Question

A fish rising vertically to the surface of water in a lake uniformly at the rate of\[2\dfrac{m}{s}\]. Observes a king fisher (bird) diving vertically towards the water at a rate of \[10\dfrac{m}{s}\] .If the refractive index of water
is\[\dfrac{4}{3}\]. What will be the actual velocity of the dive of the bird?

Answer 1

Hint:In this type of question we will take the given speed as relative speed and then we calculate the speed of bird but that speed is the image of bird seen by fish and after applying the relation for refractive index and velocity we will calculate the actual speed of the king fisher bird.

Complete step-by-step solution:
In this question fish is moving vertically upwards to the lake of water with velocity of \[2\dfrac{m}{s}\]and fish also observes that this king fisher bird is coming downwards vertically with speed of \[10\dfrac{m}{s}\]in air. This is the speed through which King fisher bird is observed by the fish under water so it is not the actual speed of the king fisher bird, it is the relative speed of the image of bird with respect to fish under water.
So in this question we have to calculate the actual speed of the bird.
So, firstly we will calculate the relative velocity of image of bird with respect to fish and it is given as: -
\[{{V}_{BF}}=10\dfrac{m}{s}\]
Where \[{{V}_{BF}}\]is representing the relative speed of bird with respect with fish under water.
Since both are moving in opposite direction of this relative velocity can be expressed as the sum of these two velocities.
So Velocity of image of bird becomes
\[\begin{align}
  & {{V}_{B}}=10-2 \\
 & {{V}_{B}}=8\dfrac{m}{s} \\
\end{align}\].
This is the velocity of image of bird seen by fish.
Since this velocity is seen by bird because there is water as medium in between.
So actual speed of bird will be calculated by using the relation of velocity and refractive index.
Let us assume the actual speed of bird is\[V\].
Let us assume refractive index of water can be represented as\[{{\mu }_{w}}\].
So actual speed is calculated by using the relation: -
\[V=\dfrac{{{V}_{B}}}{{{\mu }_{w}}}\]
\[\begin{align}
  & \Rightarrow V=\dfrac{8\times 3}{4} \\
 & \therefore V=6\dfrac{m}{s} \\
\end{align}\]
Since this is the actual speed of bird is \[6\dfrac{m}{s}\] as fish is in water so speed of bird is less than in comparison to that in air.

Note:Since refractive index is the property of medium which signifies the change in speed in different medium , whenever medium is present other than air then speed of light is always less as compared to speed of light in air and due to change in speed of light wavelength of light also changes with medium.