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