An object is 20 cm away from a concave mirror with focal length 15 cm. If the object moves with a speed of 5 m/s along the axis, then the speed of the image will be

Options

A. 45 m/s
B. 27 m/s
C. 9 m/s
D. 10 m/s

Explanation

To solve the problem step by step, we will use the mirror formula and the relationship between object speed and image speed.

### Step 1: Identify the given values
- Object distance (u) = -20 cm (the object distance is taken as negative in mirror convention)
- Focal length (f) = -15 cm (concave mirrors have a negative focal length)
- Object speed (v_o) = 5 m/s

### Step 2: Use the mirror formula to find the image distance (v)
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Substituting the known values:
\[ \frac{1}{-15} = \frac{1}{v} + \frac{1}{-20} \]

### Step 3: Rearranging the equation
Rearranging the equation gives:
\[ \frac{1}{v} = \frac{1}{-15} + \frac{1}{20} \]

### Step 4: Finding a common denominator
To add the fractions, we find a common denominator. The least common multiple of 15 and 20 is 60:
\[ \frac{1}{-15} = -\frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
Thus,
\[ \frac{1}{v} = -\frac{4}{60} + \frac{3}{60} = -\frac{1}{60} \]

### Step 5: Calculate the image distance (v)
Taking the reciprocal gives:
\[ v = -60 \text{ cm} \]

### Step 6: Use the formula for image speed
The relationship between object speed (v_o) and image speed (v_i) is given by:
\[ \frac{v_i}{v_o} = \left(\frac{v^2}{u^2}\right) \]

### Step 7: Substitute the values
Substituting the values we have:
- \( v = 60 \text{ cm} \)
- \( u = 20 \text{ cm} \)
- \( v_o = 5 \text{ m/s} \)

Now substituting into the formula:
\[ v_i = v_o \cdot \left(\frac{v^2}{u^2}\right) \]
\[ v_i = 5 \cdot \left(\frac{60^2}{20^2}\right) \]

### Step 8: Calculate the image speed
Calculating the squares:
\[ 60^2 = 3600, \quad 20^2 = 400 \]
Thus,
\[ v_i = 5 \cdot \left(\frac{3600}{400}\right) = 5 \cdot 9 = 45 \text{ m/s} \]

### Final Answer
The speed of the image is \( 45 \text{ m/s} \).

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