Question

The velocity of a particle A is 3 times the velocity of proton. If the ratio of the de Broglie wavelengths of the particle A and the proton is 3:2, the mass of the particle A is:

• A. \( \frac{2}{9} m_p \)
• B. \( \frac{2}{3} m_p \)
• C. \( \frac{2}{7} m_p \)
• D. \( \frac{3}{7} m_p \)

Hint:

Use the de Broglie wavelength formula and the relationship between velocity and mass to solve the problem.

Answer 1

The Correct Option is A

We know that the de Broglie wavelength is given by: \[ \lambda = \frac{h}{mv} \] where: - \( h \) is Planck's constant, - \( m \) is the mass of the particle, - \( v \) is the velocity of the particle. Let \( v_p \) be the velocity of the proton and \( v_A \) be the velocity of particle A. From the problem, we know: \[ v_A = 3v_p \] Let \( m_A \) be the mass of particle A. The ratio of the de Broglie wavelengths is: \[ \frac{\lambda_A}{\lambda_p} = \frac{h/(m_A v_A)}{h/(m_p v_p)} = \frac{m_p v_p}{m_A v_A} \] Substitute \( v_A = 3v_p \) into the equation: \[ \frac{m_p v_p}{m_A \cdot 3v_p} = \frac{3}{2} \] Simplifying: \[ \frac{m_p}{3 m_A} = \frac{3}{2} \] \[ m_A = \frac{2}{9} m_p \] Thus, the mass of particle A is \( \frac{2}{9} m_p \).

Answer 2

Step 1: Recall the de Broglie wavelength formula.
The de Broglie wavelength is given by:
\[ \lambda = \frac{h}{mv} \] where:
- \( \lambda \) is the de Broglie wavelength,
- \( h \) is Planck's constant,
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle.

Step 2: Let \( m_p \) be the mass of the proton and \( v_p \) be its velocity.
For proton: \( \lambda_p = \frac{h}{m_p v_p} \)
For particle A (velocity is 3 times that of proton):
\[ v_A = 3v_p,\quad \lambda_A = \frac{h}{m_A \cdot 3v_p} \]

Step 3: Given ratio of de Broglie wavelengths.
\[ \frac{\lambda_A}{\lambda_p} = \frac{3}{2} \] Substitute the expressions:
\[ \frac{\frac{h}{m_A \cdot 3v_p}}{\frac{h}{m_p v_p}} = \frac{3}{2} \Rightarrow \frac{m_p v_p}{3 m_A v_p} = \frac{3}{2} \Rightarrow \frac{m_p}{3 m_A} = \frac{3}{2} \]

Step 4: Solve for \( m_A \).
\[ \frac{m_p}{3 m_A} = \frac{3}{2} \Rightarrow m_A = \frac{2}{9} m_p \]

Step 5: Conclusion.
The mass of particle A is \( \boxed{\frac{2}{9} m_p} \).