The velocity of particel A is 0.1 m/s and that of particle B is 0.5 m/s. If the mass of particle B is five times that of particle A, then the ratio of de-Broglie wavelength associated with particles A and B is

Options

A. `25 : 1`
B. `3:4`
C. `6 : 4`
D. `5 : 2`

Explanation

To find the ratio of the de Broglie wavelengths associated with particles A and B, we will follow these steps:

### Step 1: Write the formula for de Broglie wavelength
The de Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

### Step 2: Define the momentum for both particles
The momentum \( p \) of a particle is given by:
\[ p = mv \]
where \( m \) is the mass and \( v \) is the velocity of the particle.

### Step 3: Assign variables for masses and velocities
Let the mass of particle A be \( m_A = x \) and the velocity of particle A be \( v_A = 0.1 \, \text{m/s} \).
Given that the mass of particle B is five times that of particle A, we have:
\[ m_B = 5x \]
and the velocity of particle B is \( v_B = 0.5 \, \text{m/s} \).

### Step 4: Calculate the momentum for both particles
For particle A:
\[ p_A = m_A v_A = x \cdot 0.1 = 0.1x \]

For particle B:
\[ p_B = m_B v_B = (5x) \cdot 0.5 = 2.5x \]

### Step 5: Write the expressions for the de Broglie wavelengths
Using the momentum values calculated:
\[ \lambda_A = \frac{h}{p_A} = \frac{h}{0.1x} \]
\[ \lambda_B = \frac{h}{p_B} = \frac{h}{2.5x} \]

### Step 6: Find the ratio of the de Broglie wavelengths
Now, we can find the ratio of the wavelengths:
\[ \frac{\lambda_A}{\lambda_B} = \frac{\frac{h}{0.1x}}{\frac{h}{2.5x}} = \frac{2.5x}{0.1x} \]
The \( h \) and \( x \) cancel out:
\[ \frac{\lambda_A}{\lambda_B} = \frac{2.5}{0.1} = 25 \]

### Step 7: State the final ratio
Thus, the ratio of the de Broglie wavelengths associated with particles A and B is:
\[ \lambda_A : \lambda_B = 25 : 1 \]