A smooth sphere of mass m is moving on a horizontal plane with a velocity `(3hati +hat j)` . It collides with smooth a vertical wall which is parallel to the vector `hatj` . If coefficient of restitution `e = 1/2` then impulse that acts on the sphere is

Options

A. `-9/2mhati`
B. `(-3/2hati+hatj)`
C. `3/2mhatj`
D. `(3/2mhatj+1/2mhati)`

Answer

To solve the problem, we need to analyze the motion of the sphere before and after it collides with the wall, and then calculate the impulse acting on it.

### Step-by-Step Solution:

1. **Identify Initial Velocity:**
The initial velocity of the sphere is given as:
\[ \vec{v}_i = 3\hat{i} + \hat{j} \]

2. **Understand the Collision:**
The sphere collides with a vertical wall that is parallel to the \(\hat{j}\) direction. This means that the wall will affect the \(\hat{i}\) component of the velocity but not the \(\hat{j}\) component.

3. **Determine the Components of Velocity:**
- The \(\hat{i}\) component (horizontal) before the collision is \(v_{ix} = 3\).
- The \(\hat{j}\) component (vertical) before the collision is \(v_{iy} = 1\).

4. **Apply the Coefficient of Restitution:**
The coefficient of restitution \(e\) is given as \(1/2\). The coefficient of restitution relates the velocities before and after the collision:
\[ e = \frac{v_{fy}}{v_{iy}} \]
where \(v_{fy}\) is the final velocity in the \(\hat{i}\) direction after the collision.

Since the wall does not affect the \(\hat{j}\) component, we have:
\[ v_{fy} = v_{iy} = 1 \]

For the \(\hat{i}\) component:
\[ e = \frac{-v_{fx}}{v_{ix}} \Rightarrow \frac{1}{2} = \frac{-v_{fx}}{3} \]
Solving for \(v_{fx}\):
\[ -v_{fx} = \frac{3}{2} \Rightarrow v_{fx} = -\frac{3}{2} \]

5. **Calculate the Change in Momentum:**
The initial momentum \(\vec{p}_i\) and final momentum \(\vec{p}_f\) can be expressed as:
\[ \vec{p}_i = m(3\hat{i} + \hat{j}) = 3m\hat{i} + m\hat{j} \]
\[ \vec{p}_f = m\left(-\frac{3}{2}\hat{i} + \hat{j}\right) = -\frac{3}{2}m\hat{i} + m\hat{j} \]

6. **Impulse Calculation:**
Impulse \(\vec{J}\) is defined as the change in momentum:
\[ \vec{J} = \vec{p}_f - \vec{p}_i \]
\[ \vec{J} = \left(-\frac{3}{2}m\hat{i} + m\hat{j}\right) - \left(3m\hat{i} + m\hat{j}\right) \]
\[ \vec{J} = \left(-\frac{3}{2}m - 3m\right)\hat{i} + (m - m)\hat{j} \]
\[ \vec{J} = \left(-\frac{3}{2}m - \frac{6}{2}m\right)\hat{i} + 0\hat{j} \]
\[ \vec{J} = -\frac{9}{2}m\hat{i} \]

### Final Result:
The impulse that acts on the sphere is:
\[ \vec{J} = -\frac{9}{2}m\hat{i} \]