A particle of mass m with an initial velocity `u hati+2u hatj` collides with a particle of mass 3m at rest. After collision, the two particles stick together and the combined particle moves with a velocity `v hati+v' hatj`. Which of the following is incorrect?

Options

A. `v=v'//2`
B. `v=2v'`
C. `v=u//4`
D. `v'=u//2`

Explanation

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

### Step 1: Identify the initial conditions
We have two particles:
1. Particle 1 (mass = m) with an initial velocity \( \vec{u} = u \hat{i} + 2u \hat{j} \)
2. Particle 2 (mass = 3m) at rest, so its initial velocity is \( \vec{0} \)

### Step 2: Write the momentum before the collision
The total momentum before the collision can be calculated as follows:
\[ \text{Initial Momentum} = \text{Momentum of Particle 1} + \text{Momentum of Particle 2} \]
\[ = m \vec{u} + 3m \cdot \vec{0} = m (u \hat{i} + 2u \hat{j}) = mu \hat{i} + 2mu \hat{j} \]

### Step 3: Write the momentum after the collision
After the collision, the two particles stick together, and their combined mass is \( m + 3m = 4m \). The velocity of the combined mass is given as \( \vec{v} = v \hat{i} + v' \hat{j} \). Therefore, the momentum after the collision is:
\[ \text{Final Momentum} = (4m) \vec{v} = 4m (v \hat{i} + v' \hat{j}) \]

### Step 4: Apply the conservation of momentum
According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
This gives us the equation:
\[ mu \hat{i} + 2mu \hat{j} = 4m (v \hat{i} + v' \hat{j}) \]

### Step 5: Simplify the equation
We can divide the entire equation by \( m \) (assuming \( m \neq 0 \)):
\[ u \hat{i} + 2u \hat{j} = 4(v \hat{i} + v' \hat{j}) \]

### Step 6: Compare components
Now, we can compare the coefficients of \( \hat{i} \) and \( \hat{j} \) on both sides:
1. For \( \hat{i} \):
\[ u = 4v \quad \text{(1)} \]
2. For \( \hat{j} \):
\[ 2u = 4v' \quad \text{(2)} \]

### Step 7: Solve for \( v \) and \( v' \)
From equation (1):
\[ v = \frac{u}{4} \]
From equation (2):
\[ v' = \frac{2u}{4} = \frac{u}{2} \]

### Step 8: Establish the relationship between \( v \) and \( v' \)
From the results:
- \( v = \frac{u}{4} \)
- \( v' = \frac{u}{2} \)

We can express \( v' \) in terms of \( v \):
\[ v' = 2v \]

### Step 9: Analyze the options
Now we need to check which of the given options is incorrect based on the relationships we derived.

### Conclusion
The relationships we derived are:
- \( v = \frac{u}{4} \)
- \( v' = \frac{u}{2} \)
- \( v' = 2v \)

Thus, any option contradicting these relationships would be incorrect.