A fish rising vertically to the surface of water in a lake uniformly at the rate of 3 m/s observes a king fisher bird diving vertically towards water at the rate 9 m/s vertically above it. If the refractive index of water is 4/3, find the actual velocity of the dive of the bird.

Solution

Given,
Rate at which the fist is rising = 3 m/s
Rate at which the king fisher is diving = 9 m/s
Refractive index of water = 4/3

If at any instant, the fish is at a depth ‘x’ below water surface while, the bird is at a height 'y' above the surface, then the apparent height of the bird from the surface as seen by the fish will be given by

μ = \frac{Apparent height}{Real height}

or, Apparent height = μy

So, the total apparent distance of the bird as seen by the fish in water will be h = x + μy
Now, differentiating w.r.t time, we get

\frac{dh}{dt} = \frac{dx}{dt} + μ \frac{dy}{dt}

\Rightarrow

9 = 3 + μ (\frac{dy}{dt})

\Rightarrow

\frac{dy}{dt} = \frac{6}{(4 / 3)} = 4.5 m / s .

Thus, the kingfisher bird is diving with a velocity of 4.5 m/s.

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Wave Optics

A fish rising vertically to the surface of water in a lake uniformly at the rate of 3 m/s observes a king fisher bird diving vertically towards water at the rate 9 m/s vertically above it. If the refractive index of water is 4/3, find the actual velocity of the dive of the bird.

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