A student asked me if the a spaceship with a nuclear reactor, coasting
through space, would increase speed as a result of the ongoing decrease
in mass and the conservation of momentum.
We played around with the question and re-cast it as follows.
In a stationary car out somewhere in space, suppose an electron and
positron are at rest, separated by a membrane.
We remove the membrane, the pair meet and annihilate.
Two photons are produced.
Their combined energy is given by (delta M)c^2.
Their combined momentum still has to be zero, so we conclude that the 2
photons have to have the same frequency and have to move off in opposite
directions.
The photons reach the far walls and let's assume they are absorbed. Each
exerts the same impulse on the car, which remains at rest.
Is this all correct so far? And then...
An outside observer sees that the car and the two particles are in
motion in some direction at a speed v.
This observer says that the initial momentum of the pair of particles
was 2mv (where m is the mass of each particle).
So the final momentum of the emitted photons is also 2mv. So they
cannot have equal frequency.
The photon moving in the direction of the initial velocity has greater
magnitude momentum.
The photons reach the far walls and are absorbed. The car gets a
greater impulse in the forward direction, so it speeds up. Uh oh.
Clearly, I don't know what I am talking about! Thoughts? And happy
Thanksgiving.