Linear Momentum — Question 5

Question 5

Recovering mass from momentum and energy

For a nonrelativistic particle, instruments report momentum magnitude $12\,\mathrm{kg\,m/s}$ and kinetic energy $18\,\mathrm J$.

1. Infer its mass and speed.

2. If only the momentum magnitude and energy are known, is the velocity vector determined?

Question 5 - Solution

Recovering mass from momentum and energy

(a) Derivation and calculation

Eliminate speed between $p=mv$ and $K=mv^2/2$: $$K=\frac{p^2}{2m}\Rightarrow\boxed{m=\frac{p^2}{2K}=\frac{144}{36}=4.0\ \mathrm{kg}}.$$ $$\boxed{v=\frac pm=\frac{12}{4}=3.0\ \mathrm{m/s}}.$$

(b) Result and interpretation

The measurements permit the family $$\boxed{\vec v=3.0\hat n\ \mathrm{m/s},\qquad|\hat n|=1}.$$ All members have momentum magnitude 12 kg m/s and energy 18 J. A direction measurement is needed to determine the vector.

Physical check

Substitution gives $\tfrac12(4)(3)^2=18$ J.