non-relativistic Newton's physics. Remember that at near-light speeds only the former is correct!

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Time passed in spaceship

Time passed on Earth

Time passed in the resting frame of reference. It could be an observer on Earth.

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Assuming 100% efficiency.

Maximum velocity

Note that our calculator may round velocity to the speed of light if it is really close to it.

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Voyager: The mathematics of interstellar space travel

One of the key equations needed to understand spacecraft navigation is Newton's inverse square law of gravity:

We learned this in Physics 101 in the university as engineers. Essentially, the force measured (F) is gravity’s (G) work attracting two masses (m 1 , m 2 ) that weakens/strengthens as the square of the distance between those two bodies changes. So gravity’s attraction effect is virtually infinite according to the inverse square law. The Sun’s mass is pretty huge.

Grasping this law we can further derive equations which describe the motion of the sun, the planets and the Voyager spacecraft flying between them. These equations are fairly trivial if we only consider two bodies—the Earth and the Sun. So we can find a simple solution which predicts exactly where the Earth and Sun will be at any point in time given information about their positions and velocities at some starting point. This two body problem, as it is called, is well known and fairly straightforward, but not very practical as the whole picture emerges.

Now, to simulate the reality of what Voyager will actually experience on its trip through and beyond the 11 billion miles thus far travelled, we add a third body to our equations of motion. The Voyager spacecraft, moving somewhere between and beyond the Earth and the Sun, will encounter other planets, moons and bodies as well and now we no longer have a simple analytical solution. The equations are now unsolvable! This problem is known as the three body problem —one of the most difficult problems in all of celestial mechanics.

For Voyager to travel past Neptune, for example, it must escape the gravitational pull of the Sun which has a mass so large that it still has an effect on the spacecraft so many billions of miles away. That trip would usually demand a very large rocket engine with lots of fuel (Remember that it took lots of this precious fuel, which is also added weight, just to escape Earth’s gravity pull). A flight past Neptune, more than four billion km (2.5 billion miles) away – could easily take 30 or 40 years because Voyager’s rocket engines are fighting against the tremendous forces of gravity from the Sun and the neighboring planets it will pass.

At the time Voyager was developed, NASA only had the technology to propel a spacecraft into space with, at most a few months of operational life, and so the outer planets (no less leaving our solar system) were considered out of reach.

That was an accepted fact until a 25-year-old mathematics graduate named Michael Minovitch accepted the challenge to understand and change this situation in 1961.

Minovitch was enthralled with UCLA's new IBM 7090 computer, the fastest on Earth at the time, so he decided to take on the daunting “three body problem” of the Earth/planets, Sun and Voyager reacting in space with this new tool.

The problem is predicting exactly how the gravity of the Sun and a planet will influence Voyager’s trajectory on its journey out of our Solar system.

Renowned astronomers have struggled with this problem for at least 300 years, back when they tried plotting the path that comets took as they sped through our Solar System towards the Sun.

Some of the brightest scientists at the time, like Sir Isaac Newton, hadn't been able to solve this complex problem. Minovitch had access to the IBM 7090 computer, unlike Newton and others, and endeavored to use the method of mathematical iteration to craft a solution that was not realizable without the processing power of such a computer.

In the 2-body problem, we are able to develop a set of equations that describes the orbit of two bodies in space. For example, the orbit of a planet around the sun is in the form of an ellipse, like a conical section similar to a skewed circle.

Keplers three laws of planetary motion 2 are the basic theory behind the two body problem. First and foremost of these laws is that planets orbit the sun in an elliptical shape, not a circle, as was previously thought. The other two laws of planetary motion describe numerically how the orbits behave.

All bets are off for the 3-body problem. Minovitch understood that the only way to completely describe a system of 3 bodies is to observe how their behavior unfolds. This is where the number-crunching power of the IBM 7090 came into play.

Minovitch fed rough observed data on planetary orbits into his model at first and he had a data point. He later persuaded his boss at JPL to give him more accurate data on planetary positions to refine his model. After running the iterative 1 simulations again, his solution still worked. This young novice had now made an extraordinary breakthrough in spacecraft propulsion.

The result showed that as a craft flew close to a planet orbiting the Sun; it would acquire some of that planet's orbital speed, and be sling-shot away from the Sun. Such acceleration did not have to use a single drop of rocket propellant—an amazing discovery that changed the face of space travel and exploration!

The domain of the Sun's influence is called the heliosphere: Both Voyagers are approaching the edge of this enormous balloon of charged particles thrown out into space by our Sun. (Image Courtesy of NASA)

After much doubt by Minovitch’s peers, in 1965, another summer intern named Gary Flandro, a spacecraft engineer, familiar with the intricacies of spaceflight appeared on the scene. Flandro knew that any mission to the outer planets would have to be flown as fast as possible; otherwise the craft might not last long enough to reach its destination. He drew graphs of where the planets were going to be in future years and found that Jupiter, Saturn, Uranus and Neptune would all be on the same side of the Solar System in the late 1970s.

Using a solution to the three-body problem, a single mission, launching from Earth in 1977, could sling-shot a spacecraft past all four planets within 12 years. Such an opportunity would not present itself again for another 176 years. Time was of the essence so as to not miss this window.

With further lobbying from Minovitch and high level intervention from Maxwell Hunter, who advised the President on US space policy, NASA eventually embraced Minovitch's slingshot propulsion and Flandro's idea for a “grand tour” of the planets.

The three body solution was only made possible using the most powerful computer of the day, the IBM7090 (Image courtesy of NASA)

Here is Voyager going through the “Magnetic Highway” towards the end of July 2012, a newly discovered region of our solar system. Here, it was discovered that the Sun’s magnetic field lines are connected to interstellar magnetic field lines. This allowed particles from inside the heliosphere to move away and particles from interstellar space to quickly fill into that void.

