A plane mirror is fixed at the bottom of a vessel as shown in the figure. Water of refractive index `mu` is filled up to a height h in the vessel. There is a fish at a height `h//2` from bottom of vessel and there is a bird at a height h above the surface of water in the vessel.

(a) Calculate apparent distance of bird as seen by the fish.
(b) Calculate apparent distance of image of bird in mirror as seen by the fish.
(c) Calculate apparent distance of fish as seen by the bird.
(d) Calculate apparent distance of image of fish in the mirror as seen by the bird.

Explanation

First of all we should know the following formula :
`"apparent depth" =("real depth")/(mu)`
Here `mu` is the refractive index of medium of object with respect to medium of observer. Hence if `mu` is refractive index of water with respect to air then `1//mu` will be the refractive index of air with respect to water.
As a next step for ease of calculation we can first reflect images from plane mirror without considering shift due to changing medium and then separately we can consider the shift.

(a) Apparent distance of bird as seen by the fish
`=(h)/(2)+(h)/(1//mu)=(h)/(2)+muh=h((1)/(2)+mu)`
(b) Apparent distance of image of bird as seen by the fish `=(3h)/(2)+(h)/(1//mu)=(3h)/(2)+muh=h((3)/(2)+mu)`
Note that image of water is also treated as water.
(c ) Apparent distance of fish as seen by the bird
`=h+(h//2)/(mu)=h(1+(1)/(2mu))`
(d) Apparent distance of image of fish as seen by the bird `=h+(3h//2)/(mu)=h(1+(3)/(2mu))`