This question is about converting kinetic energy into gravitational potential energy. Escape velocity is the speed required to leave the gravitational field of a mass, in this case it's a planet. In other words the body has to be moving at such a velocity that it will reach a point infinitely far away from the planet.
The gravitational potential energy (PE) at any given point is given as:
PE = -GMm/r
Where G is Newton's gravitational constant; M is the mass of the planet; m is the mass of the body moving away from the planet and r is the distance from the centre of the planet/gravitational field.
Using this formula the potential energy at a distance infinitely far away (infinity) is 0. At the surface of the planet the potential energy is:
-GMm/r
This means that in order to get from the surface to infinity there will be a gain of
GMm/r
This will come from the kinetic energy of the body escaping.
Kinetic Energy (KE) is given as:
KE = mv2/2
Where m is the mass of the moving body and v is it's velocity.Now we have all we need to solve this problem
If we set the kinetic and potential energy equal to each other:
KE = PE
mv2/2 = GMm/r
Divide by m on both sides, this gets rid of all mentions of m. That means the final answer will not depend on the mass leaving the planet at all!
v2/2 = GM/r
Rearrange:
v = sqrt(2GM/r)
sqrt() means take the square root of what is inside the bracket.
Interesting related fact:
A black hole is an object that has an escape velocity that is greater than the speed of light. This means not even light can escape the gravitational pull of a black hole!!!