Elastic Collisions in One Dimension — Question 8

Question 8

A faster target cannot be hit from behind without another event

Cart 1 is behind cart 2 on a frictionless line. Their masses are $2$ and $1\,\mathrm{kg}$ and velocities are $2$ and $5\,\mathrm{m/s}$, respectively. A calculation applies elastic-collision formulas to this state.

1. Find the formal transformed velocity pair.

2. Explain why it is not a future collision outcome for the stated freely moving carts.

Question 8 - Solution

A faster target cannot be hit from behind without another event

(a) Derivation and calculation

The center velocity is $$V_C=\frac{2(2)+1(5)}3=3\ \mathrm{m/s}.$$ Formal reversal of both center-frame velocities gives $$v_1=2V_C-u_1=4,\qquad v_2=2V_C-u_2=1\ \mathrm{m/s}.$$

(b) Result and interpretation

But the actual initial gap satisfies $$\frac{d}{dt}(x_2-x_1)=u_2-u_1=5-2=3\ \mathrm{m/s}>0.$$ The objects are already separating, so no future contact occurs in free motion: $$\boxed{v_1=2,\qquad v_2=5\ \mathrm{m/s}}.$$ The formal pair is a time-reversed incoming state, not a realizable next collision.

Physical check

Collision formulas require contact and an approaching initial configuration.