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

The Correct Option is D

To solve this problem, we need to understand the relationship between the momentum and energy of protons and photons.

1. The kinetic energy (\(K\)) of the proton is given as \(\frac{1}{8}\) of its rest mass energy. The rest mass energy (\(E_0\)) of a proton is given by Einstein’s mass-energy equivalence \(E_0 = mc^2\), where \(m\) is the mass of the proton and \(c\) is the speed of light.
2. Thus, the kinetic energy of the proton can be expressed as: \(K = \frac{1}{8} mc^2\).
3. For a proton, we know that the relationship between kinetic energy, momentum (\(P_1\)), and mass-energy equivalence is: \(K = \frac{P_1^2}{2m}\).
4. Setting the two expressions for kinetic energy equal, we have: \(\frac{1}{8} mc^2 = \frac{P_1^2}{2m}\). 
   Solving for \(P_1\), we get: \(P_1 = m c \sqrt{\frac{1}{4}}\). 
5. We simplify this to: \(P_1 = \frac{mc}{2}\).
6. A photon with the same kinetic energy as the proton will have energy \(K = \frac{1}{8} mc^2\).
7. For photons, energy (\(E\)) is related to momentum (\(P_2\)) by: \(E = P_2 c\). 
   Since \(E = \frac{1}{8} mc^2\), we have: \(P_2 = \frac{1}{8} mc\).
8. We are asked to find the ratio: \(\frac{P_1 - P_2}{P_1}\). 
   Substitute the values:
9. \(\frac{P_1 - P_2}{P_1} = \frac{\frac{mc}{2} - \frac{1}{8} mc}{\frac{mc}{2}}\).
10. Solving further: \(= \frac{\frac{4mc}{8} - \frac{1mc}{8}}{\frac{4mc}{8}}\) 
    \(= \frac{\frac{3mc}{8}}{\frac{4mc}{8}}\)\) 
    \(= \frac{3}{4}\).

Thus, the correct answer is \(\frac{3}{4}\).