Ho w Man y Ob jects Can Be Juggled

Jac k Kalv an

Original ly publishe d in 1997

I hate to break it to you aspiring n umbers jugglers,

but no h uman will ever juggle 100 balls. Only a handful

of p eople ha ve reached a lev el to thro w elev en or t welv e

ob jects in to the air, and so far, not for more than a few

seconds. No one has ev en 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

n umber of ob jects one can juggle. (The other main fac-

tors b eing 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 an y one has the hand speed

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

exp erimen t to measure the theoretical human juggling

limits - given the acceleration of the hands.

T o write the necessary equations, I deﬁne the following

v ariables:

b = num ber of balls

h = n umber of hands

f = ﬂight time of a ball from thro w to catch

τ = time betw een throws from the same hand

V

v

= v ertical throw v elocity

g = acceleration due to gravit y = 9.81 m/s

− 2

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

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

ω = av erage n um b er of balls in ﬂigh t p er arc

One can also think of r as the a v erage n um b er of balls

in a hand while juggling. omega can b e expressed as

the n um b er 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 b y how often they are thro wn: ω = f /τ .

T o simplify my analysis, I will assume balls are thro wn

and caugh t at the same height. Newtonian ph ysics tells

us the ﬂight time of a ball, f = 2 V

v

/g . Substituting this

equations for f into the second equation for omega giv es

ω = 2 V

v

/g /τ .

Since g is a constan t, w e see that omega is prop ortional

to the throwing velocity of the hand divided b y the time

b et w een thro ws. This means the num ber 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 smo othly , the n um b er of balls one can get in to

the air is appro ximately prop ortional 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 v alue for omega.

And substituting this v alue in to the equation, b/h =

ω + r , giv es an appro ximation of the maximum n um-

b er of balls one can theoretically juggle. Remember, this

maxim um n um b er of balls is calculated only from the

sp eed a juggler can p oten tially throw balls into the air.

It do es not tak e in to account accuracy of throws or the

p ossibilit y of collisions.

Since the num b er of balls juggled is prop ortional to

hand acceleration, a corollary is that the height of y our

juggling pattern is not related to y our hand acceleration.

F or example, if you juggle 5 balls high, you ha ve ab out

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 hav e longer hand motion and

a higher throwing velocity .

I b eliev e the following c hart describes ho w the deriv a-

tiv es 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 (b y shaking it

up and do wn), t wo opp osing forces act on the mass:

the acceleration force (force = mass × acceleration) and

the spring force (force = spring stiﬀness × distance

stretc hed). These forces are in equilibrium when the

spring is stretched. A mark er measures the maximum

distance the spring was stretc hed. Since the mass and

spring stiﬀness are constan t, the maxim um acceleration

is prop ortional to this distance. The distance the spring