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In this paper Jack Kalvan, a mechanical engineer and avid juggler, ...

Each hand has $b/h$ objects and so it would take $b/h \tau$ sec...

The equation for the velocity of a ball thrown in the air is

$$ ...

In 1997, Kalvan didn't have smartphones or smartwatches equipped wi...

Calibration is a matter of getting some experimental points for a l...

To illustrate the importance of accuracy when juggling let's calcul...

How Many Objects Can Be Juggled

Jack Kalvan

Originally published in 1997

I hate to break it to you aspiring numbers jugglers,

but no human will ever juggle 100 balls. Only a handful

of people have reached a level to throw eleven or twelve

objects into the air, and so far, not for more than a few

seconds. No one has even come close to juggling 13 balls.

But is this within the realm of human possibility?

Hand speed is one of the main factors that limit the

number of objects one can juggle. (The other main fac-

tors being accuracy of throws and having long enough

arms and enough space in the air for the juggling pat-

tern.) I decided to find out if anyone has the hand speed

necessary to juggle 13 or more balls. So I designed an

experiment to measure the theoretical human juggling

limits - given the acceleration of the hands.

To write the necessary equations, I define the following

variables:

b = number of balls

h = number of hands

f = flight time of a ball from throw to catch

τ = time between throws from the same hand

= vertical throw velocity

g = acceleration due to gravity = 9.81 m/s

−2

r = "dwell ratio" or fraction of time a hand is holding

a ball. My tests show r is usually about 2/3.

ω = average number of balls in flight per arc

One can also think of r as the average number of balls

in a hand while juggling. omega can be expressed as

the number of balls per hand minus the balls held in the

hand: ω = (b/h) − r .

ω is also equal to the time that balls are in flight di-

vided by how often they are thrown: ω = f/τ .

To simplify my analysis, I will assume balls are thrown

and caught at the same height. Newtonian physics tells

us the flight time of a ball, f = 2V

/g . Substituting this

equations for f into the second equation for omega gives

ω = 2V

/g/τ.

Since g is a constant, we see that omega is proportional

to the throwing velocity of the hand divided by the time

between throws. This means the number of balls in the

air while juggling is closely related to the acceleration of

the hand. Although a juggler’s hands do not necessarily

accelerate smoothly, the number of balls one can get into

the air is approximately proportional to the maximum

acceleration of one’s hands .

I figured if I measure the maximum acceleration of

a juggler’s hands with a simple accelerometer, I could

roughly calculate the juggler’s maximum value for omega.

And substituting this value into the equation, b/h =

ω + r, gives an approximation of the maximum num-

ber of balls one can theoretically juggle. Remember, this

maximum number of balls is calculated only from the

speed a juggler can potentially throw balls into the air.

It does not take into account accuracy of throws or the

possibility of collisions.

Since the number of balls juggled is proportional to

hand acceleration, a corollary is that the height of your

juggling pattern is not related to your hand acceleration.

For example, if you juggle 5 balls high, you have about

the same hand acceleration as if you juggle them low.

The difference is that to juggle high, you accelerate for a

longer time and therefore have longer hand motion and

a higher throwing velocity.

I believe the following chart describes how the deriva-

tives of the vertical hand motion relate to juggling:

I. THE JUGGLEMETER

This simple device measures the hand acceleration.

A small mass is connected to a spring inside a tube.

When the hand accelerates the device (by shaking it

up and down), two opposing forces act on the mass:

the acceleration force (force = mass × acceleration) and

the spring force (force = spring stiffness × distance

stretched). These forces are in equilibrium when the

spring is stretched. A marker measures the maximum

distance the spring was stretched. Since the mass and

spring stiffness are constant, the maximum acceleration

is proportional to this distance. The distance the spring

stretches is therefore proportional to the number of balls

potentially juggled.

II. JUGGLEMETER CALIBRATION

To calibrate the device, I attached it to the back of

a glove. This method worked well for low numbers of

balls but the glove and the...