If the de - Broglie wavelengths for a proton and for an `alpha` - particle is equal , then what is the ratio of velocities for proton and alpha particle?

Options

A. `4 : 1`
B. `2 : 1`
C. `1 : 2`
D. `1 : 4`

Explanation

To solve the problem of finding the ratio of velocities for a proton and an alpha particle given that their de Broglie wavelengths are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the de Broglie Wavelength Formula**:
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.

2. **Express Momentum**:
The momentum \( p \) can be expressed in terms of mass \( m \) and velocity \( v \):
\[ p = mv \]
Therefore, the de Broglie wavelength can be rewritten as:
\[ \lambda = \frac{h}{mv} \]

3. **Set Up the Equation for Both Particles**:
Since the de Broglie wavelengths for the proton and alpha particle are equal, we can write:
\[ \lambda_p = \lambda_{\alpha} \]
This leads to:
\[ \frac{h}{m_p v_p} = \frac{h}{m_{\alpha} v_{\alpha}} \]
where \( m_p \) and \( v_p \) are the mass and velocity of the proton, and \( m_{\alpha} \) and \( v_{\alpha} \) are the mass and velocity of the alpha particle.

4. **Cancel Planck's Constant**:
Since \( h \) is a constant and appears on both sides, we can cancel it out:
\[ \frac{1}{m_p v_p} = \frac{1}{m_{\alpha} v_{\alpha}} \]

5. **Rearrange the Equation**:
Rearranging gives us:
\[ m_{\alpha} v_{\alpha} = m_p v_p \]

6. **Substitute the Masses**:
The mass of a proton \( m_p \) is approximately 1 amu, and the mass of an alpha particle \( m_{\alpha} \) is approximately 4 amu. Substituting these values in:
\[ 4 v_{\alpha} = 1 v_p \]

7. **Find the Ratio of Velocities**:
Rearranging this gives us the ratio of velocities:
\[ \frac{v_p}{v_{\alpha}} = \frac{4}{1} \]
Thus, the ratio of the velocities of the proton to the alpha particle is:
\[ \frac{v_p}{v_{\alpha}} = 4:1 \]

### Final Answer:
The ratio of velocities for the proton and alpha particle is \( 4:1 \).