Path+of+a+baseball

I just read a very nice article by Alexander Bogomolny about the so-called parabolic sieve of primes. In that article, he pointed out a simple fact about parabolas. If the points (-a, a^2) and (b, b^2) on the parabola y = x^2 are joined by a line segment, then the segment crosses the y-axis at (0, ab). Then I realized a remarkable consequence of this fact. If one of these points moves at a constant horizontal speed and the other point is fixed, then the y-intercept also moves at a constant speed.

This would be a very useful fact for a baseball player. Imagine that you are standing in the outfield, and the batter hits the ball right towards you. The path of the baseball is a high parabolic arc. At the beginning of the ball's flight, it is too far away for you to judge its distance. But if you are standing in the ball's flight path, then it will appear as if the ball is rising at a constant speed. Knowing this information helps you to position yourself to catch the ball.

The following GeoGebra worksheet demonstrates this concept. Point B controls the apparent horizontal distance from the outfielder to the ball, and Point A is the apparent position of the ball. The red line is a graph of the apparent height of the ball as a function of time. Notice that the ball appears to be rising at a constant rate from the outfielder's point of view. Press play or drag the red X to make things go.

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 * Addendum:** After posting this, I discovered a simple proof of this fact. Choose a coordinate system so that the outfielder is at the origin. Then the equation of the parabola has the form y = ax^2 + bx, which is equivalent to y/x = ax + b. It follows immediately that if x is changing at a constant rate, then the slope is changing at a constant rate as well.

Ball Catching: An Example of Psychologically-based Behavioural Animation by M F P Gillies and N A Dodgson describes several strategies for ball catching, based on the general idea that the fielder should keep a constant rate of increase for the tangent of the angle of elevation.