Question

A point object is moving with speed $V_0$ towards a concave mirror with radius of curvature $R$, along the principal axis. The speed of the image is not defined when the object is :

• A. at focus
• B. at pole
• C. in the middle of line joining focus and centre of curvature
• D. at centre of curvature

Answer 1

The correct option is A at focus

We know from mirror formula that,
$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}=\frac{2}{R}$
Using sign convention for concave mirror :
$\frac{1}{v}+\frac{1}{-u}=\frac{1}{-f}=\frac{2}{-R}$

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

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

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

And magnification $\left(m\right)$ is given by
$m=\frac{-v}{u}$
$\Rightarrow m=\frac{\frac{-Ru}{R-2u}}{-u}=\frac{R}{R-2u}$

For motion of the object along the principal axis,
velocity of image, $V_i=-m^2{V}_o$

$\Rightarrow V_i=-{\left(\frac{R}{R-2u}\right)}^2{V}_o.........\left(1\right)$

Eq.(1) is not defined when $R-2u=0$
$\Rightarrow u=\frac{R}{2}$ [at focus]

Hence, when the object is at focus, the speed of the image is not defined.