A particle of mass 4m at rest decays into two particles of masses m and 3m having non-zero velocities. The ratio of the de Broglie wavelengths of the particles 1 and 2 is

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. 2
D. 1

Explanation

To solve the problem, we need to find the ratio of the de Broglie wavelengths of two particles resulting from the decay of a particle of mass 4m into two particles of masses m and 3m.

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have a particle of mass 4m at rest that decays into two particles of masses m and 3m.
- The initial momentum of the system is zero since the particle is at rest.

2. **Apply Conservation of Momentum**:
- According to the law of conservation of momentum, the total momentum before decay must equal the total momentum after decay.
- Before decay: \( P_i = 0 \) (since the particle is at rest).
- After decay: Let the velocities of the two particles be \( v_1 \) (for mass m) and \( v_2 \) (for mass 3m).
- The momentum of the first particle (mass m) is \( p_1 = m \cdot v_1 \).
- The momentum of the second particle (mass 3m) is \( p_2 = 3m \cdot v_2 \).

3. **Set Up the Momentum Equation**:
- For conservation of momentum:
\[ p_1 + p_2 = 0 \implies m \cdot v_1 + 3m \cdot v_2 = 0 \]
- This can be rearranged to:
\[ v_1 = -3v_2 \]
- This indicates that the particles move in opposite directions.

4. **Calculate the De Broglie Wavelengths**:
- The de Broglie wavelength \( \lambda \) is given by the formula:
\[ \lambda = \frac{h}{p} \]
- For particle 1 (mass m):
\[ \lambda_1 = \frac{h}{p_1} = \frac{h}{m \cdot v_1} \]
- For particle 2 (mass 3m):
\[ \lambda_2 = \frac{h}{p_2} = \frac{h}{3m \cdot v_2} \]

5. **Find the Ratio of the Wavelengths**:
- The ratio of the de Broglie wavelengths \( \frac{\lambda_1}{\lambda_2} \) is:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{m \cdot v_1}}{\frac{h}{3m \cdot v_2}} = \frac{3m \cdot v_2}{m \cdot v_1} = \frac{3v_2}{v_1} \]
- Substitute \( v_1 = -3v_2 \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{3v_2}{-3v_2} = -1 \]

6. **Conclusion**:
- The ratio of the de Broglie wavelengths of particles 1 and 2 is \( 1 \) (ignoring the negative sign since we are interested in the magnitude).

### Final Answer:
The ratio of the de Broglie wavelengths of the two particles is \( 1 \).