A plane mirror is placed at origin parallel of y-axis, facing the positive x-axis. An object starts from (2m, 0, 0) with a velocity of `(2hat(i)+2hat(j))m//s`. The relative velocity of image with respect to object is along

Options

A. positive x-axis
B. negative x-axis
C. positive y-axis
D. negative y-axis

Answer

To solve the problem, we need to analyze the motion of the object and its image in relation to the plane mirror. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the setup
- The plane mirror is located at the origin (0, 0) and is parallel to the y-axis, facing the positive x-axis.
- The object starts at the point (2m, 0, 0) with a velocity of \( \mathbf{v_o} = (2\hat{i} + 2\hat{j}) \, \text{m/s} \).

### Step 2: Determine the position of the image
- The image of the object in a plane mirror is located at an equal distance behind the mirror. Since the object is at (2, 0), the image will be at (-2, 0).

### Step 3: Find the velocity of the image
- The velocity of the image is equal in magnitude but opposite in direction to the velocity of the object. Therefore, if the object has a velocity \( \mathbf{v_o} = (2\hat{i} + 2\hat{j}) \), the velocity of the image \( \mathbf{v_i} \) will be:
\[ \mathbf{v_i} = -\mathbf{v_o} = (-2\hat{i} - 2\hat{j}) \, \text{m/s} \]

### Step 4: Calculate the relative velocity of the image with respect to the object
- The relative velocity \( \mathbf{v_{rel}} \) of the image with respect to the object is given by:
\[ \mathbf{v_{rel}} = \mathbf{v_i} - \mathbf{v_o} \]
Substituting the values:
\[ \mathbf{v_{rel}} = (-2\hat{i} - 2\hat{j}) - (2\hat{i} + 2\hat{j}) = -4\hat{i} - 4\hat{j} \]

### Step 5: Determine the direction of the relative velocity
- The relative velocity \( \mathbf{v_{rel}} = -4\hat{i} - 4\hat{j} \) indicates that the relative velocity is directed towards the negative x-axis and negative y-axis.

### Conclusion
- The relative velocity of the image with respect to the object is directed along the line that makes a 45-degree angle with the negative x-axis, pointing towards the third quadrant.

### Final Answer
The relative velocity of the image with respect to the object is along the direction of \( -\hat{i} - \hat{j} \).

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