Question 1

Read each question carefully. Show all your work for each part of the question. The parts within the question may not have equal weight.

A satellite of mass m is orbiting Earth in a stable circular orbit of radius R. The mass and radius of Earth are ME and RE , respectively. Express your answers to parts (a), (b), and (c) in the following in terms of m, R, ME , RE , and physical constants, as appropriate.

(a) Derive an expression for the speed of the satellite in its orbit.

(b) Derive an expression for the total mechanical energy of the satellite-Earth system in its orbit.

(c) Derive an expression for the period of the satellite's orbit.

(d) The satellite is replaced with a similar satellite that has twice the mass. The new satellite is placed into the same orbit as the original satellite. Is the new satellite's orbital speed greater than, less than, or equal to the original satellite's orbital speed?

_____ Greater than _____ Less than _____ Equal to

Justify your answer.

(e) The original satellite can be placed in many different circular orbits of radius r. On the axes, sketch a graph of the mechanical energy E of the satellite-Earth system as a function of the orbital radius r. m RE The satellite is traveling with speed v in the original orbit as shown by the dashed line in the figure. The satellite then collides with a small meteorite. The speed of the satellite is reduced to half its current value.(f) On the figure, sketch the path the satellite will follow after the collision with the meteorite