# Sharing Potential Energy

Consider the following two questions:

1. A 5 kg box, initially at rest, is released from a height of 3 meters. What is the box’s speed just before hitting the ground?

2. An electron and a positron, both initially at rest, are separated by a distance of 1 cm. What is the electron’s speed just before it collides with the positron?

Presumably you would solve both problems using conservation of energy. In 1, would you set the initial potential energy equal to the box’s final kinetic energy? Does the earth get any of the potential energy? What about in the second case? Would you set the initial potential energy equal to the electron’s final kinetic energy or equal to the sum of both particle’s final kinetic energies?

The standard solution for 1 would be KE_{box,final} = PE_{initial}. The standard solution for 2 would be KE_{e,final} + KE_{p,final} = PE_{initial}. Why the difference? In my experience, it is not discussed with students why the box gets all of the potential energy but the electron has to share the potential energy with the positron. In fact, I’m not sure I ever thought about it until recently.

Consider the first question. The box will fall for some length of time T during which the box experiences a downward force equal to mg and the earth experiences an upward force equal to mg (m is the mass of the box). During this time the box will experience an acceleration equal to F/m = g and the earth will experience an acceleration equal to F/M = mg/M (M is the mass of the earth). Let us compare the final kinetic energies of each object:

KE_{box}/KE_{earth} = 0.5m(gT)^{2}/0.5M(mgT/M)^{2} = M/m = 10^{24}

The conservation of energy equation for the first question really is KE_{box,final} + KE_{earth,final} = PE_{initial}. However, because the earth is so massive, the KE of the earth ends up being negligible compared to the KE of the box so we ignore it. The box and the earth both experience the same force but they do not share the energy equally. In the second question, the electron and the positron both have the same mass so they will share the energy equally and we must consider both objects’ KE.

Edit: If both objects are allowed to move then you would need to use both conservation of energy and conservation of momentum to solve for the final velocities. It is easy to see that a miniscule velocity for the earth is the only way to satisfy conservation of momentum. There should also be some final separation and final potential energy in the second scenario. (This is difficulty in comparing electrostatics with near-earth gravity.)