the mathematics of space rendezvous

The Mathematics

of Space Rendezvous

NASA has a variety of classroom activities designed to accompany the video. Click here to explore.

Overview

The NASA video illustrates the use of algebra and estimation in shuttle flight and rendezvous.

The accompanying slides demonstrate the problems that the astronauts pose and provide extra support as the problems are worked.

NASA has developed a learning guide to accompany this video.

Rendezvous

Have you ever missed an appointment?

How did it make you feel?

How did it make the other party feel?

Rendezvous

What is a rendezvous?See the video for definitions.

How can you plan ahead to get there on time?

How do astronauts calculate where and when to meet in space?

Math… The Language of the Universe

Mathematics is an essential part of space travel.

Mathematics even helps us travel on earth.

It is more than numbers and symbols!

Math… The Language of the Universe

What instruments did early space travelers use to calculate mathematical equations?

What instruments do space travelers use today to calculate mathematical equations?

Math aboard the Space Shuttle

The Shuttle Design The Launch Navigation The Landing The Rendezvous

NASA

Rendezvous…

Can you think of examples of rendezvous?

Rendezvous…

Is like playing frisbee catch with a dog.

Is like a quarterback throwing a timing route pass to a wide receiver.

Questions

First: What question needs to be answered?

When will the target (the Mir) be at the When will the target (the Mir) be at the rendezvous site (over Moscow)?rendezvous site (over Moscow)?

The Target

Altitude of 400 kilometers. Orbit is at 92 minutes. Incline of the orbit is 51.6’. Earth revolves once

every 24 hours.

Gather the known facts:

Find out more about the Mir.

NASA

The Target

More facts: Moscow is at 37’ E longitude. Mir’s present longitude is at 105’ E.

Questions

Review the questions again: When will the target be at the

rendezvous site? When will it pass over

the rendezvous site?

Check out where the space station is right now on this real time site.

Questions

What other information do we need to solve this problem?

Degrees the earth rotates in the time it takes Mir to make one orbit.

Set Up the Equations

360° in a full circle or around the earth.

The earth rotates once in 24 hours.

Mir’s orbit takes 92 minutes.

Recall known and previous given facts:

NASA

Set Up the Equations

The following ratio determines how far the Earth rotates with each orbit of the Mir

360° = x 24 hours 92 min

We need these in common terms, let’s change the hours into minutes.

(24 hours is 1440 minutes) Here is the problem again:

360° = x 1440 min 92 min

Questions

What other information do we need to solve this problem?

How far it is between Moscow and where the space station is currently.

The angular distance between the Mir’s present locationand its rendezvous point is 105o – 37o or 68o.

Questions

What other information do we need to solve this problem?

How long it takes Mir to make one orbit. We were given this information earlier.

Mir’s orbit takes 92 minutes.

Calculate

X = 92 min x 360o

= 23o

1440 min

The Earth rotates 23o each time Mir

circles it once.

To rendezvous at the designated location, the Earth must rotate 68

o.

Calculate

Mir must circle Earth.

orbits or orbit 323

68

If it takes 92 min per orbit, thenIf it takes 92 min per orbit, then

3 orbits x 92 min = 276 min = 4.6 hrsorbit

oo

Orbital Altitude

How does orbital altitude affect an objects travel?

SpeedOrbitRendezvous

Questions

How fast is Mir traveling in its orbit?

How fast is the shuttle traveling?

How can be at the same place and match speed? NASA

Gather the Facts

Mir’s altitude is 400 km and speed is 92 min/revolution.

The shuttle’s altitude is 298 km and speed is 89 min/revolution.

The Earth’s radius is 6,378 km.

Known Facts

Orbit is almost a circle

Circumference of a circle

C = 2r NASA

Known Facts

Equation for speed

Speed = Distance/Time

Mir’s Orbit

First the formula:Speed = C (Mir’s orbit) / R (minutes)

Then the facts:Mir orbits at a distance of 6,378 + 400 km

Find Mir’s orbit:C (Mir’s orbit) = 6,778 km x 2(pi)= 42,578 km per orbit.

Mir’s Speed

The formula again:Speed = C (Mir’s orbit) / R (minutes)

Put in Mir’s orbit:Speed = 42,587 km/92 min

Solve the equation:Speed of Mir = 463 km/min = 27,780 km/hr

Speed of the Shuttle

First the formula:Speed = C (shuttle) / R (minutes)

Find the shuttle orbital radius:637 km x 298 km = 41,947 km

Put in the facts:Speed = 41,947 km / 89 min

Calculate: Speed of the Shuttle = 471 km/min = 28,260 km/hr

Slow Down to Rendezvous

The space shuttle must slow down to rendezvous with Mir.

To slow down, the shuttle will climb to Mir’s higher orbit.

Gather the Facts

0.4 meters per second increase in velocity equals 1 km increase in altitude.

Present altitude is 298 km.

Need to increase altitude to 305 km.

Question

How many meters per second does the shuttle need to increase to rise 7 km?

NASA

Set Up the Equations

An increase of 2.8 m/s will increase the orbit

of the shuttle 7 km and actually slow it down!

Desired – current = needed increase in altitude

305 km – 298 km = 7 km

Recall: 0.4 m/s increase in velocity = 1 km increase in altitude .

7 (km) / 1 (km) x .4 (m/s) = 2.8 m/s

Shuttle Instruments

Shuttle instruments can show this process graphically.

The astronauts can also see the Mir through the windows.

Virtual view of Mir through the shuttle window.

NASA

Views from Space

The video is showing videos that were taken by the astronauts. More videos and photos are available at NASA’s web site.

Click on this slide to see more photos.

NASA

NASA

It takes Math to Rendezvous in Space

The astronauts use both: Computer calculations Math estimation skills

Try another math activity with airplanes.

Math of the Universe

Whether it is using scientific notation to measure the microscopic or astronomic, whether it is estimation or calculus, mathematics plays a big part in science.

Read more about experiments performed aboard the shuttle and space station.

Math on Earth

The video is showing some examples of everyday activities that need math. Can you think of others?

Explore a web site with activities focusing on every day math.

Math and Rendezvous

The video and slides have illustrated a few problems that the space shuttle astronauts need to estimate while on a single space flight.

Space travel and travel on earth have similarities. Work with your class to create earth examples of rendezvous.

Math and Rendezvous

Math is an important part of making meetings in space and on earth prompt and accurate.

The Technology Leadership Institute (TLI) is funded by the United States Department of Education's Preparing Tomorrow's Teachers to use Technology (PT3) Program (Catalyst grant - P342A990323).