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 mirror. Here’s a step-by-step solution:

### Step 1: Understand the setup
- The plane mirror is placed 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) and has an initial velocity of \( \vec{V_o} = 2\hat{i} + 2\hat{j} \) 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 (2m, 0, 0), the image will be at (-2m, 0, 0).

### Step 3: Determine the velocity of the image
- The velocity of the image (\( \vec{V_i} \)) is equal in magnitude but opposite in direction to the velocity of the object. Therefore, the velocity of the image is:
\[ \vec{V_i} = -\vec{V_o} = - (2\hat{i} + 2\hat{j}) = -2\hat{i} - 2\hat{j} \text{ m/s} \]

### Step 4: Calculate the relative velocity of the image with respect to the object
- The relative velocity of the image with respect to the object (\( \vec{V_{rel}} \)) is given by:
\[ \vec{V_{rel}} = \vec{V_i} - \vec{V_o} \]
Substituting the values:
\[ \vec{V_{rel}} = (-2\hat{i} - 2\hat{j}) - (2\hat{i} + 2\hat{j}) = -2\hat{i} - 2\hat{j} - 2\hat{i} - 2\hat{j} \]
\[ \vec{V_{rel}} = -4\hat{i} - 4\hat{j} \text{ m/s} \]

### Step 5: Determine the direction of the relative velocity
- The relative velocity vector \( \vec{V_{rel}} = -4\hat{i} - 4\hat{j} \) indicates that the relative velocity has components in both the negative x-direction and the negative y-direction. However, since the question asks for the direction of the relative velocity of the image with respect to the object, we focus on the x-component, which is negative.

### Conclusion
- The relative velocity of the image with respect to the object is directed along the negative x-axis.