Play at the Plate

It’s the bottom of the ninth; one out; and the winning run on third when the batter hits a lazy fly ball to left field. The runner leaves third base the instant the leftfielder makes the catch on the run 300 feet from home plate; the outfielder uses 0.6 seconds to release the ball on a direct path to the catcher. The speedy baserunner runs from third to home in 4.0 seconds.

How hard, and with what angle of inclination, must the ball be thrown to prevent the runner from scoring?

— based on “The Physics of Baseball”, Robert K. Adair, 1994 (pp. 58–61).

November 1997  ICTCM X (Chicago)  2

Play at the Plate

• Assumptions – the ball is released from a height of 8 feet above the ground – the throw arrives at home plate without bouncing

– the catcher needs 0.3 seconds to apply the tag

– the ball weighs approximately 5 oz.

– g = 9.8j m/sec^{2}

– magnitude of air resistance is proportional to the square of the speed

– ignore wind

– the umpire is in a good position to make the call

November 1997  ICTCM X (Chicago)  3

Play at the Plate

• Mathematical Formulation:

determine x′(0) and ∥x′(0)∥ where x is a solution to

mx′′ = −k∥x′∥x′ − mg
x(0) = 300i + 8j
x(3.1) = 0

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Play at the Plate

• Typical Results

Arm  Initial Release Flight
Strength Velocity Angle  Time Outcome
weak  90 mph 27.5^{∘} 3.85 sec
average 100 mph  18^{∘}  3.1 sec
strong 110 mph  13^{∘}  2.65 sec

Play at the Plate

• Further Discussion

– Why does the fielder try to make the catch while running towards home plate?

– Are there more efficient ways of getting the ball to the catcher?

– Suppose each outfielder’s maximum initial throwing velocity is known. Where should the players be positioned?