Question

A proton moving with a momentum $P _{1}$ has a kinetic energy $\frac{1}{8}$ th of its rest mass energy. Another light photon having energy equal to the kinetic energy of the possesses a momentum $P _{2} .$ Then the ratio $\frac{ P _{1}- P _{2}}{ P _{1}}$ is equal to

• A. 1
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $\frac{3}{4}$

Answer 1

Answer: (D)

For proton $v ^{2}=\frac{ c ^{2}}{4} v =\frac{ c }{2}$
$P =\sqrt{2 mE _{ k }} $
$=\sqrt{\not 2 m \frac{1}{\not 8_{4}} mc ^{2}} $
$P_{1} =\frac{ mc }{2}$
For photon
$E _{\text {photon }}= KE _{\text {proton }}$
$\frac{ h \not c}{\lambda}=\frac{1}{8} mc ^{\not 2}$
$P _{\text {photon }}=\frac{ h }{\lambda}=\frac{ mc }{8}= P _{2}$
$\frac{P_{1}-P_{2}}{P_{1}}=\frac{\frac{\not m c}{2}-\frac{\not m c}{8}}{\frac{\not m c}{2}}$
$=\frac{\frac{1}{2}-\frac{1}{8}}{\frac{1}{2}}=\frac{\frac{4-1}{8}}{\frac{1}{2}}$
$=\frac{3 \times 2}{8}=\frac{3}{4}$