A fish rising vertically upward with a uniform velocity of \(8 \mathrm{~ms}^{-1}\), observes that a bird is diving vertically downward towards the fish with the velocity of \(12 \mathrm{~ms}^{-1}\). If the refractive index of water is \(\frac{4}{3}\), then the actual velocity of the diving bird to pick the fish, will be ______ \(\mathrm{ms}^{-1}\).

JEE Main · 2023 · 13 Apr 2023 · Shift 1 · Geometrical Optics

A fish rising vertically upward with a uniform velocity of

8 ms^{- 1}

, observes that a bird is diving vertically downward towards the fish with the velocity of

12 ms^{- 1}

. If the refractive index of water is

\frac{4}{3}

, then the actual velocity of the diving bird to pick the fish, will be ______

ms^{- 1}

Answer: 3

Seen from inside the water, a length in air is magnified by the refractive index, so the apparent vertical speed of the bird is

μ

times its real speed. The fish rises through the water at its true speed

8 m/s

, while the bird's image descends at

μv

, where

v

is the bird's actual speed. The speed at which the fish sees the gap closing is the sum of the two:

8 + μv = 12.

With

μ = \frac{4}{3}

\frac{4}{3} v = 4 \Rightarrow v = 3 m/s .

The bird is actually diving at

3 m/s

JEE Main · 2023 · 13 Apr 2023 · Shift 1 · Integer answer