ADDING A NEW ROAD MAY NOT EASE TRAFFIC

We always think that adding a new road will ease traffic congestion. On the contrary, it may create a traffic jam. Let us see this paradox in detail.

    Look at the above diagram carefully.  There are two roads to go from start to end.  One via A and another via B.   First road has a narrow bridge between ‘start’ and A.  The second road has narrow bridge between B and 'End'.
     The time taken to cross the narrow bridge depends on number of vehicles.  More cars mean more time.  The time taken to cross the bridge is given by N/100, where N is the number of vehicles crossing the bridge.  Hence, the time taken to go from "start to A or B to end" is N/100 because of bridges.
    The time taken to go from A to End or start to B (hilly roads) is always 45 minutes irrespective of the traffic.

On a fine morning, 4000 vehicles try to go from start to end. Naturally, (Probability theory) half of them will go via A and another half (2000) of them will go via B.
The time taken for the drivers going via A
= 2000/100 +45 here N=2000
= 20+45 =65 minutes.
The time for the drivers going via B = 2000/100 +45 = 65 minutes. refer figure
Hence both the roads require 65 minutes time. So far, so good.

    Now the national highways department construct a new road from A to B to reduce congestion and time.  The time taken to go from A to B through the new road is practically zero.
    The highways department thinks in these lines: "suppose 1000 cars start from 'start'.  They will reach A in 1000/100 =10 minutes.
Then they will reach B in no time. From B to end, it will again take 1000/100 =10 minutes.  Totally, it will take only 20 minutes for the drivers to go from start to end", a good reduction.

But now, there is a new short route, everybody wants to use this route. Suppose 4000 drivers take the same route. What will happen? The time of travel will be 4000/100+4000/100=40 +40 =80 minutes. The time has increased from 65 to 80 minutes. Hence the new road from A to B, only increased the congestion. So, it is better to remove the new road. This is called Brass's paradox.

In Seoul, south Korea, the congestion eased, the traffic speeded up. When a motorway was removed as part of the restoration project.
In 2009, Newyork, experimented with closures of Broadway at times square and Herald Square which resulted in improved traffic flow and permanent pedestrian plazas.

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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

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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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