For particles having same K.E., the de-Broglie wavelength is

Options

A. Directly proportional to its velocity
B. Inversely proportionals to its velocity
C. Independent of its mass and velocity
D. Unpredictable

Explanation

To solve the question regarding the de Broglie wavelength for particles having the same kinetic energy, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding de Broglie Wavelength**:
The de Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is the Planck constant and \( p \) is the momentum of the particle.

2. **Momentum Definition**:
The momentum \( p \) of a particle is defined as:
\[ p = mv \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

3. **Kinetic Energy Relation**:
The kinetic energy (K.E.) of a particle is given by:
\[ K.E. = \frac{1}{2} mv^2 \]
For two particles with the same kinetic energy, we can denote their kinetic energies as \( K.E.1 \) and \( K.E.2 \):
\[ K.E.1 = K.E.2 \]

4. **Expressing Velocities**:
From the kinetic energy equation, we can express the velocities in terms of kinetic energy:
\[ v_1 = \sqrt{\frac{2K.E.1}{m_1}} \quad \text{and} \quad v_2 = \sqrt{\frac{2K.E.2}{m_2}} \]
Since \( K.E.1 = K.E.2 \), we can denote this common kinetic energy as \( K.E. \):
\[ v_1 = \sqrt{\frac{2K.E.}{m_1}} \quad \text{and} \quad v_2 = \sqrt{\frac{2K.E.}{m_2}} \]

5. **Finding the Ratio of Wavelengths**:
Now we can find the ratio of the de Broglie wavelengths of the two particles:
\[ \frac{\lambda_1}{\lambda_2} = \frac{h/p_1}{h/p_2} = \frac{p_2}{p_1} = \frac{m_2 v_2}{m_1 v_1} \]
Substituting the expressions for \( v_1 \) and \( v_2 \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{m_2 \sqrt{\frac{2K.E.}{m_2}}}{m_1 \sqrt{\frac{2K.E.}{m_1}}} \]
Simplifying this gives:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{m_2}}{\sqrt{m_1}} \cdot \frac{m_2}{m_1} \]

6. **Conclusion**:
Since the kinetic energies are the same, we conclude that the de Broglie wavelength is inversely proportional to the square root of the mass of the particles when they have the same kinetic energy.

### Final Result:
For particles having the same kinetic energy, the de Broglie wavelength is inversely proportional to the square root of their mass.