Momentum and Kinetic Energy in Collisions — Question 1

Question 1

Audit both invariants before naming a collision

Two $1\,\mathrm{kg}$ carts initially move at $4$ and $0\,\mathrm{m/s}$. Three proposed final velocity pairs are A: $(0,4)$, B: $(1,3)$ and C: $(-1,5)\,\mathrm{m/s}$. External impulse is negligible.

1. Check momentum and kinetic energy for each outcome.

2. Classify each and identify which requires release of internal energy.

Question 1 - Solution

Audit both invariants before naming a collision

(a) Derivation and calculation

For each candidate, momentum is $$P_i=1(4)+1(0)=4,\quad P_A=0+4=4,\quad P_B=1+3=4,\quad P_C=-1+5=4.$$ Thus all three satisfy the momentum constraint.

(b) Result and interpretation

Evaluate the energies separately: $$K_i=\tfrac12(4^2)=8\ \mathrm J,\quad (K_A,K_B,K_C)=\tfrac12(16,10,26)=(8,5,13)\ \mathrm J.$$ $$\boxed{A:\text{ elastic};\quad B:\text{ loses }3\ \mathrm J;\quad C:\text{ gains }5\ \mathrm J.}$$ Case C requires a source of stored internal energy.

Physical check

Equal total momentum does not imply equal kinetic energy.