Question

A fish rising vertically with speed 3 m/s to the surface of water sees a bird diving vertically towards it with speed 9 m/s. Given ${}_a{\mu }_w=\left(4/3\right)$. The actual velocity of dive of bird is

• A. 6 m/s
• B. 4 m/s
• C. 8.4 m/s
• D. 4.5 m/s

Answer 1

The correct option is D 4.5 m/s

$\begin{array}{c}Given,\\ Rateatwhichthefirstisri\sin g=3m/s\\ Rateatwhichthekingfisherisdiving=9m/s]\\ Refractiveindexofwater=\frac{4}{3}\\ Ifatanyins\tan t,thefishisatadepth{}^{\prime }{x}^{\prime }belowwatersurfacewhile,thebirdisataheight{}^{\prime }{y}^{\prime }abovethesurface,\\ thentheapparentheightofthebirdfromthesurfaceasseenbythefishwillbegivenby\\ \mu =\frac{Apparentheight}{Realheight}\\ or,Apparentheight=\mu y\\ So,thetotalapparentdis\tan ceofthebirdasseenbythefishinwaterwillbeh=x+\mu y\\ Now,differentiatingw.r.ttime,weget\\ \frac{dh}{dt}=\frac{dx}{dt}+\mu \frac{dy}{dt}\\ \Rightarrow 9=3+\mu \left(\frac{dy}{dt}\right)\\ \Rightarrow \frac{dy}{dt}=\frac{6}{\frac{4}{3}}=4.5m/s\end{array}$

Hence,

option $\left(D\right)$ is correct answer.