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 towards each other. One ball is moving at a speed of 20 m/s to the right (let's consider this as positive direction), and the other ball is moving at a speed of 25 m/s to the left (which we will consider as negative direction).

### Step 2: Define the masses and velocities
Let the mass of each ball be \( m \).
- Velocity of the first ball, \( v_1 = +20 \, \text{m/s} \)
- Velocity of the second ball, \( v_2 = -25 \, \text{m/s} \)

### Step 3: Apply the conservation of momentum
In a perfectly inelastic collision, the two objects stick together after the collision. According to the conservation of momentum:
\[ \text{Initial momentum} = \text{Final momentum} \]
The initial momentum before the collision is given by:
\[ p_{\text{initial}} = m \cdot v_1 + m \cdot v_2 = m \cdot 20 + m \cdot (-25) \]
\[ p_{\text{initial}} = m(20 - 25) = m(-5) \]

### Step 4: Calculate the final momentum
After the collision, both balls move together with a common velocity \( v_f \). The final momentum is:
\[ p_{\text{final}} = (m + m) \cdot v_f = 2m \cdot v_f \]

### Step 5: Set the initial momentum equal to the final momentum
Setting the initial momentum equal to the final momentum gives:
\[ m(-5) = 2m \cdot v_f \]
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ -5 = 2v_f \]

### Step 6: Solve for the final velocity
Now, we can solve for \( v_f \):
\[ v_f = \frac{-5}{2} = -2.5 \, \text{m/s} \]

### Conclusion
The speed of the combined mass after the collision is \( 2.5 \, \text{m/s} \) in the negative direction (indicating it moves to the left).

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

---