A particle `X` moving with a certain velocity has a debragile wave length of `1.72Å`, If particle `Y` has a mass of `50%` that of `X` and velocity `50%` that of `X`, debroglies wave length of `Y` will be `-`

Options

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

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: Determine the parameters for particle X
Given that the de Broglie wavelength of particle \(X\) is \(1.72 \, \text{Å}\) (which is \(1.72 \times 10^{-10} \, \text{m}\)), we can denote:
- Mass of particle \(X\) = \(m_X\)
- Velocity of particle \(X\) = \(v_X\)

Using the de Broglie wavelength formula for particle \(X\):

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

### Step 2: Determine the parameters for particle Y
According to the problem:
- Mass of particle \(Y\) = \(0.5 \times m_X\)
- Velocity of particle \(Y\) = \(0.5 \times v_X\)

Using the de Broglie wavelength formula for particle \(Y\):

\[ \lambda_Y = \frac{h}{m_Y v_Y} = \frac{h}{(0.5 m_X)(0.5 v_X)} = \frac{h}{0.25 m_X v_X} \]

### Step 3: Relate the wavelengths of particles X and Y
From the above equations, we can express \(\lambda_Y\) in terms of \(\lambda_X\):

\[ \lambda_Y = \frac{h}{0.25 m_X v_X} = 4 \cdot \frac{h}{m_X v_X} = 4 \lambda_X \]

### Step 4: Calculate \(\lambda_Y\)
Now substituting the value of \(\lambda_X\):

\[ \lambda_Y = 4 \cdot 1.72 \, \text{Å} = 6.88 \, \text{Å} \]

### Conclusion
Thus, the de Broglie wavelength of particle \(Y\) is:

\[ \lambda_Y = 6.88 \, \text{Å} \]

### Final Answer
The de Broglie wavelength of particle \(Y\) will be \(6.88 \, \text{Å}\).

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