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$$^{-1}$$.

Using apparent position relation: $$y_{\text{apparent}} = \mu y$$

Total apparent distance observed by fish: $$s = x + \mu y$$

Differentiating with respect to time for velocities:

$$\frac{ds}{dt} = \frac{dx}{dt} + \mu \frac{dy}{dt}$$

$$12 = 8 + \frac{4}{3} v_{\text{bird}}$$

$$4 = \frac{4}{3} v_{\text{bird}} \implies v_{\text{bird}} = 3\text{ m s}^{-1}$$