Why is it not possible for a photon to be completely absorbed by a...

Assume a photon of frequency $\nu$ is completely absorbed by a free electron initially at rest.

Let the rest mass of the electron be $m_0$ and its final velocity be $v$.

By conservation of energy:
$h\nu + m_0 c^2 = mc^2$

By conservation of linear momentum:
$\dfrac{h\nu}{c} = mv \Rightarrow h\nu = mvc$

Substituting $h\nu$ from the momentum equation into the energy equation:
$mvc + m_0 c^2 = mc^2$
$m_0 c^2 = mc(c - v)$

Using the relativistic mass formula $m = \dfrac{m_0}{\sqrt{1 - \dfrac{v^2}{c^2}}}$:
$m_0 c^2 = \dfrac{m_0 c^2 \left(1 - \dfrac{v}{c}\right)}{\sqrt{1 - \dfrac{v^2}{c^2}}}$

$1 = \sqrt{\dfrac{1 - \dfrac{v}{c}}{1 + \dfrac{v}{c}}}$

Squaring both sides:
$1 + \dfrac{v}{c} = 1 - \dfrac{v}{c} \Rightarrow \dfrac{2v}{c} = 0 \Rightarrow v = 0$

If $v = 0$, then the photon energy $h\nu = mvc = 0$, which implies no photon was present. Thus, simultaneous conservation of energy and linear momentum cannot be achieved.