Winter Olympics: Bad Math of Figure Skating Scores

According to the fab NBC video, Mathletes,  nine Figure Skating judges score competitors for the complexity of each element (eg. Triple axel or triple spin jump) and the quality of the performance producing a score out of ten.

Tessa Virtue and Scott Moir win GOLD at Pyeong Chang 2018

                                   Kailani Craine, Australia

This is a typical figure skating score card for one competitor.

The final score, however, is based  on the average for only 5 of these scores. Two are eliminated by random selection (Red Brackets). Then the top and bottom scores are removed and the remaining five scores averaged.

Screen grab NBC Mathletes

……………………………………………………

Now consider the IDENTICAL SCORE CARDS

of Skater A & B:

Skater A:

Four scores are removed. Two by the random selector (in brackets) and then the top and bottom scores (with line drawn through them)

7.00 + 7.00 + 7.00 + 6.75 + 7.00

……………………………………..

=  34.75/ 5 = 6.95

Skater B:

Four scores are removed. Two by the random selector (in brackets) and then the top and bottom scores (with line drawn through them). But this time the random selector eliminates two low scores.

The average:

7.00 + 7.25 + 7.00 + 7.00 + 7.00

……………………………………..

=  35.25/ 5 = 7.05

Same score cards but Skater B gets a higher average score than Skater A.

Skater A is, in fact, beaten by a random number selector!!!!

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WINTER OLYMPICS: How ski Jumpers Use Math to Increase their Jump Length

While air resistance has little impact on aerial skiers it is a significant factor used by ski jumpers to increase their jump distance.

The significant maths for ski jumpers is therefore X-section area.

Here is the jump at Pyeong Chang, 2018. Just imagine going down that at top speed!!!

              A ski jumper is set to jump in Pyeongchang.

                   Casey Larson USA Pyeong chang 2018

Ski jumpers increase their speed going down the ramp by reducing their X-section area:

           Lindsey Van, USA, practicing in a wind tunnel

Once they leave the ramp, ski jumpers try to increase their X-section area like Ski Divers to slow their vertical fall. But they have to land safely so they keep their skis at a minimum  angle.

Abby Hughes, USA, practicing in a wind tunnel.

          Abby Hughes, USA, in the air

Here are the X-section areas for Abby Hughes*:

Here is the formula for Air Resistance of Drag:

D = ½CApv²

Where C is the drag coefficient or constant, which depends on the shape and spin of an object. It is found by testing the object in a wind tunnel.

A is the X-section Area,

p is the density of the air and

v the velocity of the object.

More here.

As Abby Hughes has tripled here X-section area in the air she will have tripled the vertical drag during her jump. This will slow here decent.

*Mathspig calculated the X-section area by the old fashioned method of counting squares and rounding off the final count. Mathspig sized the two pics of Abby Huges so that her head was the same size in both pictures.

Winter Olympics: Why do figure skaters spin so fast?

Here is the sensational pairs team from the 2018 Winter Olympics.

But why do ice skaters spin so fast?

Here is the math!

REASON  1:

Well, mathspiggies, the girl in this video is right. Angular momentum remains constant unless external forces are applied.

L = mvr

L = angular momentum

v = linear velocity

r = separation of object

Louisa Barama, USA

Let’s have a look at this equation:

The fastest spin on ice skates was achieved by Natalia Kanounnikova (Russia) with a maximum rotational velocity of 308 RPM (rotations per minute) at Rockefeller Centre Ice Rink, New York, USA on 27 March 2006. See Guinness Book of Records.

 Record spin :  v_r = 308 RPM

Other spins include:

Mao Asada, Japan, triple Axel

Triple Axel spin v_r = 220 – 280 RPM

……………………………………………………………………………………………..

 Maximum Triple Axel spin v_r = 402 RPM

Skaters can spin faster during a triple axel jump because there is no friction from the ice slowing their spin.

To complete a quad axel, it’s estimated that the skater would have to rotate in the air at:

540 rpm.

…………………………………………………………………………………

Camel spin v_r = 90 RPM

More info here.

Kim Yuna, South Korea

…………………………………………….

REASON 2:

How can a figure skater move from

a camel spin into a very fast standing spin?

Now, mathspiggies, you must separate Linear Velocity (v₁ ) from Angular Velocity (v_r ). Linear Velocity is measured in m/sec ie. it is the speed of, say, a skaters foot around the circle. Angular Velocity is measured in either RPM (Revolutions Per Minute) or degrees or Radians per minute. Ie. It is the rate of spin. We can’t judge how many m/sec a skaters foot is moving in a circle. We can only see how fast they spin. In other words, we see their Angular Velocity. When a skaters foot is in the Camel position that foot travels in a very big circle.

But when that same foot is in a Triple Axel postion it moves in a very, very small circle.

Patrick Chan, Canada, Camel Spin

Patrick Chan, Canada, Triple Axel

By halving the radius, firstly, a skater’s Linear Velocity doubles due to the conservation of angular momentum.

Then, secondly, by halving the radius the circumference of the circle moved by , say, the skaters foot is halved.

Overall, by doubling the velocity around the circle and halving the circumference a skater increases their rotational velocity by a factor of 4.

Look at the numbers:

Camel spin v_r = 90 RPM

…………………………………………………………………………………

Triple Axel spin v_r = 4 x 90 RPM = 360 RPM

That’s about right.

