6.0 Motion of object and image

6.1 Graph between $\left( {\frac{1}{v}} \right)$ and $\left( {\frac{1}{u}} \right)$

Mirror formula is given by, $$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$

Let us differentiate the above formula wrt $t$ we get, $$ - \frac{1}{{{v^2}}}\frac{{dv}}{{dt}} - \frac{1}{{{u^2}}}\frac{{du}}{{dt}} = \frac{{df}}{{dt}}$$ As $f$ is constant, $$\begin{equation} \begin{aligned} - \frac{1}{{{v^2}}}\frac{{dv}}{{dt}} - \frac{1}{{{u^2}}}\frac{{du}}{{dt}} = 0 \\ \frac{{dv}}{{dt}} = - \left( {\frac{{{v^2}}}{{{u^2}}}} \right)\frac{{du}}{{dt}} \\\end{aligned} \end{equation} $$ where,

$\frac{{du}}{{dt}}$: Rate of change of $u$ or speed of the object when mirror is stationary

$\frac{{dv}}{{dt}}$: Rate of change of $v$ or speed of the image when mirror is stationary

$u$: Object distance from pole

$v$: Image distance from pole