How Many Objects Can Be Juggled
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
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
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
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