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Throwing a Baseball from the Outfield to Home Plate

How fast would an outfielder have to throw a ball for it to reach home plate without it going higher than the height of her head? We can solve the problem just by knowing the height at which the outfielder releases the ball, how tall she is, the distance to home plate, and some elementary physics.

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Danica McKellar solves a similar problem in the mathematics tutoring section of her site, where she concludes that an 8-foot-tall outfielder who released the ball from a height of 4 feet would have to throw it at an initial speed of 101.9 mph to get it to home plate 180 feet away. Nolan Ryan threw the fastest pitch on record, which was 100.9 mph, so it seems impossible that an outfielder would be able to do this.
For clarity, the vertical scale of the graphic is increased by a factor of four.
If the ball travels from initial height to maximum height , we know that its initial vertical speed is . With the initial speed, we can calculate the total time that the ball is in flight, that is, the time that it takes to rise from to and fall back down to the ground: . Then, the initial speed of the ball is a simple rate problem (distance = rate times time), .
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