If numerical value of mass and velocity are equal then de Broglie wavelength in terms of K.E. is

Options

A. `(mh)/(2KE)`
B. `(vh)/(2KE)`
C. both (1) and (2) are correct
D. `sqrt(vh)/(2KE)`

Answer

To solve the problem, we need to derive the de Broglie wavelength in terms of kinetic energy, given that the numerical values of mass (m) and velocity (v) are equal.

### Step-by-Step Solution:

1. **Understanding de Broglie Wavelength**:
The de Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is mass, and \( v \) is velocity.

2. **Given Condition**:
According to the problem, the numerical values of mass and velocity are equal. Therefore, we can write:
\[ m = v \]

3. **Substituting Mass in the Wavelength Formula**:
Substituting \( m \) with \( v \) in the de Broglie wavelength formula:
\[ \lambda = \frac{h}{mv} = \frac{h}{v \cdot v} = \frac{h}{v^2} \]

4. **Relating Kinetic Energy**:
The kinetic energy (K.E) of an object is given by the formula:
\[ K.E = \frac{1}{2} mv^2 \]
Since \( m = v \), we can substitute \( m \) in the kinetic energy formula:
\[ K.E = \frac{1}{2} v \cdot v^2 = \frac{1}{2} v^3 \]

5. **Expressing \( v^2 \) in Terms of Kinetic Energy**:
Rearranging the kinetic energy equation gives:
\[ v^2 = \frac{2 \cdot K.E}{m} \]

6. **Substituting \( v^2 \) Back into the Wavelength Formula**:
Now, substituting \( v^2 \) back into the wavelength formula:
\[ \lambda = \frac{h}{v^2} = \frac{h}{\frac{2 \cdot K.E}{m}} = \frac{hm}{2 \cdot K.E} \]

7. **Final Expression**:
Since \( m = v \), we can also express the wavelength as:
\[ \lambda = \frac{hv}{2 \cdot K.E} \]

### Conclusion:
Thus, the de Broglie wavelength in terms of kinetic energy, given that the numerical values of mass and velocity are equal, can be expressed as:
\[ \lambda = \frac{h}{2 \cdot K.E} \]

### Correct Options:
From the analysis, we find that both options 1 and 2 are correct, making option 3 the right answer.