An object moving at constant speed of 4m/s towards a convex mirror of focal length 1 m is at a distance of 19m. The average speed of the image is .

Options

A. `(1)/(20)` m/s
B. `(1)/(80) `m/s
C. 20 m/s
D. 80 m/s

Answer

To find the average speed of the image formed by a convex mirror when an object is moving towards it, we can follow these steps:

### Step 1: Understand the problem
We have an object moving towards a convex mirror at a constant speed of 4 m/s. The initial distance of the object from the mirror is 19 m, and the focal length of the convex mirror is 1 m. We need to find the average speed of the image formed by the mirror as the object approaches it.

### Step 2: Use the mirror formula
The mirror formula for a convex mirror is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) = focal length of the mirror (positive for convex mirrors)
- \( v \) = image distance (to be found)
- \( u \) = object distance (negative as per sign convention)

Given:
- \( f = +1 \) m
- \( u = -19 \) m (since the object is in front of the mirror)

### Step 3: Calculate the initial image distance
Substituting the values into the mirror formula:
\[ \frac{1}{1} = \frac{1}{v} + \frac{1}{-19} \]
This simplifies to:
\[ 1 = \frac{1}{v} - \frac{1}{19} \]
Rearranging gives:
\[ \frac{1}{v} = 1 + \frac{1}{19} = \frac{19 + 1}{19} = \frac{20}{19} \]
Thus, the image distance \( v \) is:
\[ v = \frac{19}{20} \text{ m} \]

### Step 4: Calculate the new object distance after 1 second
In 1 second, the object moves towards the mirror by 4 m. Therefore, the new object distance \( u' \) is:
\[ u' = - (19 - 4) = -15 \text{ m} \]

### Step 5: Calculate the new image distance
Using the mirror formula again with the new object distance:
\[ \frac{1}{f} = \frac{1}{v'} + \frac{1}{u'} \]
Substituting the values:
\[ \frac{1}{1} = \frac{1}{v'} + \frac{1}{-15} \]
This simplifies to:
\[ 1 = \frac{1}{v'} - \frac{1}{15} \]
Rearranging gives:
\[ \frac{1}{v'} = 1 + \frac{1}{15} = \frac{15 + 1}{15} = \frac{16}{15} \]
Thus, the new image distance \( v' \) is:
\[ v' = \frac{15}{16} \text{ m} \]

### Step 6: Calculate the shift in image position
The shift in image position as the object moves from the initial position to the new position is:
\[ \text{Shift} = v' - v = \frac{15}{16} - \frac{19}{20} \]
To calculate this, find a common denominator (80):
\[ \text{Shift} = \left(\frac{15 \times 5}{80}\right) - \left(\frac{19 \times 4}{80}\right) = \frac{75}{80} - \frac{76}{80} = -\frac{1}{80} \text{ m} \]

### Step 7: Calculate the average speed of the image
Since the object moves towards the mirror at 4 m/s for 1 second, the average speed of the image is equal to the shift in image position:
\[ \text{Average speed of the image} = \frac{\text{Shift}}{\text{Time}} = \frac{-\frac{1}{80}}{1} = -\frac{1}{80} \text{ m/s} \]
The negative sign indicates that the image is moving in the opposite direction to the object.

### Final Result
The average speed of the image is:
\[ \frac{1}{80} \text{ m/s} \]