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The Calculus of Juggling Ashley Bennett Stephen Bent

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3-Ball Cascade TrialTime (seconds) 10.47 20.49 30.52 40.49 50.51 Average0.496 5-ball Cascade TrialTime (seconds) 10.63 20.68 30.67 40.64 5 Average0.658

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We knew that the formula for an accelerating body is: Distance = (startingvelocity)(time) +(1/2)(acceleration)(time)(time) d = (v o )(t)+(1/2)(a)(t 2 )

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Second Degree Taylor Polynomial General Formula: P 2 (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)

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Hand 1q1 Hand Height Three Ball Cascade: t/2 =.496/2 =.248 seconds Five Ball Cascade: t/2 =.658/2 =.329 seconds

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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

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3 Ball Cascade TrialHeight (meters) 10.30 2 30.31 4 50.30 Average0.304 5 Ball Cascade TrialHeight (meters) 10.53 2 3 4 5 Average0.53

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V = V 0 + a*t Three Balls: V = (0) + (9.8) (.248) = 2.4304 meters/second Five Balls V = (0) + (9.8) (.329) = 3.2242 meters/second

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Three Balls: V = V0 + (a) (t) V = 2.4304 – (9.8*t)

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Five Balls: V = V0 + (a) (t) V = 3.2242 – (9.8*t)

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Assumptions 1: Juggling is consistent 2: Gravity is the only force 3: Stephen’s juggling height and speed is law!!!

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