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

NASA has developed a learning guide to accompany this video. 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 rendezvous site (over Moscow)?

The Target Gather the known facts: Altitude of 400 kilometers. Orbit is at 92 minutes. Incline of the orbit is 51.6’. Earth revolves once every 24 hours. NASA Find out more about the Mir.

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 Recall known and previous given facts: 360° in a full circle or around the earth. The earth rotates once in 24 hours. Mir’s orbit takes 92 minutes. 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 location and 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.

X = 92 min x 360o = 23o Calculate 1440 min The Earth rotates 23o each time Mir circles it once. To rendezvous at the designated location, the Earth must rotate 68o.

Calculate Mir must circle Earth. o o 23 68 ¸ or » 3 orbits orbit If it takes 92 min per orbit, then 3 orbits x 92 min = 276 min = 4.6 hrs orbit

Orbital Altitude How does orbital altitude affect an objects travel? Speed Orbit Rendezvous

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 C = 2r Orbit is almost a circle Circumference of a circle 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 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 An increase of 2.8 m/s will increase the orbit of the shuttle 7 km and actually slow it down!

Shuttle Instruments Shuttle instruments can show this process graphically. The astronauts can also see the Mir through the windows. NASA Virtual view of Mir through the shuttle window.

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. NASA Click on this slide to see more photos. 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.

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