`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. `3A^(@)`
B. `5.33A^(@)`
C. `6.88A^(@)`
D. `48A^(@)`

Explanation

To solve the problem, we will use the de Broglie wavelength formula, which is given by:

\[ \lambda = \frac{h}{mv} \]

where:
- \(\lambda\) is the de Broglie wavelength,
- \(h\) is Planck's constant,
- \(m\) is the mass of the particle,
- \(v\) is the velocity of the particle.

### Step 1: Write down the de Broglie wavelength for particle X.

Given that the de Broglie wavelength of particle X is \(1 \, \text{Å}\) (angstrom), we can express this as:

\[ \lambda_X = \frac{h}{m_X v_X} = 1 \, \text{Å} \]

### Step 2: Determine the mass and velocity of particle Y.

We know that:
- The mass of particle Y, \(m_Y\), is \(25\%\) of the mass of particle X, \(m_X\):

\[ m_Y = 0.25 \, m_X \]

- The velocity of particle Y, \(v_Y\), is \(75\%\) of the velocity of particle X, \(v_X\):

\[ v_Y = 0.75 \, v_X \]

### Step 3: Write down the de Broglie wavelength for particle Y.

Using the de Broglie wavelength formula for particle Y, we have:

\[ \lambda_Y = \frac{h}{m_Y v_Y} \]

Substituting the expressions for \(m_Y\) and \(v_Y\):

\[ \lambda_Y = \frac{h}{(0.25 \, m_X)(0.75 \, v_X)} \]

### Step 4: Simplify the expression for \(\lambda_Y\).

Now, we can simplify the expression:

\[ \lambda_Y = \frac{h}{0.25 \, m_X \cdot 0.75 \, v_X} = \frac{h}{0.1875 \, m_X v_X} \]

### Step 5: Relate \(\lambda_Y\) to \(\lambda_X\).

Since we know that:

\[ \lambda_X = \frac{h}{m_X v_X} \]

We can express \(\lambda_Y\) in terms of \(\lambda_X\):

\[ \lambda_Y = \frac{h}{0.1875 \, m_X v_X} = \frac{1}{0.1875} \cdot \lambda_X \]

### Step 6: Calculate \(\lambda_Y\).

Now substituting \(\lambda_X = 1 \, \text{Å}\):

\[ \lambda_Y = \frac{1}{0.1875} \cdot 1 \, \text{Å} = 5.33 \, \text{Å} \]

### Final Answer:

Thus, the de Broglie wavelength of particle Y is:

\[ \lambda_Y = 5.33 \, \text{Å} \]