Elastic Collisions in One Dimension — Question 2

Question 2

Measuring a target mass from rebound

A cart of known mass $m$ strikes a stationary target elastically and rebounds with speed $u/3$, where $u$ was its incident speed. Motion is one dimensional.

1. Infer the target mass $M$ and its final speed.

2. What fraction of initial kinetic energy reaches the target?

Question 2 - Solution

Measuring a target mass from rebound

(a) Derivation and calculation

For a stationary target in an elastic collision, $$\frac{v_1}{u}=\frac{m-M}{m+M}=-\frac13.$$ $$3m-3M=-m-M\Rightarrow\boxed{M=2m}.$$ $$\boxed{v_2=\frac{2m}{m+M}u=\frac23u}.$$

(b) Result and interpretation

The target energy fraction is $$\boxed{\eta=\frac{\tfrac12(2m)(2u/3)^2}{\tfrac12mu^2}=\frac89}.$$ The incident object retains $K_{1f}/K_i=(v_1/u)^2=1/9$, so the fractions sum to unity.

Physical check

The final momentum is $-mu/3+4mu/3=mu$.