A ball of mass 1 kg moving with velocity `vecv = 3hati + 4 hatj` collides with a wall and after collision velocity of the ball is `-2 hati + 6 hatj` . Which of the following unit vectors is perpendicular to wall?

Options

A. `1/5 (-3hati - 4hatj)`
B. `1/(sqrt(29))(5 hati + 2 hatj)`
C. `1/(sqrt(40)) -(2 hati + 6 hatj)`
D. `1/(sqrt(40))(5 hati - 2hatj)`

Explanation

To find the unit vector that is perpendicular to the wall after the collision of the ball, we need to follow these steps:

### Step 1: Determine the initial and final velocities of the ball.
- Initial velocity of the ball:
\[ \vec{v_i} = 3 \hat{i} + 4 \hat{j} \]
- Final velocity of the ball after collision:
\[ \vec{v_f} = -2 \hat{i} + 6 \hat{j} \]

### Step 2: Calculate the change in velocity.
The change in velocity (\(\Delta \vec{v}\)) is given by:
\[ \Delta \vec{v} = \vec{v_f} - \vec{v_i} \]
Substituting the values:
\[ \Delta \vec{v} = (-2 \hat{i} + 6 \hat{j}) - (3 \hat{i} + 4 \hat{j}) \]
\[ \Delta \vec{v} = -2 \hat{i} + 6 \hat{j} - 3 \hat{i} - 4 \hat{j} \]
\[ \Delta \vec{v} = (-2 - 3) \hat{i} + (6 - 4) \hat{j} \]
\[ \Delta \vec{v} = -5 \hat{i} + 2 \hat{j} \]

### Step 3: Find the magnitude of the change in velocity.
The magnitude of \(\Delta \vec{v}\) is calculated as follows:
\[ |\Delta \vec{v}| = \sqrt{(-5)^2 + (2)^2} \]
\[ |\Delta \vec{v}| = \sqrt{25 + 4} = \sqrt{29} \]

### Step 4: Calculate the unit vector in the direction of the change in velocity.
The unit vector \(\hat{n}\) in the direction of \(\Delta \vec{v}\) is given by:
\[ \hat{n} = \frac{\Delta \vec{v}}{|\Delta \vec{v}|} \]
Substituting the values:
\[ \hat{n} = \frac{-5 \hat{i} + 2 \hat{j}}{\sqrt{29}} \]

### Step 5: Identify the unit vector perpendicular to the wall.
Since the impulse acts in the direction of the change in momentum, which is the same as the change in velocity, the unit vector we found is perpendicular to the wall. Thus, the unit vector perpendicular to the wall is:
\[ \hat{n} = \frac{-5}{\sqrt{29}} \hat{i} + \frac{2}{\sqrt{29}} \hat{j} \]

### Final Answer:
The unit vector perpendicular to the wall is:
\[ \hat{n} = \frac{-5 \hat{i} + 2 \hat{j}}{\sqrt{29}} \]

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