An old Star Trek episode has the Voyager (known as “Vger” since part of the name on the body of the craft had worn away) as a type of demi-god, stranded on a remote planet somewhere in the cosmos—still with its light flashing and some power remaining to keep it functional hundreds of years in the future.

As our imaginations conjure up exciting encounters that these amazing spacecraft might stumble upon, we come to realize the power of “digitally-assisted” mathematics with a computer and some innovative thinking coupled with tenacity.

1 An “iterative process” is one in which the same rule is applied repeatedly, using the output as the input for the following iteration. For example, if we take the rule “add 1″, and apply it to 0, we get 1. If we iterate again, we get 2, then 3, and so forth.

2 http://www.science4all.org

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30 comments on “ Voyager: The mathematics of interstellar space travel ”

As a Star Trek die hard I am saddened that we are still worrying about cosmic rays, the Van Allen belt. “Dilithium” crystals and warp-drives seem light years away!

Right, its not so much of a technical issue as it is an economic one. Unfortunately, “space, the final frontier” is Expensive! Kudos to the private sector for giving it a major go!

I am not a Trekkie, but I think V'ger appeared in the first Star Trek movie, not the TV series. Also, although I'm not familiar with this particular bit of history, I think Minovitch's contribution must have been the recognition of the sling-shot idea rath

I believe you're correct on the V'ger comment …. the original series did have an episode that included an earth sourced space probe called NOMAD !!!

Let's be clear here, there are no such things as “magnetic field lines” unless you're sprinkling iron filings out there.

While a computer may have led Minovitch to discover the sling-shot effect, it certainly wasn't necessary. The sling-shot effect is an example of the two-body problem, which has a closed solution. It is a conservation of momentum issue – the spacecraft “ste

There's a good discussion of Gravity Assist and Minovitch's work–plus links to his and related original papers (quite interesting!) at: http://en.wikipedia.org/wiki/gravity_assist

The mathematical proof that “Pi” is connected directly to the “Speed of Light” is now at;

http://www.WowGoHere.com

For newbies you're right, there are no “lines”, but you could draw the “fields” with lines. Sharpies (chisel point) would not do the magnetic fields justice though. LOL

It is only an economic issue if there is no 'tangible' ROI (scientific and academic returns notwithstanding)! I think that the commercial space industry sees a potential return that would greatly offset their investment. Imagine a 'small' asteroid containi

WowGoHere.com, as a physicist/engineer I am bothered by such assertions. I will admit that I often read “alternative” views and cosmologies on a lazy Sunday afternoon. Such readings often betray a bit of the workings of the minds behind those views.

Let's see them use the same derivation to explain the orbits of Jupiter's moons.

You're right, by the time man can travel into deep space, humans will probably have evolved into a radiation-resistant species—-although we need to worry about radiation on Mars

You are correct about Minovitch's contribution. Gary Flandro, on the other hand, used Minovitch's data on planetary motion and location to determine the optimum time to launch the Voyager spacecrafts so that the planets would be in the best position for th

Emanating “Magnetic field lines” can be drawn on paper from Maxwell's equations—I know people who can visualize them in their minds—I'm not one of them

I agree, but the advantages of knowing exactly where the planets will be with respect to Voyager's planned and charted journey out of the solar system (Using the IBM computer) were absolutely essential in making sure the spacecraft would receive the maximu

In which scientific journal has this been published? When Einstein's peers review and discuss this, then I would consider reading about it

Great source of info Bill—thanks!

Oh mercy, . . . the VOLT (Circumference of spherical void produced in the aether via running a magnet through a coil) has nothing to do with “free energy” ! !

Focus on the M*A*T*H*E*M*A*T*I*C*S people !

A favorite project of mine Is a new value of “Pi” to assign It seems simpler to me To remember plain “3” Than 3.14159 So what would happen to the speed of light then?

i won't dwell on the speed of light issue… but if i recall something i read or heard about years ago correctly, there was a small town in italy a long long time ago, where the town council decided to adopt Pi=3. everything worked dandily until the masons

If you are looking for a prestigious brand replica watches, then you came to the right place. These replica watches are a mix of perfect craftsmanship and cheap price. http://www.missreplicawatches.com

“Relative scarcity pricing based on accessible reserves rather predicts that as soon as that is possible, price depression will rapidly follow.nnHowever, the same model applied to construction in space without the cost of lifting refined materials out of

“You get it John.Bass!”

“Hi Steve, I've sent a separate e-mail regarding some other points… but to respond to this specific one, Gary Flandro did not use Minovitch's calculations. Flandro plotted planetary positions on graph paper, by hand, using Her Majesty's Nautical Almanac,

“Thanks for that correction Jay!”

“You're welcome! There has been some confusion over the role played by Minovitch with regard to Voyager, as well as Minovitch's research in general. I hope my comments do not detract from the overall quality of the article. It was well-written and interest

“Hi, I was wondering if you could help me with an equation. Sorry to be off topic.”

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ORIGO Education has launched a brand new program, ORIGO Mathematics FIND OUT MORE

The Relationship Between Maths and Space Explored

Perhaps you’ve looked up at the night sky and tried to count the stars or dreamt of being an astronaut. In 1961 humans first launched into space, eight years later in 1969, man first set foot on the moon. As we now explore the outer reaches of our solar system the concept of humans landing on Mars is now a genuine goal to achieve in our lifetime – critical to all of these monumental achievements has been the use of maths.

Understanding the size and mass of planets, their gravitational forces and how to use acceleration and deceleration for rockets to explore space are just some examples of maths being used by rocket engineers, astrophysicists and astronauts.

Every aspect of space travel uses maths. Inside spaceships, astronauts routinely measure their height to study the effects of zero gravity which extends and stretches their spine. This makes them taller when in outer space. How cool is that?

