Inelastic Collisions in One Dimension — Question 7

Question 7

The restitution needed for the lighter body to reverse

A $1\,\mathrm{kg}$ cart at $u>0$ hits a stationary $3\,\mathrm{kg}$ cart. Their one-dimensional collision has $0\le e\le1$ and negligible external impulse.

1. Derive both final velocities in terms of $u$ and $e$.

2. Find the threshold for the incident cart to rebound and its velocity when $e=0.50$.

Question 7 - Solution

The restitution needed for the lighter body to reverse

(a) Derivation and calculation

Momentum and restitution give $$v_1+3v_2=u,\qquad v_2-v_1=eu.$$ Solving the linear system, $$\boxed{v_1=\frac{1-3e}{4}u,\qquad v_2=\frac{1+e}{4}u}.$$

(b) Result and interpretation

Rebound requires $v_1<0$ for $u>0$: $$1-3e<0\Rightarrow\boxed{e>1/3}.$$ At $e=0.50$, $$\boxed{v_1=-u/8,\qquad v_2=3u/8}.$$

Physical check

At the threshold $e=1/3$, the incident object stops without reversing.