A plane mirror is moving with velocity `4 (hat i) + 4 (hat j) + 8 (hat k)`. A point object in front of the mirror moves with a velocity `3 (hat i) + 4 (hat j) + 5 (hat k)`. Here, `hat k` is along the normal to the plane mirror and facing towards the object. The velocity of the image is

Options

A. `-3hat(i) - 4 hat(j) + 5 hat(k)`
B. `3 hat(i) + 4hat(j) + 11 hat(k)`
C. `-3hat(i) - 4 hat(j) + 11 hat(k)`
D. `7hat(i) + 9 hat(j) + 11 hat(k)`

Answer

To find the velocity of the image when a plane mirror is moving and a point object is also moving, we can follow these steps:

### Step 1: Understand the velocities
- The velocity of the mirror \( \vec{V_m} = 4 \hat{i} + 4 \hat{j} + 8 \hat{k} \).
- The velocity of the object \( \vec{V_o} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \).

### Step 2: Identify the normal direction
Since the \( \hat{k} \) direction is along the normal to the mirror and facing towards the object, we note that only the component of velocity along the normal (the \( \hat{k} \) component) will change upon reflection.

### Step 3: Determine the change in the normal component
The normal component of the object's velocity is \( 5 \hat{k} \). When the object moves towards the mirror, the image's normal component will be affected by the mirror's motion.

### Step 4: Calculate the new normal component
The velocity of the image \( \vec{V_i} \) can be calculated using the formula:
\[ \vec{V_i} = \vec{V_o} + 2\vec{V_m} \]
However, since the direction of the normal component reverses, we can express this as:
\[ V_{i,k} = -V_{o,k} + 2V_{m,k} \]
Substituting the values:
\[ V_{i,k} = -5 + 2 \times 8 = -5 + 16 = 11 \]
So, the \( \hat{k} \) component of the image's velocity is \( 11 \hat{k} \).

### Step 5: Combine the components
The \( \hat{i} \) and \( \hat{j} \) components of the image's velocity remain unchanged:
- \( V_{i,i} = 3 \hat{i} \)
- \( V_{i,j} = 4 \hat{j} \)

### Step 6: Write the final velocity of the image
Combining all components, we get:
\[ \vec{V_i} = 3 \hat{i} + 4 \hat{j} + 11 \hat{k} \]

### Final Answer
The velocity of the image is:
\[ \vec{V_i} = 3 \hat{i} + 4 \hat{j} + 11 \hat{k} \]

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