RAIN, UMBRELLA,AND VECTORS

First, we must know about relative velocity. Say, you are travelling in a express train at the speed of 100km/hour. In the next track another train is moving at the speed of 120km/hour. Now subtract your speed (100) from 120 km/hour. You get 20km/hour. That means, another train is running at the speed of just 20km/hour with respect to you. Hence it will apear to go slowly for you.

Say, rain is falling vertically at the speed of 4 kmph. Draw a 4 cm arrow pointing downwards. It is the vector representing rain's speed.
You are fast walking at the speed of 6 kmph towards east. Draw 6 cm arrow from the head of rain vector representing your speed. To find the relative velocity, we have to subtract. Hence reverse your vector (-6cm) towards west side. If we now add both the vectors, we will get the relative velocity.

To add, complete the triangle, the third side gives the result of addition.(triangular law of vector addition).The length of the third side is found to be 7.2 cm and it makes an angle 33.7 with the horizontal. It means, the rain is falling at the increased speed of 7.2 kmph slantingly at an angle 33.7 degrees in your view. When you run in the rain, it will splash in your face. If you hold the umbrella at the proper angle, you can save your face.
For the same reason, the rain falls slantingly across the train's glass window. Also the boat appears to go at an angle, when it crosses a flowing river.

This is the one use of vectors. When a quantity has a direction, we go for vectors. Geometry is the basis of vectors. But one can solve problems just using formulas.

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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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