Conservation of Linear Momentum — Question 8

Question 8

A wall can hide the missing momentum

A $0.30\,\mathrm{kg}$ ball strikes a wall at $+10\,\mathrm{m/s}$ and rebounds at $-6\,\mathrm{m/s}$. Treat the wall and its support as part of Earth and neglect other impulses.

1. Find the momentum change of the ball and that of Earth.

2. If Earth is assigned mass $6.0\times10^{24}\,\mathrm{kg}$, estimate its velocity change. Why does a visibly stationary wall not disprove conservation?

Question 8 - Solution

A wall can hide the missing momentum

(a) Derivation and calculation

The ball momentum change and the Earth response are $$\boxed{\Delta p_b=0.30(-6-10)=-4.8\ \mathrm{kg\,m/s}}.$$ $$\Delta p_b+\Delta P_E=0\Rightarrow\boxed{\Delta P_E=+4.8\ \mathrm{kg\,m/s}}.$$

(b) Result and interpretation

The corresponding Earth velocity change is $$\boxed{\Delta V_E=\frac{4.8}{6.0\times10^{24}}=8.0\times10^{-25}\ \mathrm{m/s}}.$$ Earth receives finite momentum but imperceptible speed because of its enormous mass. The ball alone is subject to an external wall impulse.

Physical check

Total momentum is conserved when the system boundary includes the wall and its support.