Conservation of Linear Momentum — Question 9

Question 9

Uncertain measurements test an isolation claim

Two $1.0\,\mathrm{kg}$ carts initially move at $3.0$ and $-1.0\,\mathrm{m/s}$, taken as exact. After interaction, their signed velocities lie in $[0.4,0.6]$ and $[1.0,1.2]\,\mathrm{m/s}$.

1. Can the records be consistent with zero external impulse?

2. Find the full external-impulse interval required by the measurements.

Question 9 - Solution

Uncertain measurements test an isolation claim

(a) Derivation and calculation

The exact initial momentum and bounded final momentum are $$P_i=1(3)+1(-1)=2.0\ \mathrm{kg\,m/s}.$$ $$P_f\in[0.4+1.0,0.6+1.2]=[1.4,1.8]\ \mathrm{kg\,m/s}.$$ Since $2.0\notin[1.4,1.8]$, zero external impulse is excluded.

(b) Result and interpretation

Subtract the exact initial value from both bounds: $$\boxed{J_{\mathrm{ext}}=P_f-P_i\in[-0.6,-0.2]\ \mathrm{N\,s}}.$$ Every permitted outcome requires negative impulse, assuming all uncertainty is contained in the stated final intervals.

Physical check

Even the largest allowed final momentum falls short of the initial momentum by 0.2 kg m/s.