A proton a deutron and an `alpha` particle are accelerated through potentials of `V, 2V` and `4V` respectively. Their velocity will bear a ratio

Options

A. `1:1:1`
B. `1:sqrt2:1`
C. `sqrt2:1:1`
D. `1:1:sqrt2`

Explanation

To find the ratio of velocities of a proton, a deuteron, and an alpha particle when they are accelerated through potentials of \( V \), \( 2V \), and \( 4V \) respectively, we can use the principle of energy conservation. The kinetic energy gained by a charged particle when accelerated through a potential difference is given by:

\[ \text{K.E.} = QV \]

Where \( Q \) is the charge of the particle and \( V \) is the potential difference. The kinetic energy can also be expressed in terms of mass and velocity:

\[ \text{K.E.} = \frac{1}{2} mv^2 \]

Equating the two expressions for kinetic energy, we have:

\[ \frac{1}{2} mv^2 = QV \]

From this, we can solve for velocity \( v \):

\[ v = \sqrt{\frac{2QV}{m}} \]

### Step 1: Calculate the velocity of the proton
For the proton:
- Charge \( Q_p = e \)
- Mass \( m_p = m \)
- Potential \( V_p = V \)

Thus, the velocity \( v_p \) is:

\[ v_p = \sqrt{\frac{2eV}{m}} \]

### Step 2: Calculate the velocity of the deuteron
For the deuteron:
- Charge \( Q_d = e \)
- Mass \( m_d = 2m \)
- Potential \( V_d = 2V \)

Thus, the velocity \( v_d \) is:

\[ v_d = \sqrt{\frac{2e(2V)}{2m}} = \sqrt{\frac{4eV}{2m}} = \sqrt{\frac{2eV}{m}} = v_p \]

### Step 3: Calculate the velocity of the alpha particle
For the alpha particle:
- Charge \( Q_{\alpha} = 2e \)
- Mass \( m_{\alpha} = 4m \)
- Potential \( V_{\alpha} = 4V \)

Thus, the velocity \( v_{\alpha} \) is:

\[ v_{\alpha} = \sqrt{\frac{2(2e)(4V)}{4m}} = \sqrt{\frac{16eV}{4m}} = \sqrt{\frac{4eV}{m}} = 2\sqrt{\frac{eV}{m}} = 2v_p \]

### Step 4: Determine the ratio of velocities
Now we have:
- \( v_p = \sqrt{\frac{2eV}{m}} \)
- \( v_d = v_p \)
- \( v_{\alpha} = 2v_p \)

The ratio of velocities \( v_p : v_d : v_{\alpha} \) is:

\[ 1 : 1 : 2 \]

### Final Answer
The ratio of the velocities of the proton, deuteron, and alpha particle is:

\[ 1 : 1 : 2 \]