The Linear Momentum of a System of Particles — Question 7

Question 7

Grouping momentum does not average velocities equally

Group A contains particles of total mass $2\,\mathrm{kg}$ and has center velocity $6\,\mathrm{m/s}$ to the right. Group B has total mass $6\,\mathrm{kg}$ and center velocity $2\,\mathrm{m/s}$ to the left.

1. Find the total momentum and center velocity of both groups.

2. Explain why averaging $+6$ and $-2$ is incorrect, even if both groups contain the same number of particles.

Question 7 - Solution

Grouping momentum does not average velocities equally

(a) Derivation and calculation

Each group’s momentum equals its mass times its group-center velocity: $$P_A=2(6)=12,\qquad P_B=6(-2)=-12\quad\mathrm{kg\,m/s}.$$ $$\boxed{P=P_A+P_B=0},\qquad\boxed{V_C=P/(2+6)=0}.$$

(b) Result and interpretation

The correct group average is mass weighted: $$V_C=\frac{M_AV_A+M_BV_B}{M_A+M_B}\ne\frac{V_A+V_B}{2}=2\ \mathrm{m/s}.$$ Equal particle counts do not imply equal group masses. Internal group motions are already included in each group-center velocity.

Physical check

The heavier group needs only one third the speed to balance the lighter group momentum.