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 __________ $\mathrm{ms}^{-1}$.

The bird's diving velocity is given relative to the fish. In order to find the actual velocity of the bird, we need to consider the refractive index of the water. <br/><br/> The fish sees the bird diving with a velocity of $12 \ \text{m/s}$. We can write the equation considering the velocities relative to the fish: <br/><br/> $\frac{V_{\text{b/f}}}{\frac{4}{3}} = \frac{-8}{\frac{4}{3}} + \frac{-v}{1}$ <br/><br/> Here, $V_{\text{b/f}}$ is the bird's diving velocity relative to the fish, and $v$ is the actual velocity of the bird. <br/><br/> Now, let's solve for $v$: <br/><br/> $\frac{-12}{\frac{4}{3}} = \frac{-8}{\frac{4}{3}} - v$ <br/><br/> $v = 3 \ \text{m/s}$ <br/><br/> So, the actual velocity of the diving bird to pick the fish relative to the fish is $3 \ \text{m/s}$.