To determine the velocity of a body falling from rest at a height R above the Earth's surface, we must equate the increase in kinetic energy to the decrease in potential energy. Given: Initial height h=R Radius of Earth R Acceleration due to gravity g The change in potential energy when the body falls to the Earth's surface can be given by: ∆U=‌
mgh
1+‌
h
R
Substituting h=R, we get: ∆U=‌
mgR
1+‌
R
R
=‌
mgR
2
By the conservation of energy principle, the increase in kinetic energy equals the decrease in potential energy: ‌
1
2
mv2=‌
mgR
2
Solving for v : ‌mv2=mgR ‌v2=gR ‌v=√gR Thus, the velocity of the body when it reaches the Earth's surface is v=√gR.