A proton moving with a velocity of `0.125xx10^(5) ms ^(-1)` Collides with a stationary helium atom. The velocity of the proton after the collision is

Options

A. `0.75xx10^(5)ms^(-1)`
B. `7.5xx10^(5)ms^(-1)`
C. `-0.75xx10^(5)ms^(-1)`
D. `0 ms ^(-1)`

Answer

To solve the problem of a proton colliding with a stationary helium atom, we will use the principles of conservation of momentum and the properties of elastic collisions.

### Step-by-Step Solution:

1. **Identify the masses and initial velocities:**
- Mass of the proton, \( m_p = 1 \, \text{amu} \)
- Mass of the helium atom, \( m_{He} = 4 \, \text{amu} \)
- Initial velocity of the proton, \( v_0 = 0.125 \times 10^5 \, \text{m/s} \)
- Initial velocity of the helium atom, \( v_{He0} = 0 \, \text{m/s} \)

2. **Set up the equations for elastic collision:**
- For elastic collisions, the following two equations hold:
- Conservation of momentum:
\[ m_p v_0 + m_{He} v_{He0} = m_p v_1 + m_{He} v_2 \]
- Conservation of kinetic energy (which implies the velocity of separation equals the velocity of approach):
\[ v_2 - v_1 = v_0 \]

3. **Substituting known values:**
- Since \( v_{He0} = 0 \):
\[ 1 \cdot (0.125 \times 10^5) + 4 \cdot 0 = 1 \cdot v_1 + 4 \cdot v_2 \]
- This simplifies to:
\[ 0.125 \times 10^5 = v_1 + 4v_2 \quad \text{(Equation 1)} \]

4. **Using the velocity of separation:**
- From the velocity of separation equation:
\[ v_2 - v_1 = 0.125 \times 10^5 \quad \text{(Equation 2)} \]

5. **Solving the equations:**
- Rearranging Equation 2 gives:
\[ v_2 = v_1 + 0.125 \times 10^5 \]
- Substitute \( v_2 \) from Equation 2 into Equation 1:
\[ 0.125 \times 10^5 = v_1 + 4(v_1 + 0.125 \times 10^5) \]
- Expanding this:
\[ 0.125 \times 10^5 = v_1 + 4v_1 + 0.5 \times 10^5 \]
- Combine like terms:
\[ 0.125 \times 10^5 = 5v_1 + 0.5 \times 10^5 \]
- Rearranging gives:
\[ 5v_1 = 0.125 \times 10^5 - 0.5 \times 10^5 \]
\[ 5v_1 = -0.375 \times 10^5 \]
- Therefore:
\[ v_1 = -0.075 \times 10^5 \, \text{m/s} \]

### Final Answer:
The velocity of the proton after the collision is:
\[ v_1 = -0.075 \times 10^5 \, \text{m/s} = -7500 \, \text{m/s} \]