Inelastic Collisions in One Dimension — Question 1

Question 1

A head-on capture can end at rest with no energy left in translation

A $3\,\mathrm{kg}$ cart moving at $+4\,\mathrm{m/s}$ sticks to a $2\,\mathrm{kg}$ cart moving at $-6\,\mathrm{m/s}$. The track is frictionless.

1. Find their common velocity and kinetic-energy loss.

2. Find the impulse on each cart. Explain how nonzero impulses coexist with zero system impulse.

Question 1 - Solution

A head-on capture can end at rest with no energy left in translation

(a) Derivation and calculation

For the isolated pair, $$P_i=3(4)+2(-6)=0=(3+2)V\Rightarrow\boxed{V=0}.$$ $$K_i=\tfrac12(3)(4)^2+\tfrac12(2)(6)^2=60\ \mathrm J,\qquad K_f=0.$$ $$\boxed{K_{\mathrm{loss}}=60\ \mathrm J}.$$

(b) Result and interpretation

The individual impulses follow from their momentum changes: $$\boxed{J_1=3(0-4)=-12\ \mathrm{N\,s}},\qquad\boxed{J_2=2(0+6)=+12\ \mathrm{N\,s}}.$$ Their sum is zero, although each object experiences a nonzero impulse.

Physical check

Zero system momentum does not imply zero initial kinetic energy.