Conservation of Linear Momentum — Question 1

Question 1

Recoil specified relative to the launcher

A $4.0\,\mathrm{kg}$ launcher and a $0.20\,\mathrm{kg}$ projectile are initially at rest on a frictionless horizontal track. After firing, the projectile moves right at $21\,\mathrm{m/s}$ relative to the recoiling launcher.

1. Find both laboratory velocities.

2. Why is assigning $21\,\mathrm{m/s}$ directly to the projectile in the lab incorrect?

Question 1 - Solution

Recoil specified relative to the launcher

(a) Derivation and calculation

Use laboratory momentum conservation together with the relative-speed constraint: $$4V+0.20v=0,\qquad v-V=21.$$ $$4V+0.20(V+21)=0\Rightarrow\boxed{V=-1.0\ \mathrm{m/s}},\qquad \boxed{v=V+21=20\ \mathrm{m/s}}.$$

(b) Result and interpretation

The launch specification is $$v_{\mathrm{projectile/launcher}}=v_{\mathrm{projectile/lab}}-V_{\mathrm{launcher/lab}}.$$ Assigning $v=21$ in the laboratory would give $V=-1.05$ and a separation speed of 22.05 m/s, contradicting the stated 21 m/s.

Physical check

The laboratory momenta are $-4$ and $+4$ kg m/s and cancel.