Archive for the ‘algebra’ Category
Winter Olympics 2022: Why do figure skaters spin so fast?
February 3, 2022
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 : vr = 308 RPM
Other spins include:
Mao Asada, Japan, triple Axel
Triple Axel spin vr = 220 – 280 RPM
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Maximum Triple Axel spin vr = 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.
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Camel spin vr = 90 RPM
More info here.
Kim Yuna, South Korea
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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 (vr ). 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 vr = 90 RPM
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Triple Axel spin vr = 4 x 90 RPM = 360 RPM
That’s about right.
WINTER OLYMPICS: How ski Jumpers Use Math to Increase their Jump Length
February 3, 2022
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 her 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 2022: One Rule Aerial Skiers Cannot Break
February 1, 2022The Parabola Must Be Obeyed!!!!
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
Death by Caffeine: The Maths Part 1 METRIC
November 26, 2021Death by Caffeine: The Math Part 1 USA UNITS here.
The lethal dose for caffeine is about 150 milligrams per kg of body weight. While the average person’s caffeine consumption is around 200 milligrams a day, the Mayo Clinic advises against exceeding 500 to 600 milligrams per day.
This work titled Timing is Everything is by street artist ABOVE in London.
You may want this much energy but be warned, energy drinks can kill.
It would take a ridiculously huge number of ANGRY WOMBAT energy drinks to kill a 14 year old weighing, say, 48 kg. Nevertheless, some kids are more sensitive to caffeine than others and then much lower levels of consumption of energy drinks can be fatal.
See Death by Caffeine: The Maths Part 2 METRIC
Here is the work (see below) TIMING IS EVERYTHING by street artist ABOVE in London showing the timing.
Death by Caffeine: The Math Part 1 USA UNITS
November 26, 2021Death by Caffeine: The Maths Part 1 METRIC here
The lethal dose for caffeine is about 68 milligrams per lb of body weight. Average consumption of caffeine is about 200 milligrams a day. The Mayo Clinic advises against exceeding 500 to 600 milligrams per day.
This work titled Timing is Everything is by street artist ABOVE in London.
You may want this much energy but be warned, energy drinks can kill.
It would take a ridiculously huge number of ANGRY WOMBAT energy drinks to kill a 14 year old weighing, say, 110 lb. Nevertheless, some kids are more sensitive to caffeine than others and then much lower levels of consumption of energy drinks can be fatal.
See Death by Caffeine: The Math Part 2 USA UNITS
Here is the work (see below) TIMING IS EVERYTHING by street artist ABOVE in London showing the timing.
Maths in the Real World: 10 Attention Grabbers for Middle School
September 5, 20211. Smoke Jumpers: The Amazing Maths of wildfires
USA UNITS HERE
METRIC UNITS HERE
2. The Rolling Coin Paradox!!
ROLLING COIN PARADOX HERE
3. How barcodes work!
Barcode MATHS HERE
4. Pop Song Beats and Jogging
Pop Song Beats and Jogging MATHS HERE
5. Linear Math and Linear Drumming. It’s a thing!
Linear Math and Linear Drumming. HERE
6. Powers and the Loudest Rock Band in the World
Powers and the Loudest Rock Band MATHS HERE
7. Alcohol Kills! Calculate how much would kill you!
Alcohol Kills! MATHS HERE
8. Tall Tales: Is height the most important factor in sport?
Height in Sport maths: USA UNITS HERE
Height in Sport maths: METRIC UNITS HERE
9. Mmmmm! Chocolate. Yes! It can kill you
Chocolate. Yes! It can kill you MATHS HERE
10. Random Music? You think!
Random Music?MATHS HERE
Toilet Paper FUN FACTS Maths
June 1, 2021
Mathspig is in Melbourne, Australia. We’re in Lockdown. AGAIN!
And toilet paper has disappeared off the supermarket shelves. AGAIN.
So today we are doing maths based entirely on toilet paper using information from the Toilet Paper Fun Facts website. Yes! It exists here.
Toilet Paper Fun Facts Website
Protractor Wind Speed Calculator
POP UP ANS Q 1 & 2 Yr 7&8 Toilet Paper fun facts
Pythagoras Equation Diagram here
Here is the Sine Rule (Watch 1.5 mins)
Check sine curve here:
Listen to sine curve here:
20 Gobsmacklingly AMAZING numbers … Revisited
May 25, 2021… these nos. are amaaaazing but no one could find the answers in the earlier post.
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Click on the POP-UP answers under each question.
1. How many bristles in a toothbrush?
2. What % of Oxygen in the atmosphere comes from forests?
3. How loud was the loudest recorded human scream?
4. How many apples could kill an average-sized adult?
5. How many countries in Africa?
6. What happens when you Google this number, 241543903?
7. How many vertebrae in a giraffe’s neck?
8. What is the highest recorded wind speed?
9. What is the world record high-jump height for a guinea pig?
10. How many times does England fit into Australia?
11. How much did the most expensive dessert ever cost?
14. How many burgers does McDonalds sell a second worldwide?
16. How many times was the most frequently married person in the world, married?
17. Why is Zero Zero Zero like 123?
18. What is the world record for the greatest number of snorts in 10 seconds?
19. How long would it take to grow your hair to your feet?
20. What number causes the greatest loss of money in the world?
LINKS:
1. How many bristles in a toothbrush? HERE
2. What % of Oxygen in the atmosphere comes from forests? HERE
3. 3. How loud was the loudest recorded human scream? GOOGLE IT!
4. How many apples could kill an average-sized adult? HERE
5. 5. How many countries in Africa? GOOGLE IT!
6. What happens when you Google this number: 241543903? HERE
7. How many vertebrae in a giraffe’s neck? HERE
8. What is the highest recorded wind speed? HERE and HERE
9. What is the world record high-jump height for a guinea pig? GOOGLE IT!
10. How many times does England fit into Australia? GOOGLE IT!
11. How much did the most expensive dessert ever cost? GOOGLE AGAIN.
12. Why is 70 a weird number? HERE
13. Why Google Googol? HERE
14. How many burgers does McDonalds sell a second worldwide? HERE
15. If the population of the world spread across the USA , what area would we get each?
GOOGLE THE NUMBERS!
16. How many times was the most frequently married person in the world, married?
GOOGLE HIM.
17. Why is Zero Zero Zero like 123? HERE
18. What is the world record for the greatest number of snorts in 10 seconds? HERE
19. How long would it take to grow your hair to your feet? HERE
20. 20. What number causes the greatest loss of money in the world? HERE
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