Chapter 9 Problem 36

Problem

A 0.280-kg croquet ball makes an elastic head-on collision with a second ball initially at rest. The second ball moves off with half the original speed of the first ball.

(a) What is the mass of the second ball?

(b) What fraction of the original kinetic energy ($\Delta K/K$) gets transferred to the second ball?

Solution

This is a head-on 1D collision. Call the direction of the first ball the positive direction. Let A represent the first ball, and B represent the second ball. $v_{B}=0$ and $v_{B}^{\prime }={\frac {1}{2}}v_{A}$

(a)

$v_{A}-v_{B}=-\left(v_{A}^{\prime }-v_{B}^{\prime }\right)$ $\rightarrow v_{A}^{\prime }=-{\frac {1}{2}}v_{A}$

$p_{i}=p_{f}$ $\rightarrow m_{A}v_{A}+m_{B}v_{B}=m_{A}v_{A}^{\prime }+m_{B}v_{B}^{\prime }$ $\rightarrow m_{A}v_{a}=-{\frac {1}{2}}m_{A}v_{A}+m_{B}{\frac {1}{2}}v_{A}$ $\rightarrow m_{B}=3m_{A}=3\left(0.280{\text{kg}}\right)=\left(0.840{\text{kg}}\right)$

(b)

${\frac {K_{B}^{\prime }}{K_{A}}}={\frac {{\frac {1}{2}}m_{B}v_{B}^{\prime 2}}{{\frac {1}{2}}m_{A}v_{A}^{2}}}$ $={\frac {3m_{A}\left({\frac {1}{2}}v_{A}\right)^{2}}{m_{A}v_{A}^{2}}}$ $=0.75$