We’re back with another installment of Circus Science! This time we are talking about juggling. Everybody knows at least one person who can juggle; they learned it as a party trick or at summer camp when they were a kid. They can keep up a standard three-ball cascade for a few throws, but the throws are all over the place. You go to the circus and see a professional juggler, and they also perform a three-ball cascade. But there’s a difference between the way your friend and the professional are throwing the balls; the professional’s throws are evenly spaced, while your friend’s are sporadic and often thrown way higher than necessary. Even though the only things that are the same between these two patterns are the number of balls and the number of hands throwing the balls, the patterns are the same…mathematically speaking. Let’s take a closer look at why this phenomenon is true.

Claude Elwood Shannon was an American mathematician and engineer from the twentieth century. An extremely respected figure in the science community and crusader of the Digital Age, Shannon was also a juggler. He investigated relationships in juggling patterns, which he wrote about in his 1993 paper, Scientific Aspects of Juggling. He developed a theorem which states:

b/h=(d+f)/(d+e)

Where:

b=number of balls

h=number of hands

d=dwell time; the amount of time a ball spends in the hand in between catch and throw

f=flight time; the amount of time a ball spends in the air

e=empty time; the amount of time a balls spends empty without any ball in it

This relationship is always true. That is why in the scenario stated earlier, the two juggling patterns are related because they both have the same b and h values. The other three values are all different depending on the speed of the cascade and the heights to which the balls are thrown, but they proportional the entire time.

While the proportion will always remain constant, the controlling factors of the d, f, and e values are the velocity at which the ball is thrown at and the height to which it is thrown, which are related. When the ball is thrown, it is thrown at an angle, traveling in both the vertical and horizontal planes. In order to determine the height at which the ball will travel, we need to look at just the vertical plane. The initial velocity the ball is thrown at is a combination of both the ball’s velocity in the horizontal plane and the vertical plane, and each can be determined using trigonometry. Because we are looking at height, we are only using the vertical velocity. To determine the height that the ball goes, we can use the formula:

vf2=vi2+2ad

Where:

vf=final velocity

vi=initial velocity

a=acceleration due to gravity

d=distance (height)

The initial velocity is the ball’s initial vertical velocity, which we already discussed. The ball’s final velocity is zero. The reason that this value is zero is because when we are analyzing the distance the ball travels, we only have to look at half of its total flight (the ball goes the same distance up as it does down). When the ball reaches the top of the arc, its vertical velocity is zero; it is only moving horizontally. The acceleration due to gravity is -9.81 m/s2, a universal constant (For every second the ball travels up, it slows down 9.81 meters per second. If the ball was falling down, the acceleration due to gravity would be positive and it would speed up an additional 9.81 meters per second for every second it fell.). We plug in these values and solve for d, finding the distance up the ball travels, the height of the ball’s arc.

That’s just the tip of the iceberg in regards to the physics of juggling. When you start introducing objects that spin and objects of different masses and shapes, things get a little a little bit trickier, but I’ll save that for another article. Until next time…SCIENCE!