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Example: Elastic Collision of Two Identical Carts
Cart 1 collides with stationary cart 2, which is identical. Suppose that the collision is (nearly) elastic, as it will be if the carts repel each other magnetically or interact through soft springs. In this case there is no change of internal energy. What are the final momenta of the two carts?
Facts
Initial situation: Just before collision
Final situation: Just after collision
Lacking
Final momenta of the two carts
Approximations & Assumptions
Assume there is no change of internal energy
Neglect friction and air resistance
Representations
System: Both carts
Surroundings: Earth, track, air
Solution
Since the y and z components of momentum don't change, we can work with only x components
From the momentum principle:
$$\vec{p}_f = \vec{p}_i + \vec{F}_{net} \Delta t$$
$$\vec{p}_{1xf} + \vec{p}_{2xf} = \vec{p}_{1xi} + 0$$
From the energy principle:
$$E_f = E_i + W + Q$$
$$K_{1f} + K_{2f} + E_{int1f} + E_{int2f} = K_{1i} + K_{2i} + E_{int1i} + E_{int2i}$$
$$K_{1f} + K_{2f} = K_{1i}$$
Combine momentum and energy equations:
$$\dfrac{p^{2}_{1xf}}{2m} + \dfrac{p^{2}_{2xf}}{2m} = \dfrac{(p_{1xf} + p_{2xf})^2}{2m}$$
$$p^{2}_{1xf} + p^{2}_{2xf} = p^{2}_{1xf} + 2{p_{1xf}p_{2xf} + p^{2}_{2xf}$$