Space begins around 100 kilometres above the Earth, where the shell of our Earth’s atmosphere around our planet becomes so thin that nothing can fly. Above Earth’s atmosphere, astronauts enter an environment where there is infinite blackness, no air, zero gravity, incredibly cold temperatures (-270 degrees Celsius) and nothing but the quiet void of endless space.

Our planetary system is the only system that is officially called a “solar system”. However, astronomers have discovered more than 3,000 other stars with planets orbiting them in our galaxy (and that’s just what they have found so far). Our solar system consists of eight planets that all orbit around the sun. These planets are all of different temperatures and sizes.

Uranus is the coldest planet in the solar system, with a minimum temperature of -224 degrees Celsius. Whereas Venus is the hottest, with a surface temperature of 475 degrees Celsius. Venus’ extreme heat means it is not safe to inhabit. Measuring and understanding planetary temperatures helps our understanding of our solar system and our expectation of where life may exist.

Understanding the size and mass of planets is another example of maths being brought into outer space!

Space travel involves enormous distances and a commonly used maths method called trigonometry is especially helpful to calculate the distances between planets, stars and galaxies that are measured in millions and millions of miles. In fact, these distances are so large that distance is often referred to in light years. Not a Buzz Lightyear! but the amount of distance travelled by light in a year, and light is the fastest thing known– so it’s incredibly fast, which means a light year is a really, really long way!

A key aspect of space travel is launching a rocket into orbit and a spaceship’s re-entry back into our Earth’s atmosphere to safely bring our brave astronauts home. In such circumstances, engineers use another common maths method called calculus to calculate how fast the rocket needs to accelerate to break free of Earth’s gravity and launch into space. Acceleration describes how rockets get faster and faster after take-off. The greatest acceleration happens at lift off. If a rocket is launched from the surface of the Earth, it needs to reach a speed fast enough to escape Earth’s gravity to reach space. This speed of 7.9 kilometres per second, or 28,000 kilometres per hour, is known as the orbital velocity; it corresponds to more than 20 times the speed of sound. Once in orbit the force of the rocket’s acceleration away from the Earth is balanced by the Earth’s gravity pull so that it stays at a constant distance from Earth.

What goes up must come down! So, when it’s time for our intrepid astronauts to come home they must decelerate their spaceship – which is the reverse of launching. They need to slow down so that Earth’s gravity can overcome the spaceships speed and pull it safely back to Earth. So, the spaceship will slow from an orbital speed of around 28,000 km per hour to 20,000 km per hour (still pretty fast!)

Maths is everywhere! It can be used anywhere from Earth to the outer most galaxies. Astronauts use it every day, and so do you!

If astronomy and space intrigues you, look deeper within STEM (science, technology, engineering and maths) subjects.

ORIGO Education is set on making learning fun, meaningful and accessible for everyone. Read more about our approach HERE .

Space has only started to be explored. There are endless possibilities, and mathematics is there to help. It’s to infinity and beyond!

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Space Travel: Mathematics Uncovers an Interplanetary Superhighway

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AIAA/AAS Astrodynamics Specialist Conference

David Dunham

For spacecraft departing the Earth - Moon system, lunar flybys can significantly increase the hyperbolic escape energy (C3, in km (exp 2) /sec (exp 2) ) for a modest increase in flight time. Within approx 2 months, lunar flybys can produce a C3 of approx 2. Over 4 - 6 months, lunar flybys alone can increase the C3 to approx 4.5, or they can provide for additional periapsis burns to increase the C3 from approx 2 -3 to 10 or more, suitable for planetary missions. A lunar flyby departure can be followed by additional delta -V (such as that efficiently provided by a low thrust system, eg. Solar Electric Propulsion (SEP)) to raise the Earth - relative velocity (at a ratio of more than 2:1) before a subsequent Earth flyby, which redirects that velocity to a more distant target, all within not more than a year. This paper describes the applicability of lunar flybys for different flight times and propulsion systems, and illustrates this with instances of past usage and future possibilities....

Acta Astronautica

Evandro Rocco

Giancarlo Genta

A general purpose design code for interplanetary missions is presented. The code, based on the MatLab environment, allows to deal with both impulsive propulsion (using the patched conics approach) and low continuous thrust. In the latter case, the solver developed specifically for this program is based on an indirect method. More general standard solvers, based on direct methods, like the FALCON.m code can, however, be used. The present paper shows a comparison between different approaches and methods, to evaluate the reliability and accuracy of the proposed code in different applications, by using a number of examples. Finally, some extensions of the code, which are planned for the future, are mentioned. Key-Words: Interplanetary mission design, Interplanetary trajectories, Launch opportunities, Int rplanetary mission optimization

Chit Hong Yam

Low-thrust trajectories can be modeled as a series of impulses connected by conics. We first describe a new framework for global optimization for the preliminary design of low-thrust trajectories that makes use of the impulsive delta-V transcription model. We then apply two global optimization algorithms, differential evolution and simulated annealing with adaptive neighbourhood, to solve an Earth-Mars rendezvous example problem. Test cases reveal that the two algorithms are able to produce good initial guesses for a local optimizer, which suggests a promising direction of research in automated trajectory design.

Computational Optimization and Applications

Andrea Cassioli

Paul Keaton

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Space Travel Calculator

Prepare for launch and fasten your seatbelts because we’re about to take a galactic joyride into the cosmos! 🚀✨

Formula for Space Travel:

Now, let’s boldly go where no calculator has gone before!

