A car is fitted with a convex side-view mirror of focal length 20 cm. A second car 2.8m behind the first car is overtaking the first car at a relative speed of 15 `m/s`. The speed of the image of the second car as seen in the mrror of the first one is:

Options

A. `-(1)/(15)m//s`
B. `10 m//s`
C. `15 m//s`
D. `(1)/(10) m//s`

Answer

A

Explanation

From mirror formula (using convex mirror)
`(1)/(u) + (1)/(v) = (1)/(f)` …(i)
Using `u = - 2.8 m = -280 cm, f = 20 cm`
`:. (1)/(- 280) + (1)/(v) = (1)/(20) or (1)/(v) = (1)/(20) + (1)/(280) = (15)/(280)`
or `v = (280)/(15)cm`
Differentiating (i) w.r.t `t`, we have
`-(1)/(u^(2))(d u)/(dt) - (1)/(v^(2))(dv)/(dt) = 0` or `(d v)/(dt) = -(v^(2))/(u^(2))((d u)/(dt))`
`:.` Relative velocity of image,
`v_(1) = (d v)/(dt) = -(v^(2))/(u^(2))((d u)/(dt)) = (- (280 //15^(2)))/((-280)^(2)) xx 15`
`= -(1)/(15)m//s`