# Question #dafd5

Jan 24, 2017

Let at an instant the image distance be $v$ for the real object distance $u$ during movement of a point object towards a convex mirror of focal length $F$ along its axis.

We know that the cojugate foci relation of spherical mirror is as follows.

$\textcolor{b l u e}{\frac{1}{v} + \frac{1}{u} = \frac{1}{F} \ldots \ldots \left[1\right]}$

Imposing sign convention for convex mirror
$F \to + v e \mathmr{and} u = - u \text{ for real object}$ we get equation [1] as

$\textcolor{g r e e n}{\frac{1}{v} - \frac{1}{u} = \frac{1}{F} \ldots . . \left[2\right]}$

$\textcolor{g r e e n}{\implies \frac{1}{v} = \frac{1}{u} + \frac{1}{F}}$

$\textcolor{g r e e n}{\implies \frac{1}{v} = \frac{u + F}{u F}}$

$\textcolor{g r e e n}{\implies \frac{v}{u} = \frac{F}{u + F}}$

$\textcolor{g r e e n}{\implies \frac{v}{u} = \frac{F}{u + F} < 1. \ldots . \left[3\right]}$

Now differentiating equation [2] w.r to $t$ we get

$\textcolor{v i o \le t}{- \frac{1}{v} ^ 2 \frac{\mathrm{dv}}{\mathrm{dt}} + \frac{1}{u} ^ 2 \frac{\mathrm{du}}{\mathrm{dt}} = 0. \ldots . \left[4\right]}$

$\textcolor{v i o \le t}{\implies \frac{1}{v} ^ 2 \frac{\mathrm{dv}}{\mathrm{dt}} = \frac{1}{u} ^ 2 \frac{\mathrm{du}}{\mathrm{dt}}}$

Now $\frac{\mathrm{dv}}{\mathrm{dt}} = {v}_{i} = \text{speed of image}$

And $\frac{\mathrm{du}}{\mathrm{dt}} = {v}_{o} = \text{speed of object}$

So equation [4] becomes

$\textcolor{red}{\frac{1}{v} ^ 2 \times {v}_{i} = \frac{1}{u} ^ 2 \times {v}_{o}}$

$\textcolor{red}{\implies {v}_{i} / {v}_{o} = {v}^{2} / {u}^{2.} \ldots . \left[5\right]}$

Utilising equation [3] and [5] we can say

$\textcolor{red}{{v}_{i} / {v}_{o} = {F}^{2} / {\left(u + F\right)}^{2}}$

So $\textcolor{red}{{v}_{i} < {v}_{o}} \text{ when } \left\mid u \right\mid < \left\mid F \right\mid \mathmr{and} \left\mid u \right\mid > \left\mid F \right\mid$

And $\textcolor{red}{{v}_{i} = {v}_{o}} \text{ when } \left\mid u \right\mid < \left\mid F \right\mid \mathmr{and} \left\mid u \right\mid \to 0$

Since ${\lim}_{u \to 0} \left({F}^{2} / {\left(u + F\right)}^{2}\right) = 1$

So we can support option (1) and (3) to be correct