When an object is kept at a distance of 30cm from a concave mirror, the image is formed at a distance of 10 cm. if the object is moved with a speed of 9 cm/s, find the speed (in cm/s) with which image moves.

Answer

To solve the problem step by step, we will use the mirror formula and differentiate it to find the relationship between the speeds of the object and the image.

### Step 1: Write the Mirror Formula
The mirror formula for a concave mirror is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
where:
- \( f \) is the focal length of the mirror,
- \( u \) is the object distance (negative for concave mirrors),
- \( v \) is the image distance (positive for real images).

### Step 2: Identify Given Values
From the problem, we have:
- Object distance, \( u = -30 \) cm (negative because it's a concave mirror),
- Image distance, \( v = 10 \) cm,
- Speed of the object, \( \frac{du}{dt} = -9 \) cm/s (negative because the object is moving towards the mirror).

### Step 3: Differentiate the Mirror Formula
Differentiating the mirror formula with respect to time \( t \):
\[ \frac{d}{dt}\left(\frac{1}{u}\right) + \frac{d}{dt}\left(\frac{1}{v}\right) = 0 \]
This gives us:
\[ -\frac{1}{u^2} \frac{du}{dt} - \frac{1}{v^2} \frac{dv}{dt} = 0 \]

### Step 4: Rearranging the Equation
Rearranging the equation gives:
\[ \frac{dv}{dt} = -\frac{v^2}{u^2} \frac{du}{dt} \]

### Step 5: Substitute the Known Values
Substituting the known values into the equation:
- \( u = -30 \) cm,
- \( v = 10 \) cm,
- \( \frac{du}{dt} = -9 \) cm/s.

Now substituting these values:
\[ \frac{dv}{dt} = -\frac{(10)^2}{(-30)^2} \cdot (-9) \]
Calculating:
\[ \frac{dv}{dt} = -\frac{100}{900} \cdot (-9) = \frac{100 \cdot 9}{900} = \frac{900}{900} = 1 \text{ cm/s} \]

### Step 6: Conclusion
The speed with which the image moves is:
\[ \frac{dv}{dt} = 1 \text{ cm/s} \]