## Solar Sailing

The photon comes in with momentum p and leaves with the same momentum in the opposite direction. The sail picks up a momentum of 2p in the original photon's direction. Energy conservation states that:

$$h\nu_1=h\nu_2+\frac{(2p)^2^}{2m}=p_{1}c=p_{2}c+\frac{(2p)^2^}{2m}$$

Isn't |p1|=|p2| thus violating energy conservation?
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 But is the momentum of the photon before and after truly equal. Isn't it true that $$p=\frac{h}{\lambda}$$, thus making the momentum depend on wavelength which changed due to a Doppler shift.

## Solar Sailing

It's not a Doppler shift, but the photon just loses E and p like any other projectile.
This also causes a change in its wave length.
 Energy Conservation:$$h\nu_1=h\nu_2+\frac{(2p)^2^}{2m}=p_{1}c=p_{2}c+\frac{(2p)^2^}{2m}$$ Momentum Conservation: $$p_1=p_2+p_{sail}$$ $$\frac{h\nu_{1}}{c}=\frac{h\nu_{2}}{c}+p_{sail}$$ If we assume that it is a totally elastic reflection then the photon must have momentum p_1 in the opposite direction and the sail must have momentum 2p_1 in the initial direction. This means that the wavelength of the photon should stay the same (the magnitude of the momentum is the same). However, from energy conservation we see that the photon should lose some energy and thus change its wavelength in order to give to the sail as kinetic energy. Where am I going wrong?
 If the sail recoils, the reflected photon will not have the incident momentum. Your equations would show that if you didn't assume a mistake. Trust your equations over your intuition.

 Quote by pam If the sail recoils, the reflected photon will not have the incident momentum. Your equations would show that if you didn't assume a mistake. Trust your equations over your intuition.
So then the sail momentum gain isn't really exactly 2p right? This is in order to maintain momentum conservation.

Can you explain why this isn't a simple mechanical reflection where the incident beam gets reflected with exactly the same momentum backwards and the target gains 2 times that momentum?

 Quote by pam If the sail RECOILS, the reflected photon will not have the incident momentum.
If the sail RECOILS, the reflected photon will not have the incident momentum.

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