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

(a)`21m//s`,`(b)1xx10^(-3)//"sec"`

Explanation

(a) `(1)/(v)+(1)/(u)=(1)/(f)`
differently the equation with respect to t
`-(1)/(v^(2))(dv)/(dt)-(1)/(u^(2))(dv)/(dt)=0 rArr (dv)/(dt)=-(v^(2))/(u^(2))(du)/(dt)`
`(dv)/(dt)=-m^(2)(du)/(dt) rArr (dv)/(dt)=-((1)/(10))^(2)xx(du)/(dt)`
`-0.01=-(1)/(100)(du)/(dt) rArr (du)/(dt)=+1m//s`
`vecv_(0//m)=vec(v)_(0)-vec(v)_(m) rArr vec(v)_(0)=vecv_(0//m)+vec(v)_(m)`
`vec(v)_(0)=1+20rArr vec(v)_(0)=21m//s`
(b)`m=(f)/(f-u)rArr (dm)/(dt)=-(f)/((f-u)^(2))xx-((du)/(dt))`
`(dm)/(dt)=(f)/((f-u)^(2))(du)/(dt) rArr(1)/(f)((f)/(f-u))^(2)xx(du)/(dt)`
`(dm)/(dt)=(1)/(f)xxm^(2)xx(du)/(dt) rArr(dm)/(dt)=(1)/(10)((1)/(10))^(2)xx1`
`(dm)/(dt)=1xx10^(-3)"per"//"sec"`