Question $3$

Cosmic rays are high$-$energy particles from astronomical sources that can be measured as they interact with Earth's atmosphere. Antiprotons found in cosmic rays have ch aracteristic total relativistic energies of $2$GeV, as measured in the frame $S$ in which Earth is station ary. Take an antiproton's rest $m$ ass as $0\mathrm{.}938Ge\frac{V}{c^2}$ and answer the questions below for such an antiproton.

$($a$)$ Determine the speed of the antiproton as measured in $S.$

$10$

$($b$)$ What is the momentum of the antiproton in $S?$

$($c$)$ What is the kinetic energy of the antiproton in $S?$

$($d$)$ Take the direction of the antiproton to be in the positive $x-$direction. Write down the four$-$momentum $P^{}$ for the antiproton in $S.$

$($e$)$ Transform the four$-$momentum of the antiproton to another inertial frame $S^{\text{'}},$ which is moving at speed $0\mathrm{.}4c$ in the negative $x-$direction relative to $S.$