Winter Olympics: Beaten by 0.001 seconds

 

In 2014 Winter Olympics Annette Gerristen (below) lost the Gold Medal in the 1000 m Women’s speed skate competition by 0.02 seconds.

Annette Gerritsen fromthe Netherlands

What are the speed skating times for women?

Women’s Speed Skating Calulations:

Here are the speed calcs for Yara Van Kerkhof of the Netherlands in the 2018 Winter Olympics:

If Yara lost the Gold Medal by 0.02 secs (2 hundredths of a second) what would the distance be between the Gold and Silver place getters?

When 1st and 2nd place are separated by 0.02 seconds, they are travelling at almost the same speed. So the second place contestant is:

23.8 cm behind the Gold Medalist

The 2018 Olympic Gold Medalist in the 500m Women’s Speed Skating was Arianna Fontana.

Italy’s Arianna Fontana wins the 500 m Speed Skating 2018 Olympic Gold Medal in 42.569 seconds ahead of Yara van Kerkhof of the Netherlands and Kim Boutin of Canada. 

Men’s Speed Skating:

At longer distances the men’s speed skating speed is similar to the women’s speed above.

If you lose by  0.001 secs…………

Apollo Ono (below) competed in the 1500m men’s speed skating. He has won 8 Olympic Medals.

If a speed skater lost the Gold Medal by 0.001 seconds, the smallest measured time segment at the Olympics, they would be:

1.19 cm

behind the winner. That is less than the length of a small fingernail.

WINTER OLYMPICS: One Rule Aerial Skiers Cannot Break

Aerial skiers aim for height rather than length. Their aerial flight times are much smaller than ski jumpers so air resistance has minimal impact.

In fact, there is one law the aerial skiers cannot break. It is the law of gravity.

Here is an equation for  projectile motion from Wired magazine.

Screen grab from Wired Magazine

The equation for projectile motion also applies to Motorbike Jumps and Longbow Arrows.

Here is the x-y graph for different launch angles.

trajectory wired magazine

 

You can go to this page for complete calculations. Aerial skiers twist and turn but their CENTRE OF GRAVITY must follow this graph. More on centre of Gravity at The Great Back Pack Attack ie.

The centre of gravity of Aerial Skiers must follow a

parabolic curve.

Rocky Maloney Winter X Games Aspen

1. The Real REASON why figure skaters can spin so fast

REASON  1:

Well, mathspiggies, the girl in this video is right. Angular momentum remains constant unless external forces are applied.

L = mvr

L = angular momentum

v = linear velocity

r = separation of object

Louisa Barama, USA

Let’s have a look at this equation:

The fastest spin on ice skates was achieved by Natalia Kanounnikova (Russia) with a maximum rotational velocity of 308 RPM (rotations per minute) at Rockefeller Centre Ice Rink, New York, USA on 27 March 2006. See Guinness Book of Records.

 Record spin :  v_r = 308 RPM

Other spins include:

Mao Asada, Japan, triple Axel

Triple Axel spin v_r = 220 – 280 RPM

……………………………………………………………………………………………..

 Maximum Triple Axel spin v_r = 402 RPM

Skaters can spin faster during a triple axel jump because there is no friction from the ice slowing their spin.

To complete a quad axel, it’s estimated that the skater would have to rotate in the air at:

540 rpm.

…………………………………………………………………………………

Camel spin v_r = 90 RPM

More info here.

Kim Yuna, South Korea

…………………………………………….

REASON 2:

How can a figure skater move from

a camel spin into a very fast standing spin?

Now, mathspiggies, you must separate Linear Velocity (v₁ ) from Angular Velocity (v_r ). Linear Velocity is measured in m/sec ie. it is the speed of, say, a skaters foot around the circle. Angular Velocity is measured in either RPM (Revolutions Per Minute) or degrees or Radians per minute. Ie. It is the rate of spin. We can’t judge how many m/sec a skaters foot is moving in a circle. We can only see how fast they spin. In other words, we see their Angular Velocity. When a skaters foot is in the Camel position that foot travels in a very big circle.

But when that same foot is in a Triple Axel postion it moves in a very, very small circle.

Patrick Chan, Canada, Camel Spin

Patrick Chan, Canada, Triple Axel

By halving the radius, firstly, a skater’s Linear Velocity doubles due to the conservation of angular momentum.

Then, secondly, by halving the radius the circumference of the circle moved by , say, the skaters foot is halved.

Overall, by doubling the velocity around the circle and halving the circumference a skater increases their rotational velocity by a factor of 4.

Look at the numbers:

Camel spin v_r = 90 RPM

…………………………………………………………………………………

Triple Axel spin v_r = 4 x 90 RPM = 360 RPM

That’s about right.

2. One Rule Aerial Skiers Cannot Break

Aerial skiers aim for height rather than length. Their aerial flight times are much smaller than ski jumpers so air resistance has minimal impact.

In fact, there is one law the aerial skiers cannot break. It is the law of gravity.

Here is an equation for  projectile motion from Wired magazine.

Screen grab from Wired Magazine

Here is the x-y graph for different launch angles.

trajectory wired magazine

You can go to this page for complete calculations. Aerial skiers twist and turn but their CENTRE OF GRAVITY must follow this graph. MOre on centre of Gravity at The Great Back Pack Attack ie.

The centre of gravity of Aerial Skiers must follow a

parabolic curve.

 

 

Rocky Maloney Winter X Games Aspen