Two identical balls each moving with speed v at right angle to each other collide perfectly inelastically. Their speed after collision is

Options

A. v
B. sqrt2v`
C. `(v)/(2)`
D. `(v)/(sqrt2)`

Answer

To solve the problem of finding the speed of two identical balls after a perfectly inelastic collision, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two identical balls, each with mass \( m \), moving at right angles to each other with speed \( v \). One ball is moving along the x-axis, and the other is moving along the y-axis.

2. **Setting Up the Momentum Conservation**:
Since the collision is perfectly inelastic, the two balls stick together after the collision. We will use the principle of conservation of linear momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.

3. **Calculating Initial Momentum**:
- The momentum of the first ball moving along the x-axis is:
\[ \text{Momentum}_x = mv \hat{i} \]
- The momentum of the second ball moving along the y-axis is:
\[ \text{Momentum}_y = mv \hat{j} \]
- Therefore, the total initial momentum \( \vec{P}_{initial} \) is:
\[ \vec{P}_{initial} = mv \hat{i} + mv \hat{j} \]

4. **Calculating Final Momentum**:
After the collision, the two balls combine to form a single object with mass \( 2m \). Let the velocity of this combined mass be \( \vec{V} \). The total final momentum \( \vec{P}_{final} \) is:
\[ \vec{P}_{final} = 2m \vec{V} \]

5. **Applying Conservation of Momentum**:
By conservation of momentum:
\[ \vec{P}_{initial} = \vec{P}_{final} \]
Thus, we have:
\[ mv \hat{i} + mv \hat{j} = 2m \vec{V} \]

6. **Solving for Velocity \( \vec{V} \)**:
Dividing both sides by \( 2m \):
\[ \vec{V} = \frac{1}{2} \left( v \hat{i} + v \hat{j} \right) = \frac{v}{2} \hat{i} + \frac{v}{2} \hat{j} \]

7. **Finding the Magnitude of \( \vec{V} \)**:
The magnitude of the velocity \( V \) can be calculated using the Pythagorean theorem:
\[ V = \sqrt{\left(\frac{v}{2}\right)^2 + \left(\frac{v}{2}\right)^2} = \sqrt{\frac{v^2}{4} + \frac{v^2}{4}} = \sqrt{\frac{2v^2}{4}} = \sqrt{\frac{v^2}{2}} = \frac{v}{\sqrt{2}} \]

8. **Conclusion**:
The speed of the combined mass after the collision is:
\[ V = \frac{v}{\sqrt{2}} \]

### Final Answer:
The speed of the two identical balls after the collision is \( \frac{v}{\sqrt{2}} \).