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Principles    of    Mathematics    Found    in    Cricket

In a game of cricket, there are two main points of interest where the flight of the ball is concerned. The first is the time from when the bowler releases the ball to when it is either hit or missed by the batsman. The second is the time after the collision of the ball with the bat. As the batsman's goal is to score as many runs as possible, most hits are played so that the ball is close to the ground, and is therefore harder to catch by a fieldsman. The bowler's main aim is to pitch the ball so the batsman does not hit the ball to his best ability. The flight path of the ball is such that the trajectory can be found with a simple equation. However, this does not necessarily apply to slow pitches. There is a small set of critical speeds in which pressure imbalances cause the ball to swing (deviate) to one side or the other of a bowl. These speeds are functions of several variables, including the angle of the seam, surface texture of the ball, the spin put on the ball by the bowler, and the air currents. Forces up to 30% of the weight of the ball push on the ball from the side. In a horizontal direction of motion,

m(dv/dt)=-kv2

where m is the mass of the ball, (dv/dt) is the derivative based on time, representing acceleration, and k is the side force constant. This equation is only true if the vertical motions are completely ignored. If this equation is changed to be a derivative of velocity in respect to distance rather than time, it will be:

v(dv/dx)=-(k/m)v2

where all variables remain the same, but x is the distance down the bowl that the ball is when measured. This equation can be solved to give

x=(m/k)ln(v0/v)

where ln is the natural logarithm, and v0 is the initial velocity, and all other variables remain constant. This shows the relationship of distance and velocity after a hit by the bowler. In order to find an estimate of the time of flight, separation of variables can be performed on the last equation to give

t=(m/k){(1/v)-(1/v0)}

This shows how long the ball is in the air for a particular velocity. Once each of these equations is solved using the known variable(s), the deviation of the ball from the visible path can be traced. Even the slightest variation can trick a batter's eye into missing the ball or mistiming a hit.



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Bibliography

de Mestre, Neville. The Mathematics of Projectiles in Sport. Cambridge University Press, New York: 1990. pp. 148-150.