A particle of mass M at rest decays into two particles of masses `m_1 and m_2` having velocities `V_1 and V_2` respectively. Find the ratio of de- Broglie wavelengths of the two particles

Answer

To find the ratio of the de Broglie wavelengths of the two particles resulting from the decay of a particle of mass \( M \) at rest, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength \( \lambda \) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

### Step 2: Write the expression for momentum
The momentum \( p \) of a particle can be expressed as:
\[ p = mv \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 3: Apply conservation of momentum
Since the original particle of mass \( M \) is at rest, its initial momentum is zero. According to the law of conservation of momentum, the total momentum after the decay must also be zero. Therefore, we have:
\[ M_1 V_1 + M_2 V_2 = 0 \]
This implies:
\[ M_1 V_1 = -M_2 V_2 \]

### Step 4: Express the momenta of the two particles
From the above equation, we can express the momenta of the two particles:
- Momentum of particle 1: \( p_1 = M_1 V_1 \)
- Momentum of particle 2: \( p_2 = M_2 V_2 \)

### Step 5: Find the de Broglie wavelengths
Using the de Broglie wavelength formula for both particles:
\[ \lambda_1 = \frac{h}{p_1} = \frac{h}{M_1 V_1} \]
\[ \lambda_2 = \frac{h}{p_2} = \frac{h}{M_2 V_2} \]

### Step 6: Find the ratio of the de Broglie wavelengths
Now, we can find the ratio of the de Broglie wavelengths \( \frac{\lambda_1}{\lambda_2} \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{M_1 V_1}}{\frac{h}{M_2 V_2}} = \frac{M_2 V_2}{M_1 V_1} \]

### Step 7: Substitute from conservation of momentum
From the conservation of momentum, we know that \( M_1 V_1 = -M_2 V_2 \). This means:
\[ V_2 = -\frac{M_1}{M_2} V_1 \]
Substituting this into the ratio gives:
\[ \frac{\lambda_1}{\lambda_2} = \frac{M_2 \left(-\frac{M_1}{M_2} V_1\right)}{M_1 V_1} = \frac{-M_1}{M_1} = -1 \]
Since we are interested in the absolute ratio, we can ignore the negative sign:
\[ \frac{\lambda_1}{\lambda_2} = 1 \]

### Final Answer
Thus, the ratio of the de Broglie wavelengths of the two particles is:
\[ \frac{\lambda_1}{\lambda_2} = 1 \]