Question

A particle is moving at a constant speed V from a large distance towards a concave mirror of radius R along its principal axis. Find the speed of the image formed by the mirror as a function of the distance x of the particle from the mirror.

Answer 1

Given $\frac{dx}{dt}=V$

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

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

$\frac{1}{v}=\frac{1}{x}-\frac{2}{R}=\frac{R-2x}{xR}$

$v=\frac{xR}{R-2x}$

$\frac{dv}{dt}=\frac{\left(R\frac{dx}{dt}\right)(R-2x)+xR\left(2\frac{dx}{dt}\right)}{(R-2x{)}^2}$

$\frac{dv}{dt}=\frac{R^2\frac{dx}{dt}}{(2x-R{)}^2}=\frac{R^{2}V}{(2x-R{)}^2}$