A proton and a ionized deuterium are initially at rest and are accelerated through the same potential difference. Which of the following is false concerning the final properties of the two particles ?

Options

A. a. they have different speed
B. b. they have momentum
C. c. they have same kinetic energy
D. d . none of these

Explanation

To solve the problem, we need to analyze the situation where a proton and an ionized deuterium (which is a nucleus containing one proton and one neutron) are accelerated through the same potential difference. We will determine the final properties of both particles and identify which statement is false.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- Both the proton and the ionized deuterium are initially at rest.
- They are accelerated through the same potential difference (V).

2. **Calculating the Charge**:
- The charge of a proton (p) is \( +e \).
- The charge of an ionized deuterium (D) is also \( +e \) (since it has lost one electron).

3. **Using the Work-Energy Principle**:
- The work done on each particle when they are accelerated through the potential difference is given by:
\[ W = qV \]
- For both particles, the work done will be:
\[ W_{p} = eV \quad \text{(for proton)} \]
\[ W_{D} = eV \quad \text{(for ionized deuterium)} \]

4. **Relating Work to Kinetic Energy**:
- The work done on each particle is converted into kinetic energy (KE):
\[ KE_{p} = \frac{1}{2} m_{p} v_{p}^2 = eV \]
\[ KE_{D} = \frac{1}{2} m_{D} v_{D}^2 = eV \]

5. **Finding the Masses**:
- The mass of a proton \( m_{p} \) is approximately \( 1 \, u \) (atomic mass unit).
- The mass of an ionized deuterium \( m_{D} \) is approximately \( 2 \, u \) (since it consists of one proton and one neutron).

6. **Setting Up the Kinetic Energy Equations**:
- For the proton:
\[ \frac{1}{2} m_{p} v_{p}^2 = eV \quad \Rightarrow \quad v_{p}^2 = \frac{2eV}{m_{p}} \]
- For the ionized deuterium:
\[ \frac{1}{2} m_{D} v_{D}^2 = eV \quad \Rightarrow \quad v_{D}^2 = \frac{2eV}{m_{D}} \]

7. **Comparing Velocities**:
- Since \( m_{D} = 2 m_{p} \):
\[ v_{D}^2 = \frac{2eV}{2m_{p}} = \frac{eV}{m_{p}} \quad \Rightarrow \quad v_{D} = \sqrt{\frac{eV}{m_{p}}} \]
- Therefore, \( v_{p} > v_{D} \) because the proton is lighter.

8. **Calculating Momentum**:
- Momentum is given by \( p = mv \).
- For the proton:
\[ p_{p} = m_{p} v_{p} \]
- For the ionized deuterium:
\[ p_{D} = m_{D} v_{D} = 2m_{p} v_{D} \]

9. **Final Comparison**:
- Since \( v_{p} > v_{D} \) and \( m_{D} = 2m_{p} \), the momenta will not be equal.
- Thus, the statement that they have the same momentum is false.

### Conclusion:
The false statement concerning the final properties of the two particles is that they have the same momentum.