Inelastic Collisions in One Dimension — Question 9

Question 9

A measured energy loss determines restitution

Two $2\,\mathrm{kg}$ carts approach at $+3$ and $-1\,\mathrm{m/s}$. Their isolated head-on collision reduces kinetic energy by $6\,\mathrm J$. They then separate.

1. Find the separation speed and coefficient of restitution.

2. Find each final velocity and identify the branch rejected by the separation condition.

Question 9 - Solution

A measured energy loss determines restitution

(a) Derivation and calculation

The center velocity and reduced mass are $$V_C=\frac{2(3)+2(-1)}4=1\ \mathrm{m/s},\qquad\mu=1\ \mathrm{kg}.$$ The initial relative energy is 8 J, so after losing 6 J, $$\tfrac12\mu w_f^2=8-6=2\Rightarrow w_f=2\ \mathrm{m/s},\qquad\boxed{e=2/4=0.50}.$$

(b) Result and interpretation

Select the separating branch: $$v_1+v_2=2V_C=2,\qquad v_2-v_1=2.$$ $$\boxed{v_1=0,\qquad v_2=2\ \mathrm{m/s}}.$$ The alternative $(2,0)$ has the same energy and momentum but is still approaching.

Physical check

Total energy falls from 10 J to 4 J, exactly the specified 6 J loss.