Question

An object moves with uniform velocity $\left(V_0\right)$ on the axis of concave mirror. If it moves towards the mirror. Show that when it is at a distance u from the mirror. The velocity of the image is given by $V_t={\left(\frac{R}{2u-R}\right)}^2\cdot V_o$ where R is the radius of mirror.

Answer 1

$\frac{1}{v}+\frac{1}{-u}=\frac{1}{f}$

Here, $f=\frac{-R}{2}$ since focal length of a concave mirror is negative.

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

$⟹v=\frac{uR}{R-2u}$

$\frac{du}{dt}=V_o$ is positive as it is towards the mirror that is rightward

Differentiating both sides w.r.t t-

$\frac{dv}{dt}=\frac{V_{o}R(R-2u)-uR(-2V_o)}{(R-2u{)}^2}$

$⟹V_i=\frac{R^2}{(2u-R{)}^2}{V}_o$