`A` particle X moving with a certain velocity has a debroglie wave length of `1A^(@)`. If particle Y has a mass of `25%` that of X and velocity `75%` that of X, debroglies wave length of Y will be :-

Options

A. `3 Å`
B. `5.33 Å`
C. `6.88 Å`
D. `48 Å`

Answer

To solve the problem, we need to calculate the de Broglie wavelength of particle Y based on the given information about particle X.

### Step-by-Step Solution:

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

2. **Given information for particle X**:
- The de Broglie wavelength of particle X (λ_X) is given as \( 1 \, \text{Å} \) (1 Angstrom).
- Let the mass of particle X be \( m_X \) and its velocity be \( v_X \).

Using the formula for λ_X:
\[ \lambda_X = \frac{h}{m_X v_X} = 1 \, \text{Å} \]

3. **Determine the mass and velocity of particle Y**:
- The mass of particle Y (m_Y) is \( 25\% \) of the mass of particle X:
\[ m_Y = 0.25 m_X = \frac{m_X}{4} \]
- The velocity of particle Y (v_Y) is \( 75\% \) of the velocity of particle X:
\[ v_Y = 0.75 v_X = \frac{3 v_X}{4} \]

4. **Calculate the de Broglie wavelength of particle Y**:
Using the de Broglie wavelength formula for particle Y:
\[ \lambda_Y = \frac{h}{m_Y v_Y} \]
Substituting the values of \( m_Y \) and \( v_Y \):
\[ \lambda_Y = \frac{h}{\left(\frac{m_X}{4}\right) \left(\frac{3 v_X}{4}\right)} = \frac{h}{\frac{3 m_X v_X}{16}} = \frac{16 h}{3 m_X v_X} \]

5. **Relate λ_Y to λ_X**:
Since we know that \( \lambda_X = \frac{h}{m_X v_X} \):
\[ \lambda_Y = \frac{16}{3} \lambda_X \]
Substituting \( \lambda_X = 1 \, \text{Å} \):
\[ \lambda_Y = \frac{16}{3} \times 1 \, \text{Å} = \frac{16}{3} \, \text{Å} \approx 5.33 \, \text{Å} \]

### Final Answer:
The de Broglie wavelength of particle Y is approximately \( 5.33 \, \text{Å} \).

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