T HE DROWNING CHILD , LIFEGUARD AND S NELL ’ S LAW Consider a rectangular swimming pool PQSR; see figure here. A lifeguard sitting at G outside the pool notices a child drowning at a point C. The guard wants to reach the child in the shortest possible time. Let SR be the side of the pool between G and C.
Should he/she take a straight line path GAC between G and C or GBC in which the path BC in water would be the shortest, or some other path GXC? The guard knows that his/her running speed v on ground is higher than his/her swimming speed v . Suppose the guard enters water at X. Let GX = l and XC = l .
Then the time taken to reach from G to C would be l l t v v To make this time minimum, one has to differentiate it (with respect to the coordinate of X ) and find the point X when t is a minimum. On doing all this algebra (which we skip here), we find that the guard should enter water at a point where Snell’s law is satisfied. To understand this, draw a perpendicular LM to side SR at X. Let GXM = i and CXL = r .
Then it can be seen that t is minimum when sin sin v i r v In the case of light v /v , the ratio of the velocity of light in vacuum to that in the medium, is the refractive index n of the medium. In short, whether it is a wave or a particle or a human being, whenever two mediums and two velocities are involved, one must follow Snell’s law if one wants to take the shortest time. This is called total internal reflection . When light gets reflected by a surface, normally some fraction of it gets transmitted.
The reflected ray, therefore, is always less intense than the incident ray, howsoever smooth the reflecting surface may be. In total internal reflection,