Question

A proton moving with a momentum P₁ 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 P₂. 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

Step 1: Kinetic energy of proton (Non-relativistic)

$KE_p = \frac{P_1^2}{2m_0} = \frac{1}{8} m_0 c^2$

Step 2: Solve for $P_1$

$P_1^2 = \frac{2m_0}{8} m_0 c^2 = \frac{1}{4} m_0^2 c^2$

$P_1 = \sqrt{\frac{1}{4} m_0^2 c^2} = \frac{1}{2} m_0 c$

Step 3: Momentum of photon ($P_2$)

Energy of photon $E_{photon} = KE_p = \frac{1}{8} m_0 c^2$

Momentum of photon $P_2 = \frac{E_{photon}}{c} = \frac{\frac{1}{8} m_0 c^2}{c} = \frac{1}{8} m_0 c$

Step 4: Calculate $\frac{P_1 - P_2}{P_1}$

$\frac{P_1 - P_2}{P_1} = 1 - \frac{P_2}{P_1} = 1 - \frac{\frac{1}{8} m_0 c}{\frac{1}{2} m_0 c} = 1 - \frac{1/8}{1/2} = 1 - \frac{2}{8} = 1 - \frac{1}{4} = \frac{3}{4}$

Step 5: Match to Option

Option (D) is $\frac{3}{4}$.

Final Answer: The final answer is $\boxed{\frac{3}{4}}$

Answer 2

Given:

• Proton momentum: \( P_1 \)
• Proton kinetic energy: \( K = \frac{1}{8} \) of rest mass energy (\( m_pc^2 \))
• Photon energy: Equal to proton's kinetic energy (\( E_\gamma = K \))
• Photon momentum: \( P_2 \)

Step 1: Express proton's kinetic energy

Given \( K = \frac{1}{8}m_pc^2 \), where \( m_p \) is proton rest mass.

Step 2: Relate kinetic energy to momentum (relativistically)

Total energy \( E = K + m_pc^2 = \frac{9}{8}m_pc^2 \)

Using energy-momentum relation:

\[ E^2 = (P_1c)^2 + (m_pc^2)^2 \]

\[ \left(\frac{9}{8}m_pc^2\right)^2 = (P_1c)^2 + (m_pc^2)^2 \]

\[ \frac{81}{64}m_p^2c^4 = P_1^2c^2 + m_p^2c^4 \]

\[ P_1^2c^2 = \frac{17}{64}m_p^2c^4 \]

\[ P_1 = \frac{\sqrt{17}}{8}m_pc \]

Step 3: Calculate photon momentum \( P_2 \)

Photon energy \( E_\gamma = K = \frac{1}{8}m_pc^2 \)

Photon momentum:

\[ P_2 = \frac{E_\gamma}{c} = \frac{m_pc}{8} \]

Step 4: Compute the ratio \( \frac{P_1 - P_2}{P_1} \)

\[ \frac{P_1 - P_2}{P_1} = 1 - \frac{P_2}{P_1} \]

\[ = 1 - \frac{\frac{m_pc}{8}}{\frac{\sqrt{17}}{8}m_pc} \]

\[ = 1 - \frac{1}{\sqrt{17}} \]

\[ ≈ 1 - 0.2425 ≈ 0.7575 \]