Two small spheres `A` and `B` of equal radius but different masses of `3m` and `2m` are moving towards each other and impinge directly. The speeds of `A` and `B` before collision are, respectively, `4u` and `u`. The collision is such that `B` experiences an impulse of magnitude `6mcu`, where `c` is a constant. Determine
a. the coefficient of restitution,
b. the limits for the value of `c` for which such collision is possible.

Explanation

To solve the problem step by step, we will analyze the situation involving two spheres A and B, and derive the required results.

### Step 1: Define the given parameters
- Mass of sphere A, \( m_A = 3m \)
- Mass of sphere B, \( m_B = 2m \)
- Initial velocity of sphere A, \( u_A = 4u \) (moving towards B)
- Initial velocity of sphere B, \( u_B = -u \) (moving towards A)
- Impulse experienced by B, \( J = 6mcu \)

### Step 2: Apply the conservation of linear momentum
Using the conservation of momentum before and after the collision, we can write:
\[ m_A u_A + m_B u_B = m_A v_A + m_B v_B \]
Substituting the known values:
\[ 3m(4u) + 2m(-u) = 3m v_A + 2m v_B \]
This simplifies to:
\[ 12m u - 2m u = 3m v_A + 2m v_B \]
\[ 10m u = 3m v_A + 2m v_B \]
Dividing through by \( m \):
\[ 10u = 3v_A + 2v_B \quad \text{(Equation 1)} \]

### Step 3: Apply the impulse-momentum theorem for sphere B
According to the impulse-momentum theorem:
\[ J = \Delta p_B \]
Where \( \Delta p_B \) is the change in momentum of sphere B:
\[ 6mcu = m_B v_B - m_B u_B \]
Substituting the values:
\[ 6mcu = 2m v_B - 2m(-u) \]
This simplifies to:
\[ 6cu = 2v_B + 2u \]
Dividing through by \( 2 \):
\[ 3cu = v_B + u \]
Rearranging gives:
\[ v_B = 3cu - u \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
Now we substitute \( v_B \) from Equation 2 into Equation 1:
\[ 10u = 3v_A + 2(3cu - u) \]
Expanding this:
\[ 10u = 3v_A + 6cu - 2u \]
Combining like terms:
\[ 10u + 2u = 3v_A + 6cu \]
\[ 12u = 3v_A + 6cu \]
Solving for \( v_A \):
\[ 3v_A = 12u - 6cu \]
\[ v_A = 4u - 2cu \quad \text{(Equation 3)} \]

### Step 5: Calculate the coefficient of restitution
The coefficient of restitution \( e \) is defined as:
\[ e = \frac{v_B - v_A}{u_A - u_B} \]
Substituting the values:
\[ e = \frac{(3cu - u) - (4u - 2cu)}{4u - (-u)} \]
This simplifies to:
\[ e = \frac{3cu - u - 4u + 2cu}{4u + u} \]
\[ e = \frac{5cu - 5u}{5u} \]
\[ e = c - 1 \]

### Step 6: Determine the limits for the value of \( c \)
Since the coefficient of restitution \( e \) must lie between 0 and 1:
\[ 0 \leq c - 1 \leq 1 \]
This implies:
\[ 1 \leq c \leq 2 \]

### Final Answers
a. The coefficient of restitution \( e = c - 1 \).

b. The limits for the value of \( c \) are \( 1 \leq c \leq 2 \).