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I am having trouble understanding the conservation of energy in the scenario below.

A photon of $\lambda_1$ is emitted by a source and then absorbed by object 1 in space. Object 1 can be considered to have 0 velocity relative to the emission source. Object 1 captures 100% of the emitted photon's energy and uses it to preform work according to $E=h/\lambda_1$.

The same photon if it were captured by object2, which is traveling in space at a constant velocity of 50% speed of light towards the emission source, would observe $\lambda_2<\lambda_1$. Hence object 2 would capture more energy from the same photon, and it would be more energy than was lost to the photon when it was emitted by the source.

What am I missing here?

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You're missing momentum conservation...

Intuitively, for a body to be moving at a relativistic speed, it must have very low mass, so momentum conservation, even when colliding with a photon, must be taken into account, and as you see, that would mean the absorbing body loses some kinetic energy in exchange for the amount it wins by the blueshifted photon in comparison with the static body.

Of couse, the same thing follows for more massive bodies, but isn't as visual (since one would usually approximate to 0 momentum transfer in this case).

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