Problems

practice

1. The following four statements about circular orbits are equivalent. Derive any one of them from first principles.
   • Negative kinetic energy equals half the potential energy (−K = ½U).
   • Potential energy equals twice the total energy (U = 2E).
   • Total energy equals negative kinetic energy (E = −K).
   • Twice the kinetic energy plus the potential energy equals zero (2K + U = 0).

   This is a key relationship for a larger problem in orbital mechanics known as the virial theorem.

2. Determine the minimum energy required to place a large (five metric ton) telecommunications satellite in a geostationary orbit.
   
3. 
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   A satellite of mass m is in orbit about the earth, which has mass M and radius R. (State all answers in terms of the given quantities and fundamental constants.)

   1. The satellite is initially in an elliptical orbit as shown in the diagram to the right. At perigee (the point of closest approach) the distance from the center of the satellite to the center of the earth is r_p and the speed of the satellite is v_p. At apogee (the point when it is furthest from the earth) the distance from the center of the satellite to the center of the earth is r_a. Determine v_a, the speed at apogee.
   2. As the satellite reaches perigee, its speed is changed abruptly so that the satellite enters a circular orbit of radius r_p and speed v as shown in the diagram to the right. How much work and what impulse was applied to the satellite to change its orbit?
4. Locate the L1, L2, and L3 Lagrange points for the earth-sun system using energy considerations. State your answers as distances …
   1. from the sun and earth in meters
   2. from the sun as multiples of the earth's orbital radius (AU)
   3. from the earth as multiples of the moon's orbital radius

numerical

1. Roche Limit
   1. What is the total mechanical energy of the moon at its current distance from the earth?
   2. What total mechanical energy would the moon have if it were to orbit the earth 2.4 earth radii from the earth's center?
   3. The earth loses 10²³ joule of rotational kinetic energy per century to lunar and solar tides. For the sake of argument, assume that the moon's rate of energy loss is 1% of the earth's. When will the moon reach the distance calculated in b?
   4. Speculate on the fate of the moon at this distance. How likely is it that this event would come to pass?

worksheets

1. trajectories-satellite.pdf
   The accompanying pdf file shows a satellite in a circular orbit about the earth. Sketch the new path that the satellite would take if its speed were changed abruptly in the ways described.