A thief is running away in a car with velocity of `20 m//s`. A police jeep is following him, which is sighted by thief in his rear view mirror, which is a convex mirror of focal length 10 m. He observes that the image of jeep is moving towards him with a velocity of `1 cm//s`. if the magnification of mirror for the jeep at that time is `1/10`. Find
(a) the actual speed of jeep,
(b) rate at which magnification is changing.
Assume the police's jeep is on the axis of the mirror.

Answer

21 m/s (b) `10^(-3) m//s`

Explanation

(a) `v_(tm) = -((v^(2))/(u^(2)))v_(Jm)`
`v_(tm) = -1 "cm"//"sec" implies 1 m//s = (-(1)/(100)) v_(Jm)`
`v_(Jm) = 100 "cm"//"sec" = 1 m//sec, v_(m) = 20 m//"sec"`
`implies v_(Jm) = v_(J) - v_(m)`
`implies v_(J) = v_(Jm) + v_(m) = 1 + 20 = 21 m//"sec"`
(b) `m = (v)/(u) implies (dm)/(dt) = (d)/(dt)((f)/(u+f))`
`implies (dm)/(dt) = (f) / (u + f^(2)) xx -(du)/(dt) implies (dm)/(dt) = ((f)/(u+f))^(2) (1)/(f) xx (-(du)/(dt))`
`(dm)/(dt) = ((1)/(10))^(2) xx (1)/(10) xx (-1) = =-10^(-3)`