Linear Momentum — Question 1

Question 1

Momentum points where the particle moves

A $0.50\,\mathrm{kg}$ puck has velocity $(-6,8)\,\mathrm{m/s}$ in a laboratory frame.

1. Find its momentum vector, magnitude and direction measured counterclockwise from $+x$.

2. A second puck has the same momentum but twice the mass. Compare their kinetic energies.

Question 1 - Solution

Momentum points where the particle moves

(a) Derivation and calculation

Momentum has the velocity direction: $$\boxed{\vec p=m\vec v=0.50(-6,8)=(-3,4)\ \mathrm{kg\,m/s}},\qquad \boxed{p=\sqrt{9+16}=5.0\ \mathrm{kg\,m/s}}.$$ $$\boxed{\theta=180^\circ-\tan^{-1}(4/3)=126.9^\circ}.$$

(b) Result and interpretation

At a fixed momentum magnitude, $$K=\frac{p^2}{2m},\qquad K_1=\frac{25}{2(0.50)}=25\ \mathrm J,\quad \boxed{K_2=\frac{25}{2(1.0)}=12.5\ \mathrm J}.$$ Doubling the mass halves both speed and energy at the same momentum.

Physical check

The negative horizontal and positive vertical components place the momentum in quadrant II.