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

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Let the fish be at a depth x from the surface and let the bird be at height y from the surface. So, with respect to fish y ' = μ y + x ⇒ y ' = 4 y 3 + x Differentiating with respect to time, d y ' d t = 4 d y 3 d t + d x d t ⇒ 4 y 3 + d x d t = - 12 ⇒ 4 3 · d y d t = - 12 + 8 = - 4 ⇒ d y d t = - 3 m s - 1

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