Table of Contents

Categories of Space Travel

From short interplanetary jaunts to epic journeys across the universe, space travel can be categorized into mind-boggling types:

Examples of Space Travel Calculations

Hold onto your space helmets as we calculate some whimsical space journeys:

Different Methods of Calculation

Space travel calculations can be as diverse as the cosmos itself, each with its quirks and peculiarities:

Evolution of Space Travel Calculation

The history of space travel calculations is a journey in itself:

Limitations of Accuracy

Even in the vastness of space, accuracy has its limits:

• Complexity: Space travel equations can involve complex math and relativistic principles.
• Specific Scenarios: Some methods are limited to specific scenarios, like near-light-speed travel.
• Theoretical Speculation: Hyperspace theory remains speculative and unproven.

Alternative Methods for Measurement

When it comes to space travel, alternative methods are often found in the realm of science fiction:

FAQs on Space Travel Calculator

• Can space travel be faster than the speed of light? According to current physics, traveling at or faster than the speed of light is impossible.
• What is time dilation in space travel? Time dilation is the effect where time passes differently for travelers moving at high speeds or in strong gravitational fields.
• Are wormholes real and can we use them for space travel? Wormholes are theoretical, and their existence is unproven. They remain a concept in the realm of science fiction.
• How does space travel affect time? Traveling at high speeds, as described by special relativity, can cause time dilation, where time passes more slowly for the traveler than for those at rest.
• What is the concept of hyperspace travel? Hyperspace is a theoretical concept where ships can travel faster than light by entering another dimension or space.
• Is it possible to travel back in time through space travel? Current scientific understanding suggests that traveling backward in time is highly unlikely and remains a subject of science fiction.
• How far can humans travel in space today? Human space travel is primarily limited to our solar system, with missions to Mars and beyond in planning stages.
• What is the closest star to Earth? The closest star to Earth is the Sun, which is part of our solar system. The closest star system is Alpha Centauri.
• How do scientists calculate travel time to other planets? Scientists use the principles of physics, including special relativity and rocket science, to calculate travel time to other planets.
• What is the ultimate goal of space travel? The ultimate goal of space travel is to explore, understand, and potentially colonize other planets, and to expand humanity’s presence in the universe.

Resources for Further Research

Explore more about space travel through these reputable government and educational resources:

• NASA’s Official Space Travel Page : Discover NASA’s missions, research, and the latest in space travel.
• Space.com – Space News : Stay updated with the latest news, articles, and information on space exploration.
• ESA – European Space Agency : Explore Europe’s contribution to space science, technology, and exploration.
• Caltech – Jet Propulsion Laboratory : Learn about space missions, robotics, and planetary exploration.

Katherine Johnson: Pioneering NASA mathematician

Katherine Johnson played a pivotal role in the Space Race.

Johnson's road to NASA

Naca becomes nasa, notable achievements, johnson's legacy, additional information.

Katherine Johnson was a NASA mathematician who played a key role in several NASA missions during the Space Race, including calculating the trajectory needed to get the Apollo 11 mission to the moon and back. 

As a black woman working for NASA in the 1950s and '60s, Johnson overcame social boundaries and racial discrimination. 

Her impressive career was the subject of the 2016 book and movie " Hidden Figures ." 

Related: How 'Hidden Figures' came together: Interview with author Margot Shetterly  

Johnson was born in 1918 in White Sulphur Springs, West Virginia, the youngest of four children. From a very young age, she had a fascination with numbers, which would lead her to defy all expectations throughout her life. "I counted everything. I counted the steps to the road, the steps up to church, the number of dishes and silverware I washed … anything that could be counted, I did," said Johnson according to a NASA history article . said during an interview with NASA in 2015. 

Johnson's hometown did not offer public schooling for black children past eighth grade, so her family moved 120 miles (193 kilometers) away so that she could attend high school. She graduated high school at just 14 and college at 18, securing degrees in both mathematics and French from the historically black West Virginia State College. 

While in school, her potential was clear to her teachers. One of her professors, William Schieffelin Claytor, encouraged Johnson to become a research mathematician and created a geometry class just for her. After teaching for a few years, Johnson was accepted to West Virginia University's graduate math program, and in 1939, she became the first black woman to attend the school. 

A year into her coursework, she left to raise her three daughters. Then, in 1952, a relative told her about an exciting new opportunity: The National Advisory Committee for Aeronautics (NACA), the predecessor to NASA, was hiring black women to solve math problems. Johnson applied right away. She was soon hired as a "computer" at the Langley Research Center , tasked with performing and checking calculations for flight tests.

"Before these big IBM mainframe computers came in [the NACA] really relied on human beings to do the math for calculating trajectories. Katherine really does play a role in that human computing lab," NASA historian Brian Odom told our sister publication All About History.

In addition to excelling at her work, Johnson was exceptionally curious and assertive, always questioning her colleagues and asking to be included in important meetings. When she started at NACA, Johnson and her black colleagues were required to work, eat and use restrooms separately from the white employees. But Johnson ignored the racial and gender barriers of the time and became the first woman in the Flight Research Division to be credited as an author on a research report.

"We needed to be assertive as women in those days — assertive and aggressive — and the degree to which we had to be that way depended on where you were. I had to be," Johnson said in 1999 of her time working for the NACA and later NASA.

In 1958, NACA became NASA. Johnson's time working for NASA would prove to be a more positive experience with regards to racial segregation given NASA's diversity policies. "Once NASA comes online in 1958 you see the build up of a lot of centres in the south. Today we'll think of Johnson Space Centre in Houston, Marshall Space Flight Centre in Huntsville, Alabama, and the Kennedy Space Centre in Florida, but there were other locations across the south too near New Orleans and South Mississippi. So basically what's happening is NASA is building a huge infrastructure in the time period from 1958 to 1963, at the build up for Apollo, in the Jim Crow segregated south," Odom said.

"The federal government, from the very beginning of this process, is committed to equal employment opportunity and that's something that John F Kennedy in March of 1961, with a new executive order, really makes a key part of this programme. So equal employment becomes a huge thing and for the first time in that executive order, we see that term "affirmative action". It's not just that you have to not discriminate, but you also have to have a positive plan for affirmative action, in which you demonstrate that you're taking the steps necessary to bring these jobs to African Americans," Odom added.

