At major league baseball games, it is commonplace to flash on the scoreboard a speed for each pitch. This speed is determined with a radar gun aimed by an operator positioned behind home plate. The gun uses the Doppler shift of microwaves reflected from the baseball. The gun determines the speed at some particular point on the baseball's path, depending on when the operator pulls the trigger. Because the ball is subject to a drag force due to air proportional to the square of its speed given by R = kmv^2, it slows as it travels 18.3 m toward the plate according to the formula v = vie^?kx. Suppose the ball leaves the pitcher's hand at 88.5 mi/h = 39.6 m/s. Ignore its vertical motion. Using the following equation
R = 1/2D?Av^2 where D is the drag coefficient (0.305 for a baseball), A is the cross-sectional area (4.2 10^-3 m^2 for a baseball), ? is the density (1.20 kg/m^3 for a baseball), and v is velocity, determine the speed of the pitch when the ball crosses the plate. (Take the mass of a baseball to be 0.145 kg.)
_____m/s
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