The energy and momentum of the ground

Imagine that you throw a ball in the air, and lands on the ground and stops (i.e., the collision is inelastic). What happened to the energy and momentum? The answer for energy is completely different from that for momentum, which is a trivial but interesting illustration of the differences between them. Also we get fun with infinities.

Assume a ball of mass $m$ and velocity $v$ hits the ground and stops. Before it hits, it has energy $\frac{1}{2}m v^2$ and momentum $m v$. After it hits, the energy and momentum of the ball are both zero, so by conservation they must have gone elsewhere.

Momentum is easy: the momentum went into the ground. This is possible without the ground having noticeably moved, since the ground’s mass is some huge value $M \approx \infty$. The resulting velocity of the ground is the minute $m v / M \approx 0$, but this zero is canceled out by the huge mass to get the finite momentum $M m v / M = m v$. Thus, it is perfectly consistent to model the ground as a rigid body with infinite mass, zero velocity, and nonzero momentum $p$. Only 3 new variables are required to account for this ground momentum and make our system closed with respect to conservation of momentum.

Energy is different: it is impossible to transfer energy to a large, rigid object. The kinetic energy of the ground after the impact is $$\frac{1}{2} M \left(\frac{m v}{M}\right)^2 = \frac{m^2 v^2}{2 M} \approx 0$$ In other words, $M = \infty$ and $V = 0$ give $\frac{1}{2} M V^2 = \infty 0 0 = 0$ since the zeros outnumber the ones. It is impossible to store energy in the ground by moving it as a whole.

What happens instead is that the energy goes into smaller scale phenomena, typically sound waves or heat. Both cases involve small amounts of matter moving extremely rapidly, which we can roughly characterize as mass $0$, velocity $\sqrt{\infty}$. The key is that we can store as much energy as we like in these tiny phenomena without ever making a dent in momentum, since $0 \sqrt{\infty} = 0$.

We can summarize this situation as follows:

  1. It is possible to hide momentum in a large, motionless object without expending any energy.
  2. It is possible to hide energy in a bunch of small objects without using any momentum.

The fact that energy hides in small places makes it harder to deal with in general; there a lot of small places, which is why the law of conservation of energy has been modified many times to account for new terms no one had previously noticed. Momentum is simpler: if you lose some momentum, all you have to do is look around for the gigantic object (e.g., the Earth).

comments powered by Disqus