LIGHT RAY FINDS PATH OF LEAST TIME WITHOUT GOOGLE MAP

We know, when a ray of light travels from air to water, it bends according to snell's law. Why should it bend? Why should it follow snell's law?

Say, you are coast guard. Some one cries for help from the sea. You want to reach the person as quickly as possible. Here you have to travel in two media. 1. You have to run in the land (fast). 2. You have to swim in the sea (slow). Hence you need to find the path of shortest time. So you will choose to run a maximum distance on the sand; will reach a particular point on the edge of water; than will swim towards the person in distress.
That path will never be a straight line but looks bent.

Similarly, a ray of light travels faster in air, but slower in water and glass. Hence the light takes a bent path while crossing from air to glass. This is the principle of least time. Also called Fermat's principle.
(Note: When a ray hits a water surface normally (at 90 degree), it will not bend but go straight. (In this case, there is no other path which has 'more least time' than straight line). Next why it should follow snell's law?

sin i/sin r = Refractive index of water = Speed of light in air / speed of light in water- the law
i is angle of incidence and r is angle of refraction as shown in the figure.
The angles depends on the speeds of light in the two media. Hence the speeds determine the angle of bending. Consequently the light ray takes the path of least time.
How the light sense the path of least time? It is because of the wave nature of light. The spherical wave spread from a source of light (a lamp) like a water wave on the sheet of water. These waves senses the nearby paths and finds the quickest path. This principle has been experimentally verified using Radio waves.
There can be many explanations for refraction. This is another way of looking at refraction. Also we get one more idea of 'how the nature works.'

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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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