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,
Radius of the concave mirror is R
Therefore focal length of the mirror, 
\[f = \frac{R}{2}\]
Velocity of the particle, \[V = \frac{dx}{dt}\] 
Object distance, u = −x

Using mirror equation,
`1/v + 1/u = 1/f` 
On putting the respective values we get,
`1/v + 1/-x= -2/R`
⇒ `1/v = -2/R + 1/x = ( R- 2x )/ ( Rx )`
∴ v = `(Rx) /( R - 2x`
Velocity of the image is given by V₁
`V^1= (dv)/dt = d/dt [(Rx)/ R-2x]` 

        `= [d/dx (Rx ) (R -2x)]- [d/dx ( R-2x )(Rx)]/ (R - 2x)^2`

        `= R [ (dx/dt) (R - 2x )] - [ 2dx/dt x] /( R -2x )^2`

         `=(R[(V) (R-2x)] - [ 2Vxx 0] )/( R-2x)^2`

          `= VR^2/(2x -R )^2`