= Solution
Take a stationary initial electron of <mass> $m_e$, a photon incident along $x$, and a nonrelativistic recoil <velocity> of magnitude $v$. Choose $\theta$ above $x$ for the scattered photon and $\phi$ below $x$ for the electron. <Conservation of momentum> gives
$$
\boxed{\frac{h\nu}{c}=\frac{h\nu'}{c}\cos\theta+m_ev\cos\phi,\qquad
0=\frac{h\nu'}c\sin\theta-m_ev\sin\phi.}
$$
The electron's unchanged rest <energy> cancels in <conservation of energy>, leaving
$$
\boxed{h\nu=h\nu'+\frac12m_ev^2.}
$$
Equivalently use $h\nu/c=h/\lambda$ and $h\nu=hc/\lambda$. The <nonrelativistic limit> requires $v/c\ll1$, and treating the electron as free requires its binding <energy> to be negligible on the scale of the collision.
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