A sphere of mass m is moving with a velocity `(4 hat(i)-hat(j))` m/s hits a surface and rebounds with a velocity `(hat(i)+3hat(j))` m/s. The coefficient of restitution between the sphere and the surface is k/16. find the value of k.

Answer

To solve the problem, we need to find the value of \( k \) given the coefficient of restitution \( e = \frac{k}{16} \). The coefficient of restitution is defined as the ratio of the relative velocity of separation to the relative velocity of approach along the line of impact.

### Step-by-Step Solution:

1. **Identify Initial and Final Velocities:**
- The initial velocity \( \vec{u} \) of the sphere is given as:
\[ \vec{u} = 4 \hat{i} - \hat{j} \, \text{m/s} \]
- The final velocity \( \vec{v} \) after rebounding is:
\[ \vec{v} = \hat{i} + 3 \hat{j} \, \text{m/s} \]

2. **Calculate the Change in Momentum:**
- The impulse experienced by the sphere can be calculated as:
\[ \text{Impulse} = m(\vec{v} - \vec{u}) = m\left((\hat{i} + 3\hat{j}) - (4\hat{i} - \hat{j})\right) \]
- Simplifying this gives:
\[ \text{Impulse} = m\left(-3\hat{i} + 4\hat{j}\right) \]

3. **Determine the Direction of Impulse:**
- The unit vector along the direction of impulse is given by:
\[ \hat{n} = \frac{\text{Impulse}}{|\text{Impulse}|} \]
- First, we find the magnitude of the impulse:
\[ |\text{Impulse}| = m \sqrt{(-3)^2 + 4^2} = m \sqrt{9 + 16} = m \sqrt{25} = 5m \]
- Thus, the unit vector is:
\[ \hat{n} = \frac{-3\hat{i} + 4\hat{j}}{5m} \]

4. **Calculate the Components of Velocities Along the Direction of Impulse:**
- The component of \( \vec{u} \) along \( \hat{n} \):
\[ u_1 = \vec{u} \cdot \hat{n} = (4\hat{i} - \hat{j}) \cdot \left(\frac{-3\hat{i} + 4\hat{j}}{5}\right) \]
- Calculating this gives:
\[ u_1 = \frac{1}{5} (4 \cdot -3 + (-1) \cdot 4) = \frac{1}{5} (-12 - 4) = \frac{-16}{5} \]

- The component of \( \vec{v} \) along \( \hat{n} \):
\[ v_1 = \vec{v} \cdot \hat{n} = (\hat{i} + 3\hat{j}) \cdot \left(\frac{-3\hat{i} + 4\hat{j}}{5}\right) \]
- Calculating this gives:
\[ v_1 = \frac{1}{5} (-3 + 12) = \frac{9}{5} \]

5. **Apply the Coefficient of Restitution Formula:**
- The coefficient of restitution \( e \) is defined as:
\[ e = \frac{\text{Velocity of Separation}}{\text{Velocity of Approach}} = \frac{v_1 - 0}{0 - u_1} = \frac{v_1}{-u_1} \]
- Substituting the values:
\[ e = \frac{\frac{9}{5}}{-\left(-\frac{16}{5}\right)} = \frac{9}{16} \]

6. **Relate \( e \) to \( k \):**
- From the problem, we know:
\[ e = \frac{k}{16} \]
- Setting the two expressions for \( e \) equal gives:
\[ \frac{k}{16} = \frac{9}{16} \]
- Therefore, solving for \( k \):
\[ k = 9 \]

### Final Answer:
The value of \( k \) is \( 9 \).