Elastic Collisions in One Dimension — Question 1

Question 1

Deriving the unequal-mass elastic outcome

A $2\,\mathrm{kg}$ cart moving at $5\,\mathrm{m/s}$ collides elastically in one dimension with a stationary $3\,\mathrm{kg}$ cart on a frictionless track.

1. Use momentum and kinetic energy to derive both final velocities, rejecting the no-interaction branch.

2. Check the final momentum, energy and separation speed.

Question 1 - Solution

Deriving the unequal-mass elastic outcome

(a) Derivation and calculation

Momentum and energy conservation are $$2v_1+3v_2=10,\qquad2v_1^2+3v_2^2=50.$$ Factor the changes: $2(5-v_1)=3v_2$ and $2(5-v_1)(5+v_1)=3v_2^2$. For a nonzero impulse, division yields $5+v_1=v_2$. $$2v_1+3(v_1+5)=10\Rightarrow\boxed{v_1=-1,\quad v_2=4\ \mathrm{m/s}}.$$

(b) Result and interpretation

The unchanged branch $(5,0)$ satisfies the conservation equations but is not the outgoing state after contact: $$v_2-v_1=5>0,\qquad P_f=-2+12=10,\qquad K_f=1+24=25\ \mathrm J.$$ The final relative speed reverses the initial relative velocity.

Physical check

The heavier target moves forward while the incident cart rebounds.