= Solution
Retain the backscattering geometry from part (c). With $\lambda=10\,\mathrm{nm}$, $\lambda'=\lambda+2\lambda_C$ differs from $\lambda$ by only $4.85\times10^{-4}$ of its value. <Momentum conservation> gives
$$
v\simeq\frac{h}{m_e}\left(\frac1\lambda+\frac1{\lambda'}\right)\simeq\frac{2h}{m_e\lambda},\qquad
\boxed{v\simeq1.45\times10^5\,\mathrm{m\,s^{-1}},\quad\frac vc\simeq4.85\times10^{-4}.}
$$
The Lorentz-factor correction is of order $v^2/c^2\simeq2.4\times10^{-7}$, so the electron recoil is safely nonrelativistic. Its <kinetic energy> is only about $0.060\,\mathrm{eV}$, compared with incoming photon <energy> about $124\,\mathrm{eV}$. This validates the free-electron recoil approximation's relativistic aspect; actual binding or thermal motion in matter can require separate corrections. No larger recoil occurs at another scattering angle for the same initial photon.
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