Question

An object is 20cm away from a concave mirror with focal length 15cm. If the object moves witha speed of 5m/s along the axis then the speed of the image is

Answer 1

Dear Student,
Using sign conventions for concave mirror:
object distance, u = -20 cm; focal length, f = - 15cm.
From mirror formula:
$$\begin{gathered} \frac{1}{f}=\frac{1}{v}+\frac{1}{u} \\ \frac{{\displaystyle 1}}{{\displaystyle v}}=\frac{1}{f}-\frac{1}{u}=-\frac{1}{15}+\frac{1}{20}=\frac{-4+3}{60}=-\frac{{\displaystyle 1}}{60} \\ Thus,theimagedis\tan ce,v=-60cm \\ Differentiatingthemirrorformulawithrespecttotime,t \\ \frac{{\displaystyle 1}}{{\displaystyle f}}=\frac{{\displaystyle 1}}{{\displaystyle v}}+\frac{{\displaystyle 1}}{{\displaystyle u}} \\ 0=-\frac{1}{v^2}\frac{{\displaystyle dv}}{{\displaystyle dt}}-\frac{{\displaystyle 1}}{{\displaystyle u^2}}\frac{{\displaystyle du}}{{\displaystyle dt}} \\ so,\frac{{\displaystyle dv}}{{\displaystyle dt}}=-\frac{{\displaystyle v^2}}{{\displaystyle u^2}}\frac{{\displaystyle du}}{{\displaystyle dt}}=-{\left(\frac{60}{20}\right)}^2\times 5\frac{{\displaystyle m}}{s}=-45\frac{{\displaystyle m}}{s} \end{gathered}$$
Regards