A particle of mass 25 kg at rest decay into two practicals of messas 12 kg and 13 kg having non zero velocity. The ratio of the de Broglie wavelength of two respectively practicals is

Options

A. `sqrt((12)/(13))`
B. `sqrt((13)/(12))`
C. 1
D. `13/12`

Explanation

To solve the problem step by step, we will follow the principles of conservation of momentum and the formula for de Broglie wavelength.

### Step 1: Understand the Problem
We have a particle of mass 25 kg at rest that decays into two particles of masses 12 kg and 13 kg. We need to find the ratio of their de Broglie wavelengths.

### Step 2: Apply Conservation of Momentum
Since the initial particle is at rest, its initial momentum is zero. Therefore, the total momentum after the decay must also equal zero.

Let:
- \( V_1 \) = velocity of the 12 kg particle
- \( V_2 \) = velocity of the 13 kg particle

According to the conservation of momentum:
\[ 0 = 12V_1 + 13V_2 \]
This can be rearranged to:
\[ 12V_1 = -13V_2 \]
From this, we can express \( V_1 \) in terms of \( V_2 \):
\[ V_1 = -\frac{13}{12} V_2 \]

### Step 3: Write the Formula for de Broglie Wavelength
The de Broglie wavelength \( \lambda \) is given by:
\[ \lambda = \frac{h}{p} \]
where \( p \) is the momentum of the particle, and \( p = mv \) (mass times velocity).

### Step 4: Calculate the de Broglie Wavelengths
For the 12 kg particle:
\[ \lambda_1 = \frac{h}{12V_1} \]
For the 13 kg particle:
\[ \lambda_2 = \frac{h}{13V_2} \]

### Step 5: Substitute \( V_1 \) into the Wavelength Equation
Substituting \( V_1 \) from Step 2 into the equation for \( \lambda_1 \):
\[ \lambda_1 = \frac{h}{12 \left(-\frac{13}{12} V_2\right)} = \frac{h}{-13V_2} \]
Since we are interested in the ratio, we can ignore the negative sign.

### Step 6: Find the Ratio of the Wavelengths
Now we can find the ratio of the de Broglie wavelengths:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{13V_2}}{\frac{h}{13V_2}} = \frac{h}{12V_1} \cdot \frac{13V_2}{h} = \frac{13V_2}{12V_1} \]
Substituting \( V_1 \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{13V_2}{12 \left(-\frac{13}{12} V_2\right)} = \frac{13V_2}{-13V_2} = -1 \]
Again, we take the absolute value for the ratio:
\[ \frac{\lambda_1}{\lambda_2} = 1 \]

### Step 7: Conclusion
Thus, the ratio of the de Broglie wavelengths of the two particles is:
\[ \lambda_1 : \lambda_2 = 1 : 1 \]