A small sphere of mass `m =1 kg` is moving with a velocity `(4hati-hatj) m//s`. it hits a fixed smooth wall and rebounds with velocity `(hati + 3hatj) m//s`. The coefficient of restitution between the sphere and the wall is `n//16`. Find the value of `n`.

Explanation

To solve the problem step by step, we will follow the concepts of momentum, the coefficient of restitution, and the components of velocity.

### Step 1: Identify the initial and final velocities
The initial velocity of the sphere is given as:
\[ \mathbf{V_i} = 4 \hat{i} - \hat{j} \, \text{m/s} \]
The final velocity after rebounding is:
\[ \mathbf{V_f} = \hat{i} + 3 \hat{j} \, \text{m/s} \]

### Step 2: Calculate the change in momentum
The change in momentum (\(\Delta \mathbf{P}\)) can be calculated as:
\[ \Delta \mathbf{P} = m \mathbf{V_f} - m \mathbf{V_i} \]
Since \(m = 1 \, \text{kg}\), we have:
\[ \Delta \mathbf{P} = \mathbf{V_f} - \mathbf{V_i} = (\hat{i} + 3 \hat{j}) - (4 \hat{i} - \hat{j}) \]
Calculating this gives:
\[ \Delta \mathbf{P} = \hat{i} + 3 \hat{j} - 4 \hat{i} + \hat{j} = -3 \hat{i} + 4 \hat{j} \]

### Step 3: Calculate the velocity of separation (\(V_s\))
The velocity of separation is the component of the final velocity in the direction of the wall. Assuming the wall is vertical, we consider the \(j\) component:
\[ V_s = \text{Component of } \mathbf{V_f} \text{ along } \hat{j} = 3 \, \text{m/s} \]

### Step 4: Calculate the velocity of approach (\(V_a\))
The velocity of approach is the component of the initial velocity in the direction of the wall:
\[ V_a = \text{Component of } \mathbf{V_i} \text{ along } \hat{j} = -1 \, \text{m/s} \]

### Step 5: Calculate the magnitudes of \(V_s\) and \(V_a\)
The magnitudes are:
\[ |V_s| = 3 \, \text{m/s} \]
\[ |V_a| = |-1| = 1 \, \text{m/s} \]

### Step 6: Apply the coefficient of restitution formula
The coefficient of restitution \(e\) is defined as:
\[ e = \frac{|V_s|}{|V_a|} \]
Substituting the values:
\[ e = \frac{3}{1} = 3 \]

### Step 7: Relate the coefficient of restitution to \(n\)
We are given that:
\[ e = \frac{n}{16} \]
Setting the two expressions for \(e\) equal gives:
\[ 3 = \frac{n}{16} \]
Multiplying both sides by 16:
\[ n = 48 \]

### Final Answer
The value of \(n\) is:
\[ \boxed{48} \]