# Someone stop them

Calculus Level 5

Akul picks up a large number of sand particles and arranges them in form of a straight line. Mayank meanwhile fetches a source of light like that in lighthouse which throws light only in one direction at a time. This source is rotated with an angular velocity w as shown in the figure. Mayank stands at a distance d from line AB as shown. He starts striking light at A and continues till light strikes all the particles once. As soon as light strikes a particle, It starts moving with a constant speed v in negative y- direction as shown in figure.

In the time Mayank is striking the light, Akul is striking a ray of light as shown in the figure grazing at the glass slab.

Mayank obtains a graph of final shape formed by particles by taking the first particle as origin and taking axes as shown in diagram while Akul obtains a graph of path of light taking origin at point of incidence. They see that the graph obtained by both is same.

Find variation of refractive index n as f(x) if it is known that for any particular x, n does not depend on y.

n=$$((\sum_{i=1}^m a_{i}*x^{i})+a_{0})^{1/k}$$

Give your answer as $$(\sum_{i=1}^m a_{i}*i)+a_{0}+k$$

Given

• v = 500 cm/s

• d = 1m

Details and Assumptions

• The sand particles are ideally very small and all the figures formed are to be treated as continuous systems.

• Particles don't interfere in each other's way if they don't collide ( There's nothing called friction).

• By "grazing" I mean angle of incidence is $$\frac { \pi }{ 2 }$$.

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