Collision and Impulse — Question 10

Question 10

Testing whether an external impulse is negligible

During a $0.030\,\mathrm s$ collision, two carts exchange a contact impulse of magnitude $8.0\,\mathrm{N\,s}$. Their total mass is $5.0\,\mathrm{kg}$. The magnitude of their net external horizontal force never exceeds $4.0\,\mathrm N$.

1. Bound the system momentum change and center-velocity change.

2. Is an exact zero change justified? Compare the external-impulse bound to the exchanged contact impulse.

Question 10 - Solution

Testing whether an external impulse is negligible

(a) Derivation and calculation

Integrate the bound on the net external force: $$|\Delta P|=|J_{\mathrm{ext}}|\le F_{{\mathrm{ext}},\max}\Delta t =4.0(0.030)=\boxed{0.12\ \mathrm{N\,s}}.$$ $$\boxed{|\Delta V_C|\le|\Delta P|/M\le0.12/5=0.024\ \mathrm{m/s}}.$$

(b) Result and interpretation

The external bound relative to the exchanged contact impulse is $$\boxed{\frac{0.12}{8.0}\,100\%=1.5\%}.$$ This supports approximate conservation on the internal-exchange scale, but does not establish exact zero change. It is not a relative error in total momentum, which might itself be zero.

Physical check

The two internal contact impulses cancel exactly in the system total even when each is large.