With the Soviet Union launching the Sputnik satellite in 1957, the US was determined not to fall behind and so the Space Race began. 

Johnson's passion was geometry, which was useful for calculating the trajectories of spacecraft. For NASA's 1961 Mercury mission , she knew that the trajectory would be a parabola, a type of symmetrical curve. So when NASA wanted the capsule to come down at a certain place, she was not deterred. 

"You tell me when you want it and where you want it to land, and I'll do it backwards and tell you when to take off," Johnson said according to a NASA . Subsequent orbital missions were more complicated, with more variables involving the position and rotation of the Earth , so Johnson used a celestial training device to perform her calculations.

Johnson was tasked with calculating the trajectory for Alan Shepard 's historic flight, during which he became the first American to reach space. She also confirmed the trajectory to send the first American into orbit around the Earth. By this time, NASA had begun using electronic computers to perform these tasks, but the machines could be a little temperamental. Before his Friendship 7 mission, astronaut John Glenn requested that Johnson personally recheck the calculations by hand. "If she says they're good, then I'm ready to go," Glenn said, according to Johnson’s recollection of the event .

The next challenge was to send humans to the moon , and Johnson's calculations helped sync the Apollo 11 lunar lander with the moon-orbiting command and service module to get the astronauts back to Earth. She also proved invaluable on the Apollo 13 mission, providing backup procedures that helped ensure the crew's safe return after their craft malfunctioned. She later helped to develop the space shuttle program and Earth resources satellite, and she co-authored 26 research reports before retiring in 1986.

Johnson spent the following years speaking to students about her extraordinary career, encouraging them to pursue STEM education. "Some things will drop out of the public eye and will go away," she said at the NASA Trailblazers and Legends STEM Conference in Cape Canaveral, Florida in 2010 . "There will always be science, engineering and technology. And there will always, always be mathematics. Everything is physics and math."

She and her colleagues became famous with the publication of " Hidden Figures " (William Morrow and Co., 2016) by Margot Lee Shetterly and the release of the blockbuster movie of the same name, which starred Taraji P. Henson, Janelle Monáe and Octavia Spencer as Johnson and her colleagues Mary Jackson and Dorothy Vaughan. The release of "Hidden Figures" made Johnson one of the most celebrated black women in space science and a hero for those calling for action against sexism and racism in science and engineering.

In 2015, President Barack Obama awarded Johnson the Presidential Medal of Freedom, America's highest civilian honor. And in 2016, the NASA Langley facility at which Johnson worked renamed a building in her honor : the Katherine G. Johnson Computational Research Facility. 

In 2019, Johnson told her own story for young readers in a book called " Reaching for the Moon " (Atheneum Books for Young Readers).

"Every time engineers would hand me their equations to evaluate, I would do more than what they'd asked. I'd try to think beyond their equations. To ensure that I'd get the answer right, I needed to understand the thinking behind their choices and decisions," she wrote.

"I didn't allow their side-eyes and annoyed looks to intimidate or stop me. I also would persist even if I thought I was being ignored. If I encountered something I didn't understand, I'd just ask. … I just ignored the social customs that told me to stay in my place."

Johnson died on Feb. 24, 2020, at age 101. NASA Administrator Jim Bridenstine announced her death and promised that her legacy would be remembered.

"At NASA, we will never forget her courage and leadership and the milestones we could not have reached without her," Bridenstine said. "We will continue building on her legacy."

"She played a huge role in NASA throughout her career and is someone we (NASA) really look back on fondly," Odom said.

You can learn more about Katherine Johnson’s work with NASA, or about her life and legacy with the New York Times . You can also educate your little ones about how remarkable Johnson was with this excellent picture book . 

Bibliography

Johnson, K. (2020). Reaching for the moon: The autobiography of Nasa mathematician Katherine Johnson . Atheneum Books for Young Readers, an imprint of Simon & Schuster Children's Publishing Division.

Loff, S. (2016, February 25). Mathematician Katherine Johnson at work. NASA. Retrieved November 11, 2022, from https://www.nasa.gov/image-feature/mathematician-katherine-johnson-at-work

Loff, S. (2020, February 24). Katherine Johnson Biography. NASA. Retrieved November 11, 2022, from https://www.nasa.gov/content/katherine-johnson-biography

Shetterly, M. L. (2017). Hidden figures: The American Dream and the untold story of the black women mathematicians who helped win the space race . William Morrow, an imprint of HarperCollins Publishers.

Smith, Y. (2015, November 24). Katherine Johnson: The girl who loved to count. NASA. Retrieved November 11, 2022, from https://www.nasa.gov/feature/katherine-johnson-the-girl-who-loved-to-count

Wild, F. (2020, February 24). Katherine Johnson: A lifetime of Stem. NASA. Retrieved November 11, 2022, from https://www.nasa.gov/audience/foreducators/a-lifetime-of-stem.html

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Emily is the Staff Writer at All About History magazine, writing and researching for the magazine's content. She has a Bachelor of Arts degree in History from the University of York and a Master of Arts degree in Journalism from the University of Sheffield . Her historical interests include Early Modern and Renaissance Europe, and the history of popular culture.

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FREE K-12 standards-aligned STEM

curriculum for educators everywhere!

Find more at TeachEngineering.org .

• TeachEngineering
• Space Travel

Lesson Space Travel

Grade Level: 8 (7-9)

Time Required: 15 minutes

Lesson Dependency: None

Subject Areas: Earth and Space, Science and Technology

• Print lesson and its associated curriculum

Curriculum in this Unit Units serve as guides to a particular content or subject area. Nested under units are lessons (in purple) and hands-on activities (in blue). Note that not all lessons and activities will exist under a unit, and instead may exist as "standalone" curriculum.

