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.

Explanation

Mirror formula a can be written as follows :
`(1)/(f) = (1)/(v) + (1)/(mu)`
Substituting values we get the followings :
`" "(1)/(-f) = (1)/(v) + (1)/(-x)`
`rArr" " (1)/(v) = (1)/(x) - (1)/(f)`
`rArr" "v = (fx)/(f-x)`
Here v is the image distance and its differential will represent speed of image . On differentiating the above result.
`rArr " "(dv)/(dt) = ((f-x) xx f (dx)/(dt) - fx xx (0-(dx)/(dt)))/((f-x)^(2))`
`rArr" (dv)/(dt) = (f^(2) (dx)/(dt) - xf (dx)/(dt) + xf(dx)/(dt))/((f-x^(2))`
`rArr" "(dv)/(dt) = (f^(2) (dx)/(dt))/((f-x)^(2))`
`rArr" "(dv)/(dt) = (f^(2)V)/((f-x)^(2))`