A glass ball collides with an identical ball at rest with `v_(0)=2` m/sec. If the coefficient of restitution of collision is e = 0.5, find the velocities of the glass balls just after the collision.

Answer

To solve the problem of the collision between two identical glass balls, we will use the principles of conservation of momentum and the coefficient of restitution. Here’s a step-by-step solution:

### Step 1: Define the Initial Conditions
- Let the mass of each glass ball be \( m \).
- The initial velocity of the first ball (moving towards the second) is \( v_0 = 2 \, \text{m/s} \).
- The second ball is initially at rest, so its initial velocity \( u_2 = 0 \, \text{m/s} \).

### Step 2: Apply Conservation of Momentum
The conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Thus, we can write:

\[ m v_0 + m u_2 = m v_1 + m v_2 \]

Substituting the known values:

\[ m(2) + m(0) = m v_1 + m v_2 \]

Since the masses are identical, we can cancel \( m \) from the equation:

\[ 2 = v_1 + v_2 \quad \text{(Equation 1)} \]

### Step 3: Use the Coefficient of Restitution
The coefficient of restitution \( e \) is defined as the ratio of the relative velocity of separation to the relative velocity of approach. Mathematically, it can be expressed as:

\[ e = \frac{v_2 - v_1}{v_0 - u_2} \]

Substituting the known values:

\[ 0.5 = \frac{v_2 - v_1}{2 - 0} \]

This simplifies to:

\[ 0.5 = \frac{v_2 - v_1}{2} \]

Multiplying both sides by 2 gives:

\[ v_2 - v_1 = 1 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations Simultaneously
Now we have two equations:
1. \( v_1 + v_2 = 2 \) (Equation 1)
2. \( v_2 - v_1 = 1 \) (Equation 2)

From Equation 2, we can express \( v_2 \) in terms of \( v_1 \):

\[ v_2 = v_1 + 1 \]

Substituting this expression for \( v_2 \) into Equation 1:

\[ v_1 + (v_1 + 1) = 2 \]

This simplifies to:

\[ 2v_1 + 1 = 2 \]

Subtracting 1 from both sides:

\[ 2v_1 = 1 \]

Dividing by 2:

\[ v_1 = 0.5 \, \text{m/s} \]

### Step 5: Find \( v_2 \)
Now substituting \( v_1 \) back into Equation 2 to find \( v_2 \):

\[ v_2 = v_1 + 1 = 0.5 + 1 = 1.5 \, \text{m/s} \]

### Final Answer
The velocities of the glass balls just after the collision are:
- \( v_1 = 0.5 \, \text{m/s} \) (first ball)
- \( v_2 = 1.5 \, \text{m/s} \) (second ball)