Conservation of momentum and laser powered solar sail

[Infinitum]

And does the Earth move away from the ball? If so where is the energy coming from?

Yes, the Earth moves away from the ball. The initial kinetic energy and potential energy of the Earth and ball, and also the gravitational potential energy are the only energies in play here(assuming no friction, no heating up etc). And the total mechanical energy is conserved.

 

[A.T.]

And does the Earth move away from the ball? If so where is the energy coming from?

The idea that the photons / balls have exactly the same energy after reflection is an idealized one. It is based on the assumption that the reflector has infinite mass, so it doesn't begin to move during the interaction.

In reality the ball has a certain finite interaction time during which the Earth will slightly accelerate, so the ball doesn't regain it's full speed. The same is true for a pulse of light, which will be reflected with a Doppler red-shift at the end. But I'm not sure if you can break this down to a single photon being reflected.

 

[Drakkith]

The idea that the photons / balls have exactly the same energy after reflection is an idealized one. It is based on the assumption that the reflector has infinite mass, so it doesn't begin to move during the interaction.

In reality the ball has a certain finite interaction time during which the Earth will slightly accelerate, so the ball doesn't regain it's full speed. The same is true for a pulse of light, which will be reflected with a Doppler red-shift at the end. But I'm not sure if you can break this down to a single photon being reflected.

Now that makes sense.

 

[D H]

So I think that is right. The ship will move in any case, even if only momentarily.

Try calculating how little the ship will move in the absorbed case. It is essentially nothing. Even in the case of a perfect reflector (and there is no such thing), one is better off dispensing with the mirror. Even a perfect laser beam is not parallel. The beam will have diverged somewhat by the time it hits the mirror, so the reflection will not be as effective as just firing the laser opposite the desired direction of travel. Bottom line: Ditch the mirror. It's a Rube Goldberg device that adds nothing.

Propulsion via a ship-borne laser is a terrible choice for propulsion. Not much oomph and too much mass for the laser and all the associated machinery. The only possible exception is a laser powered by matter / anti-matter annihilation. This is pure sci-fi, however.

 

[haruspex]

The same is true for a pulse of light, which will be reflected with a Doppler red-shift at the end. But I'm not sure if you can break this down to a single photon being reflected.

You can. On an earlier thread I calculated the Doppler shift and found that the photon's lost energy matched the mirror's gain in kinetic energy.

 

[nemesiswes]

Yeah I figured that it would be more efficient to just fire the laser out the back then use mirrors or some absorbing material. But I just wanted to know if it was possible theoretically, not if it was useful or not and apparently it is possible, just not very useful, lol.