Two identical balls moving in opposite directions with speed 20 m/s and 25 m/s undergo head on perfectly inelastic collision. The speed of combined mass after collision is

Options

A. 20 m/s
B. 22.5 m/s
C. 25 m/s
D. 2.5 m/s

Answer

To solve the problem of two identical balls undergoing a perfectly inelastic collision, we will use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Understand the scenario
We have two identical balls moving in opposite directions with speeds of 20 m/s and 25 m/s. Since they are identical, we can denote their mass as \( m \).

### Step 2: Assign directions and velocities
Let’s assume:
- Ball 1 (moving to the right) has a velocity \( v_1 = 20 \, \text{m/s} \).
- Ball 2 (moving to the left) has a velocity \( v_2 = -25 \, \text{m/s} \) (negative because it is in the opposite direction).

### Step 3: Write the momentum before the collision
The total momentum before the collision can be calculated as:
\[ p_{\text{initial}} = m \cdot v_1 + m \cdot v_2 = m \cdot 20 + m \cdot (-25) \]
\[ p_{\text{initial}} = 20m - 25m = -5m \]

### Step 4: Write the momentum after the collision
In a perfectly inelastic collision, the two balls stick together after the collision. The combined mass after the collision is \( 2m \) and let the velocity of the combined mass be \( v \). Thus, the momentum after the collision is:
\[ p_{\text{final}} = (2m) \cdot v \]

### Step 5: Apply the conservation of momentum
According to the conservation of momentum:
\[ p_{\text{initial}} = p_{\text{final}} \]
Substituting the values we calculated:
\[ -5m = 2m \cdot v \]

### Step 6: Solve for \( v \)
To find the velocity \( v \), we can divide both sides of the equation by \( 2m \):
\[ v = \frac{-5m}{2m} = -\frac{5}{2} = -2.5 \, \text{m/s} \]

### Step 7: Interpret the result
The negative sign indicates that the combined mass is moving in the direction of the original velocity of the second ball (to the left).

### Final Answer
The speed of the combined mass after the collision is \( 2.5 \, \text{m/s} \) in the left direction.

---