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 step by step, we will use the principles of conservation of momentum and the definition of the coefficient of restitution.

### Step 1: Understand the given information
- We have two identical glass balls.
- The first ball (Ball 1) is moving with an initial velocity \( v_0 = 2 \, \text{m/s} \).
- The second ball (Ball 2) is at rest, so its initial velocity \( u_2 = 0 \, \text{m/s} \).
- The coefficient of restitution \( e = 0.5 \).

### Step 2: Define the final velocities
Let:
- \( v_1 \) = final velocity of Ball 1 after the collision.
- \( v_2 \) = final velocity of Ball 2 after the collision.

### Step 3: Apply the conservation of momentum
According to the conservation of momentum:
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]
Since the masses are identical (let's denote them as \( m \)), we can simplify:
\[ u_1 + 0 = v_1 + v_2 \]
Substituting \( u_1 = v_0 = 2 \, \text{m/s} \):
\[ 2 = v_1 + v_2 \quad \text{(Equation 1)} \]

### Step 4: Apply the coefficient of restitution
The coefficient of restitution is defined as:
\[ e = \frac{\text{velocity of separation}}{\text{velocity of approach}} \]
For our case:
\[ 0.5 = \frac{v_2 - v_1}{u_1 - u_2} \]
Substituting \( u_1 = 2 \, \text{m/s} \) and \( u_2 = 0 \):
\[ 0.5 = \frac{v_2 - v_1}{2 - 0} \]
This simplifies to:
\[ 0.5 = \frac{v_2 - v_1}{2} \]
Multiplying both sides by 2:
\[ 1 = v_2 - v_1 \quad \text{(Equation 2)} \]

### Step 5: 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 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:
\[ v_1 = 0.5 \, \text{m/s} \]

Now substituting \( v_1 \) back into Equation 2 to find \( v_2 \):
\[ v_2 = 0.5 + 1 = 1.5 \, \text{m/s} \]

### Step 6: Final velocities
The final velocities of the glass balls after the collision are:
- \( v_1 = 0.5 \, \text{m/s} \) (Ball 1)
- \( v_2 = 1.5 \, \text{m/s} \) (Ball 2)

### Summary
- Ball 1 moves with a velocity of \( 0.5 \, \text{m/s} \) in the positive direction.
- Ball 2 moves with a velocity of \( 1.5 \, \text{m/s} \) in the same direction.