A particle of mass 3m at rest decays into two particles of masses m and 2m having non-zero velocities. The ratio of the de Broglie wavelengths of the particles `((lamda_1)/(lamda_2))` is

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. 2
D. None of these

Answer

To solve the problem, we need to analyze the decay of a particle with mass \(3m\) into two particles with masses \(m\) and \(2m\). We will use the principles of conservation of momentum and the de Broglie wavelength formula.

### Step-by-Step Solution:

1. **Understand the Problem**:
- A particle of mass \(3m\) is initially at rest.
- It decays into two particles with masses \(m\) and \(2m\).
- We need to find the ratio of the de Broglie wavelengths of these two particles.

2. **Apply Conservation of Momentum**:
- Before decay, the total momentum \(P_{\text{initial}} = 0\) (since the particle is at rest).
- After decay, let the velocities of the particles be \(v_1\) (for mass \(m\)) and \(v_2\) (for mass \(2m\)).
- According to conservation of momentum:
\[ 0 = mv_1 + 2mv_2 \]
- Rearranging gives:
\[ mv_1 = -2mv_2 \quad \Rightarrow \quad v_1 = -2v_2 \]

3. **Calculate the Momenta**:
- The momentum of the first particle (mass \(m\)):
\[ P_1 = mv_1 = m(-2v_2) = -2mv_2 \]
- The momentum of the second particle (mass \(2m\)):
\[ P_2 = 2mv_2 \]

4. **Find the De Broglie Wavelengths**:
- The de Broglie wavelength \(\lambda\) is given by:
\[ \lambda = \frac{h}{P} \]
- For the first particle:
\[ \lambda_1 = \frac{h}{P_1} = \frac{h}{-2mv_2} = \frac{h}{2mv_2} \]
- For the second particle:
\[ \lambda_2 = \frac{h}{P_2} = \frac{h}{2mv_2} \]

5. **Calculate the Ratio of the Wavelengths**:
- Now, we find the ratio:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{2mv_2}}{\frac{h}{2mv_2}} = 1 \]

6. **Conclusion**:
- The ratio of the de Broglie wavelengths of the two particles is:
\[ \frac{\lambda_1}{\lambda_2} = 1 \]
- Since this value does not match any of the provided options, the answer is "none of this."