The de - Broglie wavelength of a particle moving with a velocity `2.25 xx 10^(8) m//s` is equal to the wavelength of photon. The ratio of kinetic energy of the particle to the energy of the photon is (velocity of light is `3 xx 10^(8) m//s`

Options

A. `1//8`
B. `3//8`
C. `5//8`
D. `7//8`

Explanation

To solve the problem, we need to find the ratio of the kinetic energy of a particle to the energy of a photon, given that their de-Broglie wavelengths are equal.

### Step-by-Step Solution:

1. **Understand the de-Broglie wavelength**:
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. For a particle with mass \( m \) moving with velocity \( v \), the momentum \( p \) is:
\[ p = mv \]
Thus, the de-Broglie wavelength can be rewritten as:
\[ \lambda = \frac{h}{mv} \]

2. **Energy of the photon**:
The energy (\( E \)) of a photon is given by:
\[ E = \frac{hc}{\lambda} \]
where \( c \) is the speed of light.

3. **Equating the wavelengths**:
Since the de-Broglie wavelength of the particle is equal to the wavelength of the photon, we have:
\[ \lambda = \frac{h}{mv} = \frac{hc}{E} \]

4. **Kinetic energy of the particle**:
The kinetic energy (\( K.E. \)) of the particle is given by:
\[ K.E. = \frac{1}{2} mv^2 \]

5. **Substituting for \( m \)**:
From the de-Broglie wavelength equation, we can express \( m \) in terms of \( \lambda \) and \( v \):
\[ m = \frac{h}{\lambda v} \]
Now substituting this into the kinetic energy formula:
\[ K.E. = \frac{1}{2} \left(\frac{h}{\lambda v}\right) v^2 = \frac{hv}{2\lambda} \]

6. **Finding the ratio of kinetic energy of the particle to the energy of the photon**:
Now we can find the ratio:
\[ \text{Ratio} = \frac{K.E. \text{ of particle}}{E \text{ of photon}} = \frac{\frac{hv}{2\lambda}}{\frac{hc}{\lambda}} = \frac{v}{2c} \]

7. **Substituting the values**:
Given \( v = 2.25 \times 10^8 \, \text{m/s} \) and \( c = 3 \times 10^8 \, \text{m/s} \):
\[ \text{Ratio} = \frac{2.25 \times 10^8}{2 \times 3 \times 10^8} = \frac{2.25}{6} = \frac{3}{8} \]

### Final Answer:
The ratio of the kinetic energy of the particle to the energy of the photon is:
\[ \frac{3}{8} \]