A is a thin walled sphere at rest made up of glass. The radius of the sphere is 1m and it is filled with a transparent liquid of refractive index μ . S is the luminous source moving directly towards a

Question

A is a thin walled sphere at rest made up of glass. The radius of the sphere is 1m and it is filled with a transparent liquid of refractive index μ . S is the luminous source moving directly towards a plane vertical mirror M. A fish in the liquid is moving towards S. The eye of the fish and S are collinear & perpendicular to mirror M. At the moment S is 3m away from the centre of the sphere fish observes that image of S due to reflection is moving with speed of 13m/s. If speed of the fish relative to the sphere is 10 m/s and μ = 1.5 then find the speed (in m / s ) of the source at that instant. The system is placed in air.

TestHub · Accepted Answer

Refraction at the sphere surface (fish viewing S): $\frac{1.5}{v} + \frac{1}{6} = \frac{1.5 - 1}{1} \Rightarrow v = 4.5$ m.

Image speed from longitudinal magnification: $v_{I} = \frac{v ^{2}}{u ^{2}} \cdot \frac{v _{0}}{μ} = \frac{4. 5 ^{2}}{6 ^{2}} \cdot \frac{v _{0}}{1.5} = 3$ m/s.

Including the mirror reflection doubling and the fish's own 10 m/s approach: $v_{0} = \frac{3 \times 3}{2} \times \frac{36 \times 4}{9 \times 9} = 8$ m/s

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