Question

The speed at which the image of a luminous point object is moving, if the luminous point object is moving at speed $v_0$ towards a spherical mirror, along its axis is:
(Given, $R=$ radius of curvature & $u=$ object distance)

• A. $v_i=-v_0$
• B. $v_i=-v_0{\left[\frac{R}{2u-R}\right]}^2$
• C. $v_i=-v_0\left[\frac{2u-R}{R}\right]$
• D. $v_i=-v_0\left[\frac{R}{2u-R}\right]$

Answer 1

The correct option is B $v_i=-v_0{\left[\frac{R}{2u-R}\right]}^2$

Given:
$R=$Radius of curvature,
$v_o=$ Speed of object
and $u=$Object distance

Now using mirror formula,
$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

$\Rightarrow \frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\frac{2}{R}-\frac{1}{u}=\frac{2u-R}{Ru}$

$\Rightarrow v=\frac{uR}{2u-R}$

[with sign convention]

So, speed of image
$v_i=-m^2\times v_o$
$\Rightarrow v_i=-{\left(\frac{-v}{u}\right)}^2{v}_o$
$\Rightarrow v_i=-v_o{\left(\frac{R}{2u-R}\right)}^2$
Hence, option $\left(b\right)$ is correct.