Catching a ball is a simple matter of differential calculus

Differential calculus is used to calculate rates of change. I think it only applies to the physical world, not political change. In fact, I don’t think there are any rules that apply to politics in spite of the fact that some people claim there is such a thing as political science. I think that is stretching the term science which, I guess, is pretty much what politicians do anyway. Stretch the meaning of words, I mean.

I finally understood differential calculus — sort of — when I compared it to baseball, the only sport I ever really played. I played in the Old Timers League on the Pepsi Cola team with Tex Tolman. Tex was a lefty pitcher. When he pitched, I had to play third base because I could sometimes catch the ball when he tried to pick runners off at third. I hated that. 

Anyway, when a ball is thrown into the air, its vertical speed is slowed by gravity until it reaches a point where gravity is stronger than its vertical motion. Then the ball begins to fall, accelerating the vertical speed at approximately the same rate as during the ascent. If you divide the ascending arc into intervals, you can calculate the rate of decline in vertical motion and predict the end point of the arc. See, if you can catch a ball, you can do calculus!

Actually this same calculation can also be used to determine when a class will completely finish taking a multiple choice exam. Simply set the timer when the test begins. Then check the time when the first student finishes. In half of the time it took the first student to finish, half the rest of the class will finish the test. In half of that time again, half of the remaining students will have finished. This pattern continues until the last, irritating, overachieving, conscientious “A” student takes the remaining time in the period, blows differential calculus right out of the water and makes you both late for lunch. 

However, there is a catch when calculating baseballs (pun intended). If you are trying to catch a ball, you do not see the arc from the side, but see it from straight on. So how do you calculate the change in altitude? It turns out that our bodies calculate it automatically from the degree of angle at which we tilt our heads to watch the ball. 

When a ball is tossed to little children, they usually miss it. This is often because they have their arms wide open. Children also tend to shut their eyes when they see the ball coming. This makes catching the ball more difficult. So we teach children to “keep their eye on the ball.” To do so they have to tilt their head.

With practice, we learn that the speed and degree of head tilting correlates with where we should be to catch the ball. This works very well up to about 30-degree angles. So when our eyes follow the ball, the tilt of head changes. The rate of change going up helps us predict the rate of change coming down. 

When the arc is higher, the rate of change appears to slow down in the final stages because of the steep angle at which the ball is declining. The rate of head tilt slows down, so we think the ball is approaching more slowly than it is. This is why high fly-balls are harder to catch than shallower arcs. 

Apparently we use a similar mechanism for tracking balls we have to catch on the run. This also seems to involve the rotation of our heads to the side at the same time we are adjusting the angle of tilt. But all that changes with speed, so we have to adjust speed continually as we adjust the angle and tilt of our head. I don’t think I really understand that part of differential calculus.

Gary McCallister, .(JavaScript must be enabled to view this email address), is a professor of biology at Colorado Mesa University.


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