Question

A point object is moving on a principle axis of a concave mirror of focal length $24cm$ towards the mirror. When it is at a distance of $60cm$ from the mirror, its velocity is $9cm/sec$. What is the velocity of the image at that instant?

• A. $5cm/sec$
• B. $12cm/sec$
• C. $4cm/sec$
• D. $9cm/sec$

Answer 1

Answer: (C)

$4cm/sec$

Given that,

Focal length $f=24cm$

Distance from the mirror$u=60cm$

Now, by using mirror formula

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}......\left(I\right)$

$\frac{1}{24}=\frac{1}{v}+\frac{1}{60}$

$\frac{1}{v}=\frac{1}{40}$

$v=40cm$

Now, on differentiating of equation (I)

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$

$0=-\frac{1}{v^2}\frac{dv}{dt}-\frac{1}{u^2}\frac{du}{dt}$

$\frac{dv}{dt}=\frac{-v^2}{u^2}\frac{du}{dt}$

$\frac{dv}{dt}=\frac{-40\times 40}{60\times 60}\times 9$

$\frac{dv}{dt}=-4cm/s$

Negative sign indicate that the velocity of the image is away from the mirror

Hence, the velocity of the image is $4cm/s$