The Calculus of Juggling

Ashley Bennett

Stephen Bent

3-Ball CascadeTrial Time (seconds)

1 0.47

2 0.49

3 0.52

4 0.49

5 0.51

Average 0.496

5-ball CascadeTrial Time (seconds)

1 0.63

2 0.68

3 0.67

4 0.64

5 0.64

Average 0.658

We knew that the formula for an accelerating body is:

Distance = (startingvelocity)(time)

+(1/2)(acceleration)(time)(time)

d = (vo)(t)+(1/2)(a)(t2)

Second Degree Taylor Polynomial

General Formula:

P2(x) = f(a) + f’(a)(x-a) + ½f’’(a)(x - a)2

Specifically:

d = (first derivative) (change in time) + (½) (second derivative)*(change in time)* (change in time)

Hand 1q1 Hand

He

igh

t

Three Ball Cascade:t/2 = .496/2 = .248 seconds

Five Ball Cascade:t/2 = .658/2 = .329 seconds

• Three Ball Cascade:

d = (0)(.248) + (1/2)(9.8)(.2482) = .30317 meters

• Five Ball Cascade:

d = (0)(.329) + (1/2) (9.8) (.3292) = .53038 meters

3 Ball CascadeTrial Height (meters)

1 0.30

2 0.30

3 0.31

4 0.31

5 0.30

Average 0.304

5 Ball CascadeTrial Height (meters)

1 0.53

2 0.53

3 0.53

4 0.53

5 0.53

Average 0.53

V = V0 + a*t

• Three Balls:V = (0) + (9.8) (.248) = 2.4304 meters/second

• Five BallsV = (0) + (9.8) (.329) = 3.2242 meters/second

Three Balls:V = V0 + (a) (t)

V = 2.4304 – (9.8*t)

t

V

Five Balls:V = V0 + (a) (t)

V = 3.2242 – (9.8*t)

t

V

Assumptions

• 1: Juggling is consistent

• 2: Gravity is the only force

• 3: Stephen’s juggling height and speed is law!!!