• Into Space!
• Solar Sails: The Future of Space Travel
• My Moon Colony
• Earth Impact

TE Newsletter

Engineering connection, learning objectives, more curriculum like this, introduction/motivation, associated activities, lesson closure, vocabulary/definitions, user comments & tips.

Space travel is made possible by engineers. From mechanical engineers who design the components for spacecraft to biomedical engineers who design ways to care for astronauts' health while traveling in space, people from almost every discipline of engineering work together to further space exploration.

After this lesson, students should be able to:

• Identify potential reasons why people want to travel to space.
• Describe how different types of engineers each contribute to space travel.
• Give examples of what the future may hold for space travel.

Educational Standards Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards. All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN) , a project of D2L (www.achievementstandards.org). In the ASN, standards are hierarchically structured: first by source; e.g. , by state; within source by type; e.g. , science or mathematics; within type by subtype, then by grade, etc .

International technology and engineering educators association - technology.

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State Standards

Colorado - science.

(Note: Prior to beginning this lesson introduction, guide students through a brief pre-lesson assessment activity on journaling, as described in the Assessment section.)

Let us begin by having you share your journal entries on your journey through space. What do you imagine it would be like to travel into space? What questions do you have about the solar system and outer space? (Call on students to share their ideas.) Great ideas and questions! Do you have any idea how long you think a person could live in outer space? What kind of equipment would she or he need to survive? Today we will be studying a space project that many engineers have been working on that makes it possible to travel to and live in space.

But first let's think about why people would want to travel to space in the first place. You wrote in your journal about things you might explore in space. Tell me about your investigation plans. (Call on students to share their ideas. Write ideas on the board.) Good! What are other reasons that space scientists might want to explore outer space? (Ask some of the following questions if students did not already mention these ideas. Write ideas or groups of topics on the board.)

• Evidence of life: "Do you think life exists on other planets? How could we find out?"
• Gravity-free experiments: "Why might scientists be interested in performing experiments in a gravity-free environment?"
• Earth's origins: "In what ways could the exploration of the moon and our solar system tell us about our own planet's origin?"
• Colonizing other planets: "Do you think we could someday colonize the moon or another planet? Why would we want to? What would we need to figure out to make that possible?"

Excellent! Now we have an idea of why people might like to travel to space. But , who actually makes space travel possible? Engineers are the creative problem solvers who help shape the future of space travel. People from nearly every discipline —or branch of study—of engineering work together to further space exploration. Aerospace engineers design the craft, mechanical engineers design the components for the craft, electrical and computer engineers design the computer systems, and chemical engineers design the rocket fuel, among the many others who have a hand in a spacecraft's launch into outer space. What role do you think biomedical engineers have in space travel? (Answer: they design the medical devices to monitor and keep healthy the traveling astronauts.)

Let's look at an example of a space project that sounds like something out of a science fiction movie, but is actually a real project that engineers from around the world have worked on together to build for our use today! The International Space Station, or ISS for short, is designed to help scientists study questions such as the ones you came up with during your journaling activity. Astronauts—who have trained for months, if not years—live in the ISS for up to about six months at a time! Can you imagine?! While aboard the ISS, astronauts perform experiments that can only be done in space. Also, another purpose during their stay is to discover what is necessary to live in space for an extended period of time, so that longer journeys into the solar system can be planned (and carried out), such as to Mars or beyond.

The ISS is slightly bigger than a football field, and the living area for the 3-6 on-board astronauts is about the size of an average three-bedroom house (see Figure 2). How do you think life might be different for the astronauts who live there, based on the fact that they are in space and in such confined quarters? (This might be a good time to visit NASA's Interactive Space Station Reference Guide at https://www.nasa.gov/mission_pages/station/multimedia/index.html , which provides detailed visual answers to this question, as well as video clips showing how astronauts eat, sleep and exercise while on board the ISS.)

The ISS is not only an adventure in space living, it is also an adventure in science and engineering. The ISS is one of the most complex engineering projects ever designed. Engineers had to solve many difficult problems, such as how to get such a massive object to orbit the Earth, how to design ways to get astronauts on and off the space station, and how to provide electrical power to the station. Following the lesson students can expand their knowledge by combining design and creative thinking with the associated activity Solar Sails: The Future of Space Travel in which they will design, construct and test model solar sails made of foil that move cardboard tube satellites on a string through “space,” as a way to learn about new innovations, the universal laws of motion, and the transfer of energy from wave to mechanical energy. 

Lesson Background and Concepts for Teachers

What is the International Space Station?

The International Space Station , or ISS for short, began in-space assembly in 1998 and is still under construction (as of fall 2008, at the time of this writing). The largest international space research station in history, sixteen countries including the U.S., Russia, Canada, Japan and Brazil joined together to conduct science experiments and research in space to determine whether human beings have a future in space. Research topics include life in low gravity, physics, Earth observation.

The ISS is about 25% larger than a football field and weighs more than one million pounds. Orbiting 240 miles above Earth and traveling at a speed of about 28,000 kilometers an hour, the ISS circles the Earth about 16 times a day. The station is able to observe 85% of the planet and 95% of the Earth's population from its variable and frequent orbit. Although small, the living quarters give the crew a great view of space and the Earth.

Crew and supplies can access the station during NASA's scheduled missions to replenish needed items (food, research supplies, etc.), accept visitors and exchange astronauts who are living in the space station. Electricity is provided by almost an acre of solar panels. NASA provides an excellent online tour of the ISS via the Interactive Space Station Reference Guide ( https://www.nasa.gov/mission_pages/station/main/index.html ).

