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 step by step, we will use the principle of conservation of momentum.

### Step 1: Define the velocities and directions
Let:
- Ball A has a mass \( m \) and is moving with a velocity of \( 6 \, \text{m/s} \) in the x-direction. We can represent this as:
\[ \vec{V_A} = 6 \hat{i} \, \text{m/s} \]
- Ball B has a mass \( m \) and is moving with a velocity of \( 8 \, \text{m/s} \) in the y-direction. We can represent this as:
\[ \vec{V_B} = 8 \hat{j} \, \text{m/s} \]

### Step 2: Write the momentum before the collision
The total momentum before the collision can be calculated as:
\[ \vec{P_{\text{initial}}} = m \vec{V_A} + m \vec{V_B} = m (6 \hat{i}) + m (8 \hat{j}) = m (6 \hat{i} + 8 \hat{j}) \]

### Step 3: Combine the masses after the collision
Since the collision is perfectly inelastic, both balls stick together after the collision. The combined mass is:
\[ M_{\text{combined}} = m + m = 2m \]

### Step 4: Write the momentum after the collision
Let the final velocity of the combined mass be \( \vec{V} \). The momentum after the collision is:
\[ \vec{P_{\text{final}}} = (2m) \vec{V} \]

### Step 5: Apply conservation of momentum
According to the conservation of momentum:
\[ \vec{P_{\text{initial}}} = \vec{P_{\text{final}}} \]
This gives us:
\[ m (6 \hat{i} + 8 \hat{j}) = (2m) \vec{V} \]

### Step 6: Simplify the equation
We can divide both sides by \( m \) (assuming \( m \neq 0 \)):
\[ 6 \hat{i} + 8 \hat{j} = 2 \vec{V} \]

### Step 7: Solve for \( \vec{V} \)
Now, divide both sides by 2 to find \( \vec{V} \):
\[ \vec{V} = 3 \hat{i} + 4 \hat{j} \]

### Step 8: Calculate the magnitude of the final velocity
The magnitude of the velocity \( V \) can be calculated using the Pythagorean theorem:
\[ |\vec{V}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{m/s} \]

### Final Answer
The final velocity of the combined mass after the perfectly inelastic collision is:
\[ \boxed{5 \, \text{m/s}} \]