# Activities

• • • ##### Subject Area

• Math: Calculus: Other Functions

• ##### Author 9-12

60 Minutes

• TI-86
• ##### Other Materials
This is Activity 9 from the EXPLORATIONS Book:
Differential Equations With The TI-86

## Space Trajectories

#### Activity Overview

In this activity, students will explore different models for the motion of satellites. They work with two-and three-body problems, and consider both artificial and natural satellites.

#### Before the Activity

• See the attached PDF file for detailed instructions for this activity
• Print pages 95 - 104 from the attached PDF file for your class
• #### During the Activity

Distribute the pages to the class.

A Geosynchronous Orbit:

• Enter the values for a satellite in geosynchronous orbit with velocity in m/s, time in seconds and radius in meters
• Convert the second order system of two equations in two unknown functions into a first order system in four unknowns.
• Set initial conditions
• Change settings to view the plotting position as it moves two revolutions
• Use equally scaled axes to make the plot resemble a circle

• A Shuttle Orbit:
• Use the same equation as in the first example and find the radius and the velocity needed for a low earth orbit
• Change the initial conditions and the window parameters and graph the new orbit
• Use the circle command to plot the surface of the earth
• Solve the differential equation

• Hohmann Transfer Orbit:
• Calculate the parameters and plot the hohmann transfer orbit needed to go from a low-earth shuttle orbit to a geosynchronous orbit
• Compute the two changes in velocity needed (by rocket booster) δv1 and δv2
• Compute the total period time for the transfer orbit
• Plot the transfer orbit using the circle command

• An Orbit Near Lagrange point:
• Enter the set of differential equations that model the movement of the earth, moon, and an artificial satellite
• Verify the stationary point for this system
• Change settings to view the resulting trajectory

• A Trip From Near Earth to Near the Moon:
• Set initial conditions and store location of the earth and the moon to create lists
• Use stat scatter plot to mark location of two larger bodies
• Plot the restricted three-body trajectory
• #### After the Activity

Students complete the problems on the exercise page.

Review student results:

• As a class, discuss questions that appeared to be more challenging
• Re-teach concepts as necessary