Collisions in Two Dimensions — Question 3

Question 3

Inferring an unseen target mass from two components

A $2\,\mathrm{kg}$ puck initially moves at $(6,0)\,\mathrm{m/s}$ toward a stationary target of unknown mass $M$. After collision the incident puck has $(2,2)\,\mathrm{m/s}$ and the target has $(2,-1)\,\mathrm{m/s}$. External impulse is negligible.

1. Infer $M$ separately from each momentum component and test consistency.

2. Determine whether the collision is elastic.

Question 3 - Solution

Inferring an unseen target mass from two components

(a) Derivation and calculation

The component equations independently determine the unknown mass: $$x:\quad2(6)=2(2)+M(2)\Rightarrow M=4\ \mathrm{kg},$$ $$y:\quad0=2(2)+M(-1)\Rightarrow\boxed{M=4\ \mathrm{kg}}.$$ The two measurements are consistent with zero external impulse.

(b) Result and interpretation

Compare the kinetic energies using squared vector magnitudes: $$K_i=\tfrac12(2)(6)^2=36\ \mathrm J,$$ $$K_f=\tfrac12(2)(2^2+2^2)+\tfrac12(4)(2^2+(-1)^2)=18\ \mathrm J.$$ $$\boxed{L=18\ \mathrm J};\qquad\text{the collision is inelastic}.$$

Physical check

The final vertical momenta $+4$ and $-4$ kg m/s cancel.