The Linear Momentum of a System of Particles — Question 9

Question 9

Opposite momentum histories hide an interaction

Two particles form a closed system with $p_{1x}=5+3t$ and $p_{2x}=7-3t$ in SI units. Assume no external horizontal forces act on either particle.

1. Find the force on each and the total momentum.

2. If the masses are $1$ and $3\,\mathrm{kg}$, find the center velocity and explain whether both speeds are constant.

Question 9 - Solution

Opposite momentum histories hide an interaction

(a) Derivation and calculation

Differentiate each particle momentum: $$\boxed{F_{1x}=3\ \mathrm N},\qquad\boxed{F_{2x}=-3\ \mathrm N}.$$ $$\boxed{P_x=(5+3t)+(7-3t)=12\ \mathrm{kg\,m/s}}.$$ The forces form a cancelling internal pair under the stated assumptions.

(b) Result and interpretation

The center and particle velocities are $$\boxed{V_C=12/(1+3)=3\ \mathrm{m/s}},\qquad v_1=5+3t,\quad v_2=7/3-t.$$ The particle speeds generally change, and particle 2 reverses when $t=7/3$ s. Constant total momentum does not require constant individual momenta.

Physical check

The mass-weighted accelerations are $1(3)+3(-1)=0$.