Elastic Collisions in One Dimension — Question 6

Question 6

A sequence of equal-mass impacts transmits velocity

Three identical carts A, B and C lie in that order on a frictionless line with positive gaps. Initially A moves right at $6\,\mathrm{m/s}$ while B and C are at rest. Each collision is isolated, one dimensional and elastic.

1. Find the velocities after A hits B and after B hits C.

2. Explain why no later collisions occur and calculate the total impulse received by B over the sequence.

Question 6 - Solution

A sequence of equal-mass impacts transmits velocity

(a) Derivation and calculation

An elastic collision of identical masses exchanges their velocities: $$(6,0,0)\xrightarrow{A-B}(0,6,0)\xrightarrow{B-C}(0,0,6)\ \mathrm{m/s}.$$ $$\boxed{v_A=0,\quad v_B=0,\quad v_C=6\ \mathrm{m/s}}.$$

(b) Result and interpretation

The two impulses on B are $$J_{B,1}=m(6-0)=+6m,\qquad J_{B,2}=m(0-6)=-6m.$$ $$\boxed{J_{B,\mathrm{net}}=0}.$$ B is accelerated and then stopped. Zero net impulse over both collisions does not mean zero impulse at each contact; the final velocities produce no further approach.

Physical check

All the initial momentum and kinetic energy end in C.