Energy of a Spacecraft
Very far from earth(at R=\infty), a spacecraft has run out of fuel and its kineticenergy is zero. If only the gravitational force of the earth wereto act on it (i. e., neglect the forces from the sun and other solarsystem objects), the spacecraft would eventually crash into theearth. The mass of the earth is M_e and its radius is R_e. Neglect air resistance throughout this problem, sincethe spacecraft is primarily moving through the near vacuum ofspace.
Part A
Find the speed s_e of the spacecraft when it crashes into the earth.
Express the speed in terms ofM_e, R_e, and the universal gravitational constant G.
s_e =

Part B
Now find the spacecraft's speed when itsdistance from the center of the earth is R=\alpha R_{\rm e}, where \alpha \ge 1.
Express the speed in terms ofs_e and alpha.
s_alpha =