Question

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 :

Answer 1

Given object is at a distance $(u=-20cm)$ from a concave mirror of focal length $f=-15cm$.

We shall find image (v) distance using mirror formula:

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

$\Rightarrow \frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\left(\frac{1}{-15}\right)-\left(\frac{1}{-20}\right)$

$=\frac{-4+3}{60}=-\frac{1}{60}$

$\Rightarrow v=-60cm$

now consider object and image velocities be $v_{o}kvi$ respectively, where

$v_o=\frac{du}{dt}\times vi=\frac{dv}{dt}$

where $v_o=5m/s$ (given)

From mirror formula,

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

different wrt time

$-\frac{1}{v^2}\frac{dv}{dt}-\frac{1}{u^2}\frac{du}{dt}=0$ (as $f$ is constant)

$\Rightarrow -\frac{1}{v^2}{v}_1^n=\frac{1}{u^2}{v}_o=0$

$\Rightarrow v_1=-\left(\frac{v^2}{u^2}\right)v_o$

Putting values of v, u and v_o from above:

$\Rightarrow VI=-\left(\frac{{60}^2}{{20}^2}\right)\times 5=-45m/s$

$\therefore$ Speed of image $=45m/s$ ans.