History of Space Travel

The earliest rockets were similar to current day fireworks and used in 1232 during the war between the Chinese and Mongols. These firework-like missiles were a simple solid-propellant rocket. Using gunpowder stuffed into a tube that was capped only at one end, the rocket could be ignited. The rapid burning of the powder produced gas that escaped through the open end, producing thrust. The stick that the rocket was attached to acted as a simple guidance system. Refer to the associated activity Into Space! to give students an introductory learning of basic physics concepts on how rockets work as they design, build and test model rockets using camera film canisters and antacid tablets. 

Manned Spacecraft

Space travel for human beings did not develop until several centuries later when, in 1961, Russian Yuri Gagarin became the first person to orbit the Earth. The next major milestone in space travel came in 1969 when the three-man Apollo 11 mission landed on the moon — resulting in the first humans who walked on the moon!

Soon after the Russians and Americans sent people into space, engineers also started working on spacecraft that would house astronauts for longer periods so that they could plan for extended trips and perform scientific experiments. These predecessors to the International Space Station include the Apollo-Soyuz, the first international spacecraft; Skylab, the first American craft for long-term use, and Mir, the Russian space station that held international scientists during 1986 to 1996.

Unmanned Spacecraft

Unmanned spacecraft are an integral part of the discovery of our solar system and beyond. Some satellites observe the Sun, solar system, and/or the universe (such as the Hubble telescope or Viking, the Mars probe), and other satellites observe our planet from above (for weather forecasting, etc.). Still other satellites observe and sample specific environments or are used solely for the purpose of benefiting humanity (like GPS and communication).

Watch this activity on YouTube

Space travel can help us learn about the origins of the universe and our own planet. Also, it provides a unique opportunity to do research in a gravity-free environment, which is a great advance toward our knowledge of space and space-related issues. Engineers help make space travel possible by working together to solve a wide range of design problems. Someday they may even find a way to help us travel to other planets, or perhaps even live among the stars!

chemical reaction: A process whereby one type of substance is chemically converted to another substance involving an exchange of energy.

gravity: The natural force of attraction between any two massive bodies.

rocket: A vehicle that moves by ejecting fuel.

star: A huge burning sphere of gas, made up of roughly 90% hydrogen and 10% helium.

thrust: The forward-directed force on a rocket in reaction to the ejection of fuel.

Pre-Lesson Assessment

Journaling : Have students take out a sheet of paper. Explain that they will be writing in their journals for three minutes on a topic you will provide. They should not worry about the quality of their writing (organization, grammar or spelling), but should simply try to get their thoughts down. They should write nonstop for the full three minutes. Topic: Imagine that you are planning a trip through our solar system. What do you imagine you will see and feel as you travel through space? What kinds of things might you explore? What questions do you have about your upcoming journey or about the solar system or outer space in general? (You may want to write the topic questions on the board, under a pulled-down projector screen, in advance of class.)

Post-Introduction Assessment

Discussion Topic : International Space Station astronauts set up a program that enables them to talk to students around the world. Think of a few questions that you would ask an ISS astronaut, given the chance. In particular, think about questions concerning why people go to space and what the future of space travel holds. Give students a few minutes to write down their responses, and then have them share them with the class.

Lesson Summary Assessment

Acrostic Poem : Encourage students to synthesize and evaluate their learning by having them write an acrostic poem. Instructions : To make an acrostic poem, write the words space travel vertically on a piece of paper. Then, use each letter in those words as the first letter of a different word or phrase related to space travel. For example, S = scientific experiments, P = probing the universe, A = aerospace engineers, etc.)

Lesson Extension Activities

Learn more about the shuttle and its role in building and transporting crew to and from the ISS. Have students research various aspects of this topic and present their findings to the class.

Have students build a scale model of the International Space Station using scaled drawing templates from NASA; see: http://www.nasa.gov/mission_pages/station/multimedia/scalemodel/.

Have students research how the ISS is built, and what it would be like to live on it, using NASA's Interactive Space Station Reference Guide at: http://www.nasa.gov/externalflash/ISSRG/. Present their research to the class.

Ask students to consider how scientists and engineers made the International Space Station successfully orbit the Earth. Discuss factors that could influence its ability to orbit. Then have students try to create their own orbiting satellite using the online simulation "My Solar System" at: http://phet.colorado.edu/en/simulation/my-solar-system.

Build a scale model of objects in the solar system, with help from a NASA website at: http://solarscience.msfc.nasa.gov/papers/hathadh/SolarSystemModel.pdf. If any other students/group is creating a scale model of the ISS, work in tandem to demonstrate for other students what the ISS looks like traveling through space.

Students acquire a basic understanding of the science and engineering of space travel as well as a brief history of space exploration. They learn about the scientists and engineers who made space travel possible and briefly examine some famous space missions.

Students are introduced to the International Space Station (ISS) with information about its structure, operation and key experiments.

Students learn about the physical properties of the Moon. They compare these to the properties of the Earth to determine how life would be different for people living on the Moon.

The purpose of this lesson is to teach students how a spacecraft gets from the surface of the Earth to Mars. Students first investigate rockets and how they are able to get us into space. Finally, the nature of an orbit is discussed as well as how orbits enable us to get from planet to planet — spec...

