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 and finding the velocity of the proton after the collision, we will use the principles of conservation of momentum and the coefficient of restitution. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the masses and initial velocities
- Let the mass of the proton be \( m \).
- The mass of the helium atom is \( 4m \) (since helium has an atomic mass of 4 amu, and 1 amu is approximately equal to the mass of a proton).
- The initial velocity of the proton is \( V = 0.125 \times 10^5 \, \text{m/s} \).
- The initial velocity of the helium atom is \( 0 \, \text{m/s} \) (since it is stationary).

### Step 2: Apply conservation of momentum
According to the law of conservation of momentum:
\[ \text{Initial momentum} = \text{Final momentum} \]
This can be expressed as:
\[ mV + 4m \cdot 0 = mV_1 + 4mV_2 \]
Simplifying this gives:
\[ mV = mV_1 + 4mV_2 \]
Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ V = V_1 + 4V_2 \quad \text{(Equation 1)} \]

### Step 3: Apply the coefficient of restitution
For elastic collisions, the coefficient of restitution \( e \) is 1. The equation for the coefficient of restitution is given by:
\[ V_2 - V_1 = e(V - 0) \]
Substituting \( e = 1 \):
\[ V_2 - V_1 = V \quad \text{(Equation 2)} \]

### Step 4: Solve the equations simultaneously
Now we have two equations:
1. \( V = V_1 + 4V_2 \)
2. \( V_2 - V_1 = V \)

From Equation 2, we can express \( V_2 \) in terms of \( V_1 \):
\[ V_2 = V + V_1 \]

Substituting this into Equation 1:
\[ V = V_1 + 4(V + V_1) \]
Expanding this gives:
\[ V = V_1 + 4V + 4V_1 \]
Combining like terms:
\[ V = 5V_1 + 4V \]
Rearranging gives:
\[ V - 4V = 5V_1 \]
\[ -V = 5V_1 \]
Thus,
\[ V_1 = -\frac{V}{5} \]

### Step 5: Substitute the value of \( V \)
Now substitute \( V = 0.125 \times 10^5 \):
\[ V_1 = -\frac{0.125 \times 10^5}{5} \]
Calculating this gives:
\[ V_1 = -0.025 \times 10^5 = -0.75 \times 10^5 \, \text{m/s} \]

### Conclusion
The velocity of the proton after the collision is:
\[ V_1 = -0.75 \times 10^5 \, \text{m/s} \]