Two balls of masses m each are moving at right angle to each other with velocities 6 m/s and 8 m/s respectively. If collision between them is perfectly inelastic, the velocity of combined mass is

Options

A. 15 m/s
B. 10m/s
C. 5 m/s
D. 2.5 m/s

Answer

To solve the problem of finding the velocity of the combined mass after a perfectly inelastic collision between two balls of mass \( m \), moving at right angles to each other with velocities \( 6 \, \text{m/s} \) and \( 8 \, \text{m/s} \), we can follow these steps:

### Step 1: Understand the initial conditions
We have two balls:
- Ball 1 with mass \( m \) moving along the x-axis with a velocity of \( 6 \, \text{m/s} \).
- Ball 2 with mass \( m \) moving along the y-axis with a velocity of \( 8 \, \text{m/s} \).

### Step 2: Calculate the initial momentum in the x-direction
The initial momentum of Ball 1 in the x-direction is given by:
\[ p_{x, \text{initial}} = m \cdot v_{x} = m \cdot 6 = 6m \]

### Step 3: Calculate the initial momentum in the y-direction
The initial momentum of Ball 2 in the y-direction is given by:
\[ p_{y, \text{initial}} = m \cdot v_{y} = m \cdot 8 = 8m \]

### Step 4: Apply conservation of momentum
Since the collision is perfectly inelastic, the two balls stick together after the collision. The total momentum before the collision must equal the total momentum after the collision.

Let \( V_x \) and \( V_y \) be the velocities of the combined mass in the x and y directions respectively after the collision. The total mass after the collision is \( 2m \).

#### In the x-direction:
\[ p_{x, \text{final}} = 2m \cdot V_x \]
Setting the initial momentum equal to the final momentum:
\[ 6m = 2m \cdot V_x \]
Dividing both sides by \( 2m \):
\[ V_x = \frac{6m}{2m} = 3 \, \text{m/s} \]

#### In the y-direction:
\[ p_{y, \text{final}} = 2m \cdot V_y \]
Setting the initial momentum equal to the final momentum:
\[ 8m = 2m \cdot V_y \]
Dividing both sides by \( 2m \):
\[ V_y = \frac{8m}{2m} = 4 \, \text{m/s} \]

### Step 5: Calculate the magnitude of the resultant velocity
The resultant velocity \( V \) can be found using the Pythagorean theorem:
\[ V = \sqrt{V_x^2 + V_y^2} = \sqrt{(3 \, \text{m/s})^2 + (4 \, \text{m/s})^2} \]
Calculating:
\[ V = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{m/s} \]

### Conclusion
The velocity of the combined mass after the perfectly inelastic collision is \( 5 \, \text{m/s} \).

---