GEO- POSITIONING SYSTEM

One fine morning, you wake up in the middle of dense forest. You do not know where you are? But luckily, you have a FM radio set. Tune the radio, you hear one station feebly. You know the location of that radio station. The maximum range of that FM station is 100 km (say). You must be at the edge of 100 km since the FM signals are weak. Now, you draw a circle with that radio station as the center and 100 km as radius. You must be somewhere on the edge of the circle.

Now try for another station. Luck favors you again. You get second station that too feebly. Now this station as center, you draw second circle of same radius. Now the two circles meet at two points. You may be on any one of the point. Still the information is not enough. Try your luck again. The God of luck is still with you. You get third FM station some what clearly. Draw third circle as usual. The three circles meet at one point. Thank God, that is your location. This is known as 'triangular method'. This technique is used in Geo-positioning system . (GPS)

There are 31 satellites in GPS orbits now. A minimum of 4 satellites will be in contact with GPS receiver you have. Each will send the signals to your receiver and get it back in milliseconds. Then the satellite calculate the distance between you and itself using the time. Next it will draw an imaginary sphere around it using calculated distance as radius. (No circles in space, only spheres)

You can be anywhere on the surface of the sphere. Each satellite will draw its own calculated sphere. Earth sphere can also ,be taken into account. Using the triangulation method you will be located accurately by GPS. That is, all the spheres will meet at one point where you are exactly located.

GPS now have accuracy of 5 meters. It will come down to centimeters in future.
GPS is the boon to modern day navigation.

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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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