Inelastic Collisions in One Dimension — Question 10

Question 10

A passive collision permits a bounded final velocity

A $2\,\mathrm{kg}$ cart moving at $4\,\mathrm{m/s}$ overtakes a $1\,\mathrm{kg}$ cart moving at $1\,\mathrm{m/s}$. The collision is isolated, has no internal energy release, and ends with separation or equal velocities.

1. Find the allowed interval of the lighter cart's final velocity.

2. Find the corresponding interval of kinetic-energy loss and identify both limiting cases.

Question 10 - Solution

A passive collision permits a bounded final velocity

(a) Derivation and calculation

Let $w=v_2-v_1\geq0$. Conservation gives $$2v_1+v_2=2(4)+1(1)=9\Rightarrow v_1=3-w/3,\quad v_2=3+2w/3.$$ A passive collision requires $w\leq u_1-u_2=3$, hence $$\boxed{3\leq v_2\leq5\ \mathrm{m/s}}.$$

(b) Result and interpretation

With $\mu=2/3$ kg, $$\boxed{L=\tfrac12\mu(3^2-w^2)=\frac{9-w^2}{3}\ \mathrm J\in[0,3]\ \mathrm J}.$$ The endpoints are sticking $(3,3)$ with loss 3 J and elastic separation $(2,5)$ with zero loss.

Physical check

The center velocity remains 3 m/s throughout the allowed interval.