A ball is approaching a convex mirror of focal length 30 cm with speed 20 `m//s`. Calculate the speed of its image when the ball was at 5 m from the mirror?

Explanation

We have mirror equation
`1/f=1/u+1/v`
`1/v=1/f-1/u`
`v=(fu)/(u-f)`
Differentiate w.r.t .t.
`(dv)/(dt)=(f)/((u-f))(du)/(dt)-(fu)/((u-f)^2)(du)/(dt)`
`(dv)/(dt)=(du)/(dt)[(f)/(u-f)-(fu)/((u-f)^2)]`
Here: `(du)/(dt)= 20ms^(-1)`, f = 30cm=0.3m
u = -5m
`(dv)/(dt)=20[(0.3)/(-5-0.3)-(0.3(-5))/((-5-0.3)^2)]`
`=20[(90.3)/(-5.3)+(1.5)/((-5.3)^2)]`
= 20[-0.0566 + 0.0534]
`(dv)/(dt)=20[-0.0032]= -0.064 ms^(-1)`