A spherical mirror from image at distance 10 cm of an object at distance 30 cm from mirror. If object start moving with velocity 9cm/sec. Find velocity of image

Options

A. -9cm
B. -4cm
C. -1cm
D. -3cm

Answer

To solve the problem, we need to find the velocity of the image when the object is moving towards the mirror. We will use the mirror formula and the concept of differentiation to find the relationship between the velocities of the object and the image.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Object distance (u) = -30 cm (the negative sign indicates that the object is in front of the mirror)
- Image distance (v) = 10 cm (positive for real images formed by concave mirrors)
- Velocity of the object (du/dt) = 9 cm/s (the object is moving towards the mirror, so this value is positive)

2. **Use the Mirror Formula:**
The mirror formula is given by:
\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
where \( f \) is the focal length of the mirror. However, we will differentiate this formula directly without needing the focal length.

3. **Differentiate the Mirror Formula:**
Differentiate the equation with respect to time (t):
\[ \frac{d}{dt}\left(\frac{1}{v}\right) + \frac{d}{dt}\left(\frac{1}{u}\right) = 0 \]
Using the chain rule, we have:
\[ -\frac{1}{v^2} \frac{dv}{dt} - \frac{1}{u^2} \frac{du}{dt} = 0 \]

4. **Rearranging the Equation:**
Rearranging gives us:
\[ \frac{dv}{dt} = -\frac{v^2}{u^2} \frac{du}{dt} \]

5. **Substituting the Known Values:**
Substitute \( u = -30 \) cm, \( v = 10 \) cm, and \( \frac{du}{dt} = 9 \) cm/s into the equation:
\[ \frac{dv}{dt} = -\frac{(10)^2}{(-30)^2} \cdot 9 \]
\[ \frac{dv}{dt} = -\frac{100}{900} \cdot 9 \]
\[ \frac{dv}{dt} = -\frac{1}{9} \cdot 9 \]
\[ \frac{dv}{dt} = -1 \text{ cm/s} \]

6. **Conclusion:**
The velocity of the image (dv/dt) is -1 cm/s, indicating that the image is moving away from the mirror.

### Final Answer:
The velocity of the image is -1 cm/s.