Discovery Communications, Discovery Education, Classroom Resources, "Space-Age Living: Building the International Space Station," accessed October 23, 2008, http://school.discoveryeducation.com/schooladventures/spacestation/basics/how.html

Dubson, Michael. University of Colorado, Physics Education Technology, Simulations, "My Solar System," accessed October 23, 2008, http://phet.colorado.edu/en/simulation/my-solar-system

Hathaway, Dr. David H., National Aeronautics and Space Administration, "Scale Model of the Solar System," June 1998, accessed October 23, 2008, http://solarscience.msfc.nasa.gov/papers/hathadh/SolarSystemModel.pdf

National Aeronautics and Space Administration, United Space Alliance and The Boeing Company, Online Shuttle Press Kit, The International Space Station, June 3, 1999, accessed October 23, 2008, http://www.nasa.gov/news/media/presskits/index.html

National Aeronautics and Space Administration, accessed October 23, 2008, http://spaceflight.nasa.gov/station/

National Aeronautics and Space Administration, accessed October 23, 2008, http://spaceflight.nasa.gov/history/

National Aeronautics and Space Administration, International Space Station, An Interactive Reference Guide, accessed October 23, 2008. https://www.spaceflight.nasa.gov/station/reference/index.html

National Aeronautics and Space Administration, International Space Station, Multimedia, "Scale Model Drawing Package," August 15, 2008, accessed October 23, 2008, http://www.nasa.gov/mission_pages/station/multimedia/scalemodel/index.html

Space Station, "Current Missions," October 23, 2008, accessed October 23, 2008, http://www.nasa.gov/mission_pages/station/main/index.html

Newman, Phil. National Aeronautics and Space Administration, StarChild, "The Solar System," accessed October 14, 2008. http://starchild.gsfc.nasa.gov/docs/StarChild/solar_system_level2/solar_system.html

Contributors

Supporting program, acknowledgements.

The contents of these digital library curricula were developed by the Integrated Teaching and Learning Program under National Science Foundation GK-12 grant no. 0338326. However, these contents do not necessarily represent the policies of the National Science Foundation, and you should not assume endorsement by the federal government.

Last modified: March 10, 2021

A research center claims that 24% of adults in a certain country would travel into space on a commercial flight if they could afford it. In a random sample of 1200 adults in that country, 28% say that they would travel into space on a commercial flight if they could afford it. Atalpha =0.05 , is there enough evidence to reject the research center's claim? Complete parts (a) through (c) below. (a) Identify the claim and state H_0 and H_a. ldentify the claim in this scenario. Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or a decimal. Do not round.) A. No more than □ : % of adults in the country would travel into space on a commercial flight if they could afford it B. At least □ % of adults in the country would travel into space on a commercial flight if they could afford it C. The percentage adults in the country who would travel into space on a commercial flight if they could afford it is not □ %. D. □ % of adults in the country would travel into space on a commercial flight if they could afford it. Let p be the population proportion of successes, where a success is an adult in the country who would travel into space on a commercial flight if they could afford it. State H_0 and H_a Select the correct choice below and fill in the answer boxes to complete your choice (Round to two decimal places as needed.) A. H_0:p≤ □ B. H_0:p>□ C. H_0:p!= □ H_a:p>□ H_a:p≤ □ H_a:p=□ E. D. H_0:p=□ H_0:p H_0:p≥ □ F. H_a:p!= □ H_a:p≥ □ H_a:p

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May, 2018: The Insight Mission to Mars

On May 5, 2018, the Insight mission will be launched for a 200-day trip to Mars. Upon landing, its sensitive seismometers will explore the crust of Mars. Here are a few resources you can download from the embedded link that feature this historic event! [Enter here]

August, 2018: The Parker Solar Probe

Between July 30 and August 19, 2018, the Parker Solar Probe will be launched for an encounter with the sun's corona. Here are a few resources you can download from the embedded link that feature this historic event!

• The PSP Heat Shield - To prevent its delicate scientific instruments from over-heating, the entire spacecraft will hide behind a heat shield
• Exploring the Pressure of Sunlight - The spacecraft will reach a distance of only 7 million kilometers from the sun, where the sunlight pressure is 0.0047 Newtons per square meter. This pressure of sunlight has to be accounted for in order to predict an accurate orbit for the spacecraft near the sun!
• Exploring Fractions Using Mass Limits - When engineers design a spacecraft, they have to follow strict mass limits so that the total mass of the spacecraft isnt more than the rocket can carry into orbit.
• Exploring Orbital Speeds - The spacecraft will be placed in a series of elliptical orbits around the sun. As the spacecraft travels around the elliptical path, its speed changes from a very fast pace nearest the sun (called the perihelion), to a very slow speed at its farthest point (called the aphelion).
• Exploring Linear Equations - The spacecraft consists of 11 separate systems and the fuel for the thrusters. All of these systems have to work together to create a functioning spacecraft that can conduct scientific research.
• Exploring Solar Energy - NASAs mission to the sun will be powered by two different solar panel systems. The first system will operate when the spacecraft is between the orbit of Earth and Mercury. These panels will be retracted behind the heat shield to avoid damage. A second solar panel system will then take over and carefully poke out from behind the heat shield to generate just enough electricity to run the 480-watt spacecraft as the spacecraft travels to 9.5 million kilometers from the sun. Why do we need two systems?
• Solar Heating of the Spacecraft - Any surface in space near the sun will heat up as it absorbs solar energy. It will absorb sunlight at all wavelengths, but it can only cool off by emitting infrared radiation, which we experience as heat radiation making the body warm to the touch.
• Solar Proton Storms - Solar proton storms are intense bursts of high-energy protons created as dense clouds of plasma are ejected from the sun. The spacecraft will have a ring-side seat to observe how these events are caused.
• Investigating Elliptical Orbits - The Parker Solar Probe spacecraft will make 7 fly-bys of the planet Venus in order to arrive at its final elliptical orbit. This final orbit will have a perihelion of 7 million km, and an aphelion at the orbit of Venus located 110 million km from the sun. Spacecraft orbits are a great way to review the geometric properties of ellipses!
• The Interplanetary Dust Hazard - During the 24 solar orbits of the mission, the spacecraft will encounter a thermal environment that is 50 times more severe than any previous spacecraft. It will also travel through a dust environment previously unexplored, and be subject to particle hypervelocity impacts at speeds much faster than anything previously encountered by NASA spacecraft.

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