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The Math of Figure Skating

Ms. Faulkner Pre-Calc 5 Math Project -- Applying Pre-calculus/Calculus to Figure Skating
by

Kristen K

on 31 October 2015

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Transcript of The Math of Figure Skating

OVERVIEW
Figure skating applies math through:
- vertical motion and using integrals to find the position function
- graphing derivatives
- triangles and trigonometry
- use of parametric functions and their derivatives
- the application and understanding of friction, torque, and inertia
- conservation of angular momentum and use of angular velocity
The Math of
Figure Skating

ANGULAR MOMENTUM: L = I

ANGULAR VELOCITY: = /t

POSITION: s(t) = gt + v t + s

INERTIA: I = mr

TORQUE: T = I(d /dt)


CONSERVATION OF ANGULAR MOMENTUM: I = I
Mao Asada at the 2010 Olympics
ANGULAR MOMENTUM
Angular momentum determines how fast the skater spins. A skater generates angular momentum by pushing off of the ice with his or her foot. (This push off the ice also creates vertical velocity.)
SOME IMPORTANT EQUATIONS:
2
VELOCITY
Velocity is the speed and direction at which something moves. Angular velocity applies when the object is moving around a point and is defined as the rate of change of angular displacement. Linear velocity applies when the object is traveling straight.
INERTIA
TORQUE
In this case, inertia is the resistance to rotational movement and applies as the skater spins and jumps.
Torque is the force that makes something rotate. According to the law of conservation of angular momentum, when no external torque acts on an object or closed system of objects, there can be no change in angular momentum.
BREAKING IT DOWN
1
1
2
2
Deriving the Position Function
The position function is represented by the equation
s(t) = (1/2)gt + v t + s

First, deriving the velocity function by integrating the acceleration function:
The Indefinite Integral
f'(x) dx = f(x)
The POWER RULE tells us that x dx = + C:
n
x
n+1
______
n+1
s'(t) = gt + v
Introduction to Integrals
So, knowing that s"(t) (the acceleration function) is the derivative of s'(t) (the velocity function), which is the derivative of s(t) (the position function) ...
POSITION FUNCTION: s(t) = -4.9t + v t
2
And next, deriving the position function:
s(t) = (1/2)gt + v t + s
2
Conservation of Angular
Momentum
Since torque is the time derivative of angular momentum, when there is no net torque (0), angular momentum is conserved (constant).
In a jump, there truly is no torque, while in spins there is friction. Ice has a friction coefficient of 0.02, which is pretty negligible for the short amount of time that a skater spins.
SOME CALCULATIONS
INERTIA: I = mr
ANGULAR VELOCITY: = /t
ANGULAR MOMENTUM: L = I
CONSERVATION OF ANGULAR MOMENTUM:
I = I
1
1
2
2
2
THE CAMEL SPIN
I = mr
2
m = 45 kg
r = 0.791 m
L = I
= /t
= 2 rotations
t = 1.57 s
I = 28.16 kgm
= 1.27 rps
2
L = 35.76 kgm rev/sec
2
Let's check...
L = 35.76 kgm rev/sec
2
L = I
I = mr
2
m = 45 kg
r = 0.611 m
I = 16.80 kgm
2
= L/I
= 2.13 rps
= 2 rotations
t = 0.93 seconds
= 2.15 rps
= /t
Because of the law of conservation of momentum, a skater spins more quickly when he or she goes from a larger radius to a smaller one.
A Brief History:
very old pastime, primitive skates made from animal bone dating back to 3000 BC in Switzerland, and Southern Finland, used as method of communication and to traverse frozen canals
wood edges added by Dutch in 13th or 14th century, then in 1572, iron: cut instead of glide on ice
skate <-- schaats ("leg bone" or "shank bone")
commercialized in England: clubs and ice rinks built
in many places, though not all, skating was considered to be a higher-class activity (Roman Empire, France, Great Britain, China vs. Netherlands)
father of modern figure skating: Jackson Haines -- incorporated ballet and dance moves
international competitions began in the 1890s, International Skating Union founded 1892
at the 1908 Summer Olympics in London, skating became the first winter sport to be included in the Olympics
United States Figure Skating Association formed in 1921
Some Background on Scoring

2002 Winter Olympics scandal led to upheaval of scoring system: stopped using 6.0 scoring system and instituted ISU Judging System, which is used in all international competitions nowadays

6.o scoring system: skaters were judged on technical merit or required elements in addition to presentation. Each judge gave a score from 0.0 to 6.0. The free skate was weighed more than the short program.

ISU Judging System/Code of Points:
technical panel: specialist determines base value of element, controller inputs deductions
judging panel: 9 judges, 1 referee. The trimmed mean of GOE scores (-3 to 3) for elements is added to the base value of the elements to calculate the scores. Other components are judged from 1-10, in 0.25 increments. Judges can also deduct points for music, costume, or prop violations, while the referee can deduct for time violations or interruptions.
the free skate is worth twice as much as the short program
More complex jumps, as well as jumps with more rotations, have higher base point values: toeloop<salchow<loop<flip<lutz<axel
0
0
2
This equation can be derived using integrals.
0
0
KNOWN:
acceleration (g = -9.8 m/s )
initial position (s = 0 m)
0
2
0
Maximum height reached in jump: 0.66 m
WHY IT ALL MATTERS
The Triangle
Creating the
Parametric Equation
Introduction to Derivatives of Parametric Functions
It takes Asada 0.735 seconds to complete the jump.
The maximum height she reaches is 0.66 meters.
0
s(0.3675) = 0.66 m = -4.9(0.3675) + v (0.3675)
0
2
v = = 3.59667 m/s
0
0.66 + 4.9(0.3675)
2
________________
0.3675
~
~
3.6 m/s
POSITION FUNCTION: s(t) = -4.9t + 3.6t
2
UNKNOWN:
initial velocity (v = ? m/s)
0
is like an ANTIDERIVATIVE
sin37 = 3.6/h
h = 3.6/sin37
h = 5.98
tan37 = 3.6/x
x = 3.6/tan37
x = 4.78
Resultant Velocity
Horizontal Velocity
Vertical position: y = -4.9t + 3.6t
Horizontal position: x = 4.78t
Parametric Equation:
Vertical Velocity
2
t =
x
4.78
____
y = -4.9( ) + 3.6( )
x
4.78
____
x
4.78
____
dy
dt
__
2
Derivative of vertical position function:
Derivative of horizontal position function:
Derivative of parametric equation:
dy
dx
__
dt
__
dx
dy
dt
__
dt
__
dx
___
=
horizontal tangent line where = 0
dy
dt
__
Derivatives of Parametric Equations
= -0.214x + 0.753x
2
Checking the work:
(-0.429x + 0.753) dx = = -0.214x + 0.753x
-0.429x
_______
1+1
1+1
+
0.753x
_______
0+1
0+1
2
Using Integrals and the Power Rule:
Using Graphs:
1
2
2
0
0
_
Skate Design:
slightly concave -- "hollow ground"
supports and laces keep ankle from rolling
Team USA
Skating Jumps
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