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

Explanation

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 Given Information
- The object is moving towards a convex mirror with a constant speed of \(4 \, \text{m/s}\).
- The focal length \(f\) of the convex mirror is \(1 \, \text{m}\).
- The initial distance of the object from the mirror \(u\) is \(19 \, \text{m}\) (we will take it as negative in the mirror formula).

### Step 2: Use the Mirror Formula
The mirror formula 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 determined)
- \(u\) = object distance (negative for real objects)

### Step 3: Calculate Initial Image Distance \(v_1\)
Substituting the values into the mirror formula:
\[ \frac{1}{1} = \frac{1}{v_1} + \frac{1}{-19} \]
This simplifies to:
\[ 1 = \frac{1}{v_1} - \frac{1}{19} \]
Rearranging gives:
\[ \frac{1}{v_1} = 1 + \frac{1}{19} = \frac{19 + 1}{19} = \frac{20}{19} \]
Thus,
\[ v_1 = \frac{19}{20} \, \text{m} \]

### Step 4: Calculate Object Distance After 1 Second
After 1 second, the object moves towards the mirror:
\[ u = - (19 - 4 \times 1) = -15 \, \text{m} \]

### Step 5: Calculate New Image Distance \(v_2\)
Using the mirror formula again with the new object distance:
\[ \frac{1}{1} = \frac{1}{v_2} + \frac{1}{-15} \]
This simplifies to:
\[ 1 = \frac{1}{v_2} - \frac{1}{15} \]
Rearranging gives:
\[ \frac{1}{v_2} = 1 + \frac{1}{15} = \frac{15 + 1}{15} = \frac{16}{15} \]
Thus,
\[ v_2 = \frac{15}{16} \, \text{m} \]

### Step 6: Calculate the Average Speed of the Image
The average speed of the image can be calculated using the change in image distance over the time interval:
\[ \text{Average Speed} = \frac{v_2 - v_1}{\Delta t} \]
Where \(\Delta t = 1 \, \text{s}\):
\[ \text{Average Speed} = \frac{\frac{15}{16} - \frac{19}{20}}{1} \]

### Step 7: Find a Common Denominator
To subtract the fractions, find a common denominator (which is 80):
\[ \frac{15}{16} = \frac{75}{80}, \quad \frac{19}{20} = \frac{76}{80} \]
Thus,
\[ \text{Average Speed} = \frac{75/80 - 76/80}{1} = \frac{-1}{80} \, \text{m/s} \]

### Final Answer
The average speed of the image is:
\[ \text{Average Speed} = \frac{1}{80} \, \text{m/s} \]