A body of mass 3 kg moving with a velocity `(2hati+3hatj+3hatk)` m/s collides with another body of mass 4 kg moving with a velocity `(3hati+2hatj-3hatk)` m/s. The two bodies stick together after collision. The velocity of the composite body is

Options

A. `1/7(4hati+6hatj-3hatk)`
B. `1/7(18hati+17hatj-3hatk)`
C. `1/7(6hati+4hatj-6hatk)`
D. `1/7(9hati+8hatj-6hatk)`

Answer

To find the velocity of the composite body after the collision, we will use the law of conservation of momentum. Here’s the step-by-step solution:

### Step 1: Identify the masses and velocities of the two bodies
- Mass of the first body, \( m_1 = 3 \, \text{kg} \)
- Initial velocity of the first body, \( \vec{V_1} = 2 \hat{i} + 3 \hat{j} + 3 \hat{k} \, \text{m/s} \)
- Mass of the second body, \( m_2 = 4 \, \text{kg} \)
- Initial velocity of the second body, \( \vec{V_2} = 3 \hat{i} + 2 \hat{j} - 3 \hat{k} \, \text{m/s} \)

### Step 2: Write the conservation of momentum equation
According to the law of conservation of linear momentum, the total momentum before the collision is equal to the total momentum after the collision. Thus, we can write:
\[ m_1 \vec{V_1} + m_2 \vec{V_2} = (m_1 + m_2) \vec{V} \]
where \( \vec{V} \) is the velocity of the composite body after the collision.

### Step 3: Substitute the known values into the equation
Substituting the values we have:
\[ 3(2 \hat{i} + 3 \hat{j} + 3 \hat{k}) + 4(3 \hat{i} + 2 \hat{j} - 3 \hat{k}) = (3 + 4) \vec{V} \]

### Step 4: Calculate the left-hand side
Calculating the momentum contributions from both bodies:
- For the first body:
\[ 3(2 \hat{i} + 3 \hat{j} + 3 \hat{k}) = 6 \hat{i} + 9 \hat{j} + 9 \hat{k} \]
- For the second body:
\[ 4(3 \hat{i} + 2 \hat{j} - 3 \hat{k}) = 12 \hat{i} + 8 \hat{j} - 12 \hat{k} \]

Now, adding these two results:
\[ (6 \hat{i} + 9 \hat{j} + 9 \hat{k}) + (12 \hat{i} + 8 \hat{j} - 12 \hat{k}) = (6 + 12) \hat{i} + (9 + 8) \hat{j} + (9 - 12) \hat{k} \]
This simplifies to:
\[ 18 \hat{i} + 17 \hat{j} - 3 \hat{k} \]

### Step 5: Set the equation equal to the right-hand side
Now we have:
\[ 18 \hat{i} + 17 \hat{j} - 3 \hat{k} = 7 \vec{V} \]

### Step 6: Solve for \( \vec{V} \)
To find \( \vec{V} \), we divide both sides by 7:
\[ \vec{V} = \frac{1}{7}(18 \hat{i} + 17 \hat{j} - 3 \hat{k}) \]
This gives us:
\[ \vec{V} = \frac{18}{7} \hat{i} + \frac{17}{7} \hat{j} - \frac{3}{7} \hat{k} \]

### Final Answer
The velocity of the composite body after the collision is:
\[ \vec{V} = \frac{18}{7} \hat{i} + \frac{17}{7} \hat{j} - \frac{3}{7} \hat{k} \, \text{m/s} \]