An object of mass m is dropped from height h above a planet of mass m and radius r .

find an expression for the object's speed as it hits the ground.
express your answer in terms of the variables m, m,h, r and appropriate constants.

Explanation:

The initial mechanical energy of the object, when it is located at height h above the the planet, is just gravitational potential energy:

where

G is the gravitational constant

M is the mass of the planet

m is the mass of the object

R is the radius of the planet

h is the altitude of the object

When the object hits the ground, its mechanical energy will sum of potential energy and kinetic energy:

where

v is the speed of the object at the ground

Since the mechanical energy is conserved, we can write

and solving for v, we find

the part of the wave he is travelling on is the crest