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 momentum of the system
The initial momentum of the system can be calculated using the formula:
\[ \text{Initial Momentum} = \text{momentum of particle 1} + \text{momentum of particle 2} \]

For the first particle (mass \( m \) with initial velocity \( u \hat{i} + 2u \hat{j} \)):
\[ \text{Momentum of particle 1} = m \cdot (u \hat{i} + 2u \hat{j}) = m u \hat{i} + 2m u \hat{j} \]

For the second particle (mass \( 3m \) at rest):
\[ \text{Momentum of particle 2} = 3m \cdot 0 = 0 \]

Thus, the total initial momentum is:
\[ \text{Initial Momentum} = m u \hat{i} + 2m u \hat{j} + 0 = m u \hat{i} + 2m u \hat{j} \]

### Step 2: Identify the final momentum of the system
After the collision, the two particles stick together, so the total mass becomes \( 4m \) and moves with a velocity \( v \hat{i} + v' \hat{j} \):
\[ \text{Final Momentum} = \text{Total mass} \times \text{Velocity} = 4m (v \hat{i} + v' \hat{j}) = 4m v \hat{i} + 4m v' \hat{j} \]

### Step 3: Apply the conservation of momentum
According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
Thus, we can equate the two:
\[ m u \hat{i} + 2m u \hat{j} = 4m v \hat{i} + 4m v' \hat{j} \]

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

### Step 5: Compare coefficients
From the equation, we can compare the coefficients of \( \hat{i} \) and \( \hat{j} \):
1. For \( \hat{i} \):
\[ u = 4v \]
\[ v = \frac{u}{4} \]

2. For \( \hat{j} \):
\[ 2u = 4v' \]
\[ v' = \frac{u}{2} \]

### Step 6: Establish the relationship between \( v \) and \( v' \)
From the equations derived:
- \( v = \frac{u}{4} \)
- \( v' = \frac{u}{2} \)

Now, we can express \( v' \) in terms of \( v \):
\[ v' = 2v \]

### Step 7: Identify the incorrect statement
Now we need to check the statements provided:
1. \( v' = 2v \) (True)
2. \( v = \frac{u}{4} \) (True)
3. \( v' = \frac{u}{2} \) (True)
4. \( v = 2v' \) (False, as \( v' = 2v \))

Thus, the incorrect statement is:
\[ v = 2v' \]