A particle moving with a velocity of (`4hati-hatj` ) mis strikes a fixed smooth wall and finally moves with a velocity of (`3hati + 2hatj`) m/s. The coefficient of restitution between the wall and the particle in the collision will be

Options

A. `7/3`
B. `3/7`
C. `sqrt(13/17)`
D. `sqrt(17/13)`

Answer

To find the coefficient of restitution (e) between the wall and the particle after the collision, we can follow these steps:

### Step 1: Identify the initial and final velocities
The initial velocity of the particle before striking the wall is given as:
\[ \mathbf{v_i} = 4\hat{i} - \hat{j} \, \text{m/s} \]

The final velocity of the particle after striking the wall is given as:
\[ \mathbf{v_f} = 3\hat{i} + 2\hat{j} \, \text{m/s} \]

### Step 2: Calculate the change in velocity
The change in velocity (Δv) can be calculated as:
\[ \Delta \mathbf{v} = \mathbf{v_f} - \mathbf{v_i} \]
Substituting the values:
\[ \Delta \mathbf{v} = (3\hat{i} + 2\hat{j}) - (4\hat{i} - \hat{j}) \]
\[ \Delta \mathbf{v} = (3 - 4)\hat{i} + (2 + 1)\hat{j} \]
\[ \Delta \mathbf{v} = -\hat{i} + 3\hat{j} \]

### Step 3: Calculate the velocity of separation
The velocity of separation is the component of the final velocity in the direction of the wall. The wall is smooth, so it only affects the component of velocity perpendicular to its surface.

To find the velocity of separation, we need to project the final velocity onto the normal direction of the wall. In this case, we can assume the wall is vertical, so the normal direction is along the x-axis (i.e., \(\hat{i}\)).

The final velocity in the direction of the normal is:
\[ v_{f,n} = \mathbf{v_f} \cdot \hat{i} = 3\hat{i} + 2\hat{j} \cdot \hat{i} = 3 \, \text{m/s} \]

The initial velocity in the direction of the normal is:
\[ v_{i,n} = \mathbf{v_i} \cdot \hat{i} = 4\hat{i} - \hat{j} \cdot \hat{i} = 4 \, \text{m/s} \]

### Step 4: Calculate the coefficient of restitution
The coefficient of restitution (e) is defined as the ratio of the relative speed after collision to the relative speed before collision along the normal direction:
\[ e = \frac{v_{f,n}}{v_{i,n}} \]

Substituting the values we calculated:
\[ e = \frac{3}{4} \]

### Final Answer
Thus, the coefficient of restitution between the wall and the particle is:
\[ e = \frac{3}{4} \]

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