Two bodies of masses m and 2m are moving along positive x and y axes respectively with equal speed `4 ms^-1`. They collide at origin and stick together. The final velocity of combined mass is

Options

A. `(4hati+8hatj) ms^-1`
B. `(4/3hati+frac{8}{3}hatj)ms^-1`
C. `(frac{4}{3}hati+8hatj)ms^-1`
D. `(4hati+frac{8}{3}hatj)ms^-1`

Answer

To solve the problem, we will use the principle of conservation of momentum. Here are the steps to find the final velocity of the combined mass after the collision:

### Step 1: Identify the initial velocities of the two masses
- The first body has a mass \( m \) and is moving along the positive x-axis with a velocity of \( 4 \, \text{m/s} \). Therefore, its velocity vector is:
\[ \vec{v_1} = 4 \hat{i} \]
- The second body has a mass \( 2m \) and is moving along the positive y-axis with a velocity of \( 4 \, \text{m/s} \). Therefore, its velocity vector is:
\[ \vec{v_2} = 4 \hat{j} \]

### Step 2: Calculate the initial momentum of each body
- The momentum of the first body is given by:
\[ \vec{p_1} = m \cdot \vec{v_1} = m \cdot (4 \hat{i}) = 4m \hat{i} \]
- The momentum of the second body is given by:
\[ \vec{p_2} = 2m \cdot \vec{v_2} = 2m \cdot (4 \hat{j}) = 8m \hat{j} \]

### Step 3: Calculate the total initial momentum before the collision
- The total initial momentum \( \vec{p_{initial}} \) is the vector sum of the momenta of both bodies:
\[ \vec{p_{initial}} = \vec{p_1} + \vec{p_2} = 4m \hat{i} + 8m \hat{j} \]

### Step 4: Determine the total mass after the collision
- After the collision, the two bodies stick together. The total mass \( M \) of the combined body is:
\[ M = m + 2m = 3m \]

### Step 5: Apply conservation of momentum
- According to the conservation of momentum, the total initial momentum is equal to the total final momentum:
\[ \vec{p_{initial}} = \vec{p_{final}} \]
- The final momentum can be expressed as:
\[ \vec{p_{final}} = M \cdot \vec{v_{final}} = 3m \cdot \vec{v_{final}} \]

### Step 6: Set up the equation for final velocity
- Equating the total initial momentum to the total final momentum gives us:
\[ 4m \hat{i} + 8m \hat{j} = 3m \cdot \vec{v_{final}} \]
- Dividing both sides by \( 3m \) (assuming \( m \neq 0 \)):
\[ \vec{v_{final}} = \frac{4m \hat{i} + 8m \hat{j}}{3m} = \frac{4}{3} \hat{i} + \frac{8}{3} \hat{j} \]

### Step 7: Conclusion
- The final velocity of the combined mass after the collision is:
\[ \vec{v_{final}} = \frac{4}{3} \hat{i} + \frac{8}{3} \hat{j} \, \text{m/s} \]