Gravitation Potential Energy — Question 3

Question 3

Falling from a finite radius releases less energy than falling from infinity

A particle is released from rest at $r=3R$ outside a spherical body of mass $M$. It falls radially to the surface $r=R$. Neglect all non-gravitational forces.

1. Find its impact speed.

2. Compare with the speed acquired from rest at infinity and explain the difference.

Question 3 - Solution

Falling from a finite radius releases less energy than falling from infinity

(a) Derivation and calculation

Conservation between release and impact gives $$K_i+U_i=K_f+U_f,\qquad 0-\frac{GMm}{3R}=\frac12mv^2-\frac{GMm}{R}.$$ Therefore $$\frac12v^2=\frac{2GM}{3R},\qquad \boxed{v=\sqrt{\frac{4GM}{3R}}}.$$

(b) Result and interpretation

For release from rest at infinity, $$0=\frac12mv_\infty^2-\frac{GMm}{R},\quad v_\infty=\sqrt{2GM/R}.$$ $$\boxed{\frac v{v_\infty}=\sqrt{\frac{4/3}{2}}=\sqrt{2/3}\simeq.816497}.$$ The finite release supplies only $2/3$ of the kinetic energy available from infinity.

Physical check

The test mass cancels, and the finite-start impact speed is below the surface escape speed.