Momentum and Kinetic Energy in Collisions — Question 9

Question 9

The same energy and momentum admit two time directions

Two $1\,\mathrm{kg}$ carts have total momentum $6\,\mathrm{kg\,m/s}$ and total kinetic energy $10\,\mathrm J$. Cart 1 is to the left of cart 2.

1. Find all velocity pairs consistent with those totals.

2. Which pair represents separation and which represents approach?

Question 9 - Solution

The same energy and momentum admit two time directions

(a) Derivation and calculation

Momentum and energy give $$v_1+v_2=6,\qquad v_1^2+v_2^2=20.$$ $$(v_1-v_2)^2=2(20)-6^2=4\Rightarrow\boxed{(v_1,v_2)=(4,2)\text{ or }(2,4)\ \mathrm{m/s}}.$$

(b) Result and interpretation

For object 1 on the left, the gap evolves as $$\frac{d}{dt}(x_2-x_1)=v_2-v_1.$$ The pair $(4,2)$ is approaching, while $(2,4)$ is separating. Thus the outgoing state is $\boxed{(2,4)\ \mathrm{m/s}}$.

Physical check

Conservation laws alone do not distinguish incoming from outgoing branches.