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:sqrt(2) :1`
C. `sqrt(2) :1:1`
D. `1:1:sqrt(2)`

Answer

To find the ratio of the velocities of a proton, a deuteron, and an alpha particle when they are accelerated through potentials of \( V \), \( 2V \), and \( 4V \) respectively, we can follow these steps:

### Step 1: Understand the Work-Energy Principle
The work done on a charged particle when it is accelerated through a potential difference \( V \) is equal to the change in kinetic energy. This can be expressed as:
\[ W = Q \Delta V \]
where \( Q \) is the charge of the particle and \( \Delta V \) is the potential difference.

### Step 2: Set Up the Kinetic Energy Equation
The kinetic energy gained by the particle can be expressed as:
\[ \Delta KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the particle and \( v \) is its final velocity.

### Step 3: Relate Work Done to Kinetic Energy
From the work-energy principle, we have:
\[ Q \Delta V = \frac{1}{2} m v^2 \]
Rearranging gives:
\[ v^2 = \frac{2Q \Delta V}{m} \]

### Step 4: Calculate the Charge and Mass for Each Particle
- **Proton**: Charge \( Q = +e \), Mass \( m = 1 \, \text{amu} \)
- **Deuteron**: Charge \( Q = +e \), Mass \( m = 2 \, \text{amu} \)
- **Alpha Particle**: Charge \( Q = +2e \), Mass \( m = 4 \, \text{amu} \)

### Step 5: Substitute Values for Each Particle
1. **For Proton** (accelerated through \( V \)):
\[ v_p^2 = \frac{2eV}{1} = 2eV \]
Thus, \( v_p = \sqrt{2eV} \).

2. **For Deuteron** (accelerated through \( 2V \)):
\[ v_d^2 = \frac{2e(2V)}{2} = 2eV \]
Thus, \( v_d = \sqrt{2eV} \).

3. **For Alpha Particle** (accelerated through \( 4V \)):
\[ v_{\alpha}^2 = \frac{2(2e)(4V)}{4} = 4eV \]
Thus, \( v_{\alpha} = \sqrt{4eV} = 2\sqrt{eV} \).

### Step 6: Find the Ratios of Velocities
Now we can find the ratio of the velocities:
\[ \frac{v_p}{v_d} = \frac{\sqrt{2eV}}{\sqrt{2eV}} = 1 \]
\[ \frac{v_d}{v_{\alpha}} = \frac{\sqrt{2eV}}{2\sqrt{eV}} = \frac{1}{2} \]
Thus, the ratio of the velocities \( v_p : v_d : v_{\alpha} \) is:
\[ 1 : 1 : 2 \]

### Final Ratio
The final ratio of the velocities of the proton, deuteron, and alpha particle is:
\[ 1 : 1 : 2 \]