Question

[Differential Equations]

This is a physics application of differential equations relating to Newton's Law of Gravitation. See the comments for the prompt.

Answer 1

An object with mass "m" is released from rest at a distance of "a" meters above the earth's surface (See figure below). Use Newton's Universal Law of Gravitation to show that the object impacts the earth surface with a velocity determined by the following equation:
\[v=\sqrt{\frac{ 2agR }{ a+R }}\]where "g" is the acceleration due to gravity at the earth's surface and "R" is the radius of the earth. Ignore any effects due to the earth's rotation and atmosphere. HINT: On the earth's surface, explain why [mg = GMm/R^2], where "M" is the mass of the earth and "G" is the universal gravitational constant.