OpenStax College Physics for AP® Courses, Chapter 29, Problem 10 (Test Prep for AP® Courses)

This is College Physics Answers with Shaun Dychko. A metal sphere is suspended in a vacuum and it's being hit by some photons we are told that N number of photons are going to hit the sphere and there are two scenarios to consider: in one scenario, the sphere reflects the photon back with the same wavelength that it had originally and in the other scenario, there's no reflection and instead the photon is absorbed by the sphere. So our job is to find an expression for the momentum of the sphere in each scenario and also classify them as elastic or inelastic. So we'll do the categorization part first— that's the easy part—so this is elastic in this scenario with reflection and whereas when the photon is absorbed that is inelastic— this is for when objects stick together is an inelastic collision— so without reflection is inelastic and with reflection is elastic. Okay! So we have to take the total initial momentum and that's going to equal the total final momentum. Let's consider the 'with reflection' scenario first: we have the initial momentum of the light, which is denoted by this subscript γ, the initial momentum of the light then is the number of photons multiplied by the momentum for each photon, which is h over λ— h being Planck's constant— and then the momentum of the sphere initially is zero so this term disappears and then after the collision, we have a momentum for the light after collision equal to the same magnitude— N photons times Planck's constant over λ— but now it's going in the opposite direction so we need a negative sign— that's the negative direction to the left; we are taking to the right to be positive— and so that's an expression for P γ prime and then we want to know what is this momentum of this sphere after collision? So we will subtract this P γ prime from both sides and so we have momentum of the sphere after collision is the momentum of the light before collision minus the momentum of the light after the collision. So that's Nh over λ—momentum before collision—minus the momentum after collision, which is negative Nh over λ and so this negative and a negative together makes a positive so that's 2 times the number of photons times Planck's constant over λ is going to be the momentum of the sphere after the collision and this is elastic. Okay! So without reflection, we have, you know, the same conservation of momentum here and the momentum of the sphere initially is zero and the momentum of the light after collision is also zero because the photons are stuck to the sphere. When these photons are absorbed by the sphere since the photons don't have any mass, they won't contribute to the momentum of this photon-sphere combination after the collision and so this term for the photon momentum after collision is going to be zero so we will have just the momentum of the sphere after collision. And so this means the momentum of the sphere after collision is equal to the initial momentum of the light, which is the number of photons times Planck's constant over wavelength and this is an inelastic collision and you can see the elastic collision results in 2 times the greater momentum for the sphere compared with the inelastic case.