Conservation of Linear Momentum — Question 10

Question 10

Conservation gives a family until another condition is supplied

Two isolated carts of masses $2$ and $3\,\mathrm{kg}$ have initial velocities $5$ and $0\,\mathrm{m/s}$. Only momentum conservation is assumed for their interaction.

1. Express all final velocity pairs allowed by momentum conservation alone. Give two examples.

2. Explain what extra physical information is needed to select an actual outcome.

Question 10 - Solution

Conservation gives a family until another condition is supplied

(a) Derivation and calculation

One momentum equation defines a line of possible final states: $$2v_1+3v_2=2(5)+3(0)=10.$$ $$\boxed{v_2=\frac{10-2v_1}{3}},\qquad (v_1,v_2)=(2,2)\ \text{or}\ (-1,4)\ \mathrm{m/s}.$$

(b) Result and interpretation

The example energies differ despite equal momentum: $$K_{(2,2)}=\tfrac12(5)(2)^2=10\ \mathrm J,\qquad K_{(-1,4)}=\tfrac12(2)(1)+\tfrac12(3)(16)=25\ \mathrm J.$$ Sticking, elastic energy conservation, restitution, or another interaction law is needed to choose an outcome. Momentum alone also permits states requiring internal energy release.

Physical check

Both examples reproduce the same total momentum of 10 kg m/s.