Velocity of a particle of mass `2 kg` change from `vecv_(1) =-2hati-2hatjm/s` to `vecv_(2)=(hati-hatj)m//s` after colliding with as plane surface.

Options

A. The angle made by the plane surface with the positive x-axis is `tan^(-1)(1/3)`
B. The angle made by the plane surface with the positive `x`-axis is `tan^(-1)(1/3)`
C. The direction of change in momentum makes an angle `tan^(-1)(1/3)` with the positive `x`-axis.
D. The direction of change in momentum makes an angle `90^(@) + tan^(-1)(1/3)` with the plane surface.

Explanation

To solve the problem, we need to analyze the change in momentum of the particle before and after the collision with the plane surface. Here are the steps to find the change in momentum and the impulse experienced by the particle.

### Step 1: Identify the initial and final velocities
The initial velocity of the particle is given as:
\[ \vec{v}_1 = -2\hat{i} - 2\hat{j} \, \text{m/s} \]
The final velocity after the collision is:
\[ \vec{v}_2 = \hat{i} - \hat{j} \, \text{m/s} \]

### Step 2: Calculate the initial momentum
The momentum \( \vec{p}_1 \) before the collision can be calculated using the formula:
\[ \vec{p}_1 = m \vec{v}_1 \]
where \( m = 2 \, \text{kg} \). Thus,
\[ \vec{p}_1 = 2 \, \text{kg} \cdot (-2\hat{i} - 2\hat{j}) = -4\hat{i} - 4\hat{j} \, \text{kg m/s} \]

### Step 3: Calculate the final momentum
The momentum \( \vec{p}_2 \) after the collision is given by:
\[ \vec{p}_2 = m \vec{v}_2 \]
Thus,
\[ \vec{p}_2 = 2 \, \text{kg} \cdot (\hat{i} - \hat{j}) = 2\hat{i} - 2\hat{j} \, \text{kg m/s} \]

### Step 4: Calculate the change in momentum
The change in momentum \( \Delta \vec{p} \) is given by:
\[ \Delta \vec{p} = \vec{p}_2 - \vec{p}_1 \]
Substituting the values we calculated:
\[ \Delta \vec{p} = (2\hat{i} - 2\hat{j}) - (-4\hat{i} - 4\hat{j}) \]
\[ \Delta \vec{p} = 2\hat{i} - 2\hat{j} + 4\hat{i} + 4\hat{j} \]
\[ \Delta \vec{p} = (2 + 4)\hat{i} + (-2 + 4)\hat{j} = 6\hat{i} + 2\hat{j} \, \text{kg m/s} \]

### Step 5: Conclusion
The change in momentum of the particle after colliding with the plane surface is:
\[ \Delta \vec{p} = 6\hat{i} + 2\hat{j} \, \text{kg m/s} \]