$(10$ pts$)$ In $1923,$ Compton demonstrated the particle nature of light through experiments. When X$-$

rays with an energy $0$ are incident on carbon T$,$ electrons are scattered at an angle $$ with the x$-$

axis, and the scattered X$-$rays are measured at an angle

$.$

$($a$)(5$pt$)$ Show that the kinetic energy of recoil the electron is

$$

$0$

$=$

$0$

$2(1$ cos $)$

$1+0$

$2(1$ cos $)$