Question

. A spaceship of mass m has its engines switched off and is moving in a circular orbit at height R above the surface of a planet of mass M and radius R. a) Derive an expression for total mechanical energy E of the orbiting spaceship, in terms of G, m, M and R. b) Derive an expression for the minimum speed V the spaceship would need to escape from this orbit into deep space, in terms of system parameters. (The engines can’t fire for the whole trip; they can only give the spaceship one boost so it obtains this velocity. Ignore all other celestial objects.)

Answer 1

Note that GmM/r^2 = ma = MV^2/r

After dividing we have that the velocity of the spaceship will be

V = √GM/R

Now to find the mech. Energy,

But mechanical energy = 1/2 MV^2

Therefore mechanical energy of the spaceship will be

(a) mech. Energy= 1/2m(√GM/R)^2

(b) escape velocity V = √GM/R already calculated above.