Elastic Collisions in One Dimension — Question 4

Question 4

A moving wall can increase particle kinetic energy

A massive wall moves left at constant $U=-2\,\mathrm{m/s}$. A $0.50\,\mathrm{kg}$ ball moving right at $u=6\,\mathrm{m/s}$ collides normally and elastically in the wall frame. Treat the wall velocity as unaffected.

1. Find the outgoing laboratory velocity.

2. Find the ball's energy change and explain the energy source.

Question 4 - Solution

A moving wall can increase particle kinetic energy

(a) Derivation and calculation

Transform to the wall frame, reverse the elastic relative velocity, and transform back: $$u'=u-U=6-(-2)=8\ \mathrm{m/s},\qquad v'=-8\ \mathrm{m/s}.$$ $$\boxed{v=v'+U=-8-2=-10\ \mathrm{m/s}}.$$

(b) Result and interpretation

The ball gains laboratory kinetic energy: $$\boxed{\Delta K=\tfrac12(0.50)[(-10)^2-6^2]=16\ \mathrm J}.$$ Equivalently, with $v=2U-u$, $$\Delta K=2mU(U-u)=16\ \mathrm J.$$ The approaching wall or the drive maintaining its motion supplies this energy.

Physical check

Elasticity conserves relative kinetic energy in the effectively fixed wall frame.