The Maths of Cycling Bikes come in different styles, depending on their intended use
racer mountain bike bmx
What do they look like?
fast and efficient can go off-road small & manoeuvrable Advantages comfortable and sensitive steering less parts to go wrong easy to ride uncomfortable only low speeds possible riding position slower and heavier Disadvantages uncomfortable riding dangerous for novices inefficient on roads position if seated
thin wheels and tyres thick tyres with deep much smaller frame with minimal tread tread to maximise and wheels make What features to reduce friction grip and stability tricks easier to do aggressive, hunched-over low saddle so it help them suit relaxed, upright riding their purpose? riding position to reduce isn’t in the way air resistance and position to increase maximise power output comfort and allow rider only one easy gear to from legs to look up more easily accelerate quickly Gears
cassette
chain ring
Most modern bikes have a chain ring with 2 or 3 cogs and a cassette with 5 to 9 cogs Each combination of cog on the chain ring and cog on the cassette gives a different gear eg a bike with a double chain ring and 6-cog cassette has 2 x 6 = 12 gears Gear ratios The size of the two cogs can be measured cassette by how many teeth they have The ratio of the chain ring size divided by the cassette cog size gives the number of times the wheel will rotate for one turn of the pedals eg for a typical racer with a double chain ring chain ring
which combination of chain ring and cassette chain ring size gear gear are the lowest gear ratio (easiest to pedal) lowest – 34/27 ratios 34 50 and the highest gear ratio (hardest to pedal)? highest – 50/12 27 1.26 1.85
which different combinations of chain ring 24 1.42 2.08 and cassette gear are almost the same ratio? 34/18 and 50/27 21 1.62 2.38 for every 100 rotation of the pedals, how many 18 1.89 2.78 rotations do the wheels make in the highest gear? 100 x 4.12 = 412 cassette size cassette 15 2.27 3.33 12 2.83 4.12 for every 100 rotation of the wheels, how many rotations do the pedals make in the lowest gear? 100 ÷ 1.26 = 79 Gear ratio and speed
The speed of the bike is determined by the gear ratio, the size of the wheels and how fast the pedals are turning
Distance travelled by wheel in 1 hour = wheel circumference x revolutions per hour x gear ratio
eg a cyclist is in a gear with a ratio of 4. The bike has wheels with a 2m circumference. using the different gears and varying They are pedalling at 50 revolutions per minute. your pedalling speed enables you to What is their speed in miles per hour? reach a range of speeds: Distance travelled revolutions per minute by wheel in 1 hour = 2 x 50 x 60 x 4 speeds 60 120 = 24000m mph (steady (very fast ≈ 15 miles pedalling) pedalling)
1 mile ≈ 1600 metres 1 4.95 9.9 2 9.9 19.8
of course, you can only reach the ratio 3 14.85 29.7 higher speeds if have the power to 4 pedal very fast in a high gear... 19.8 39.6 Gear ratio and speed b) Here are the gear ratios for my new bike: 1a) Can you complete the table? Gear 1st 2nd 3rd 4th 5th gear chain ring size Gear ratio 1.2 1.5 2 2.5 4 ratios 30 40 i) For every 60 rotation of the pedals, how many 30 1 1.333... rotations do the wheels make in 2nd gear? 20 1.5 2 60 x 1.5 = 90 16 1.875 2.5 ii) For every 40 rotation of the wheels, how many rotations do the pedals make in 5th gear? cassette size cassette 10 3 4 40 ÷ 4 = 10
2a) Chris Hoy’s bike has a chain ring with 51 teeth and a cassette with 14 teeth. How many rotations do his wheels make for every 70 rotation of the pedals?
51 14 70 255 b) The wheels of Chris Hoy’s bike have a circumference of 1.98m. If he is pedalling at 120 revolutions per minute, what is his speed, in miles per hour?
51 1.9812060 14 1600 32.5 mph (1dp) Mr Walker’s bike has wheels with a circumference of 2m. He wants to achieve a speed of 20mph.
Find two different combinations of gear ratio and cadence (pedalling speed in revolutions per minute) that would achieve this.
The gear ratio r cannot be more than 5 as this becomes too hard to push. The cadence c cannot be more than 200 as you cannot pedal faster than this!
32000 800 2c60 r 1600 20 c 120r 3r
eg r 3 c 89 r 4 c 67 The gear ratio required to The bike used by Guy Martin when he broke the British speed record had only a gear with an even break speed records is higher ratio of 15, achieved by a system of cogs
much higher than those 112.94 mph found on normal bikes
In 1962, Jose Meiffret broke the Given that he pedalled with a cadence of 97 rpm motor-paced cycling World Record and his bike had wheels with a circumference of on a bike with a gear ratio of 8.7! 2.07 metres, how fast did he go?
Guy Martin wants to break the World Record and build a bike capable of 200 mph How would you design his bike?! Power The power P W for a cyclist to maintain a speed This is a cubic relationship v mph on level ground, using an average adult + between speed and power: bike of 90kg, is given by the following formula: P 0.0165v3 2.09v Eg what power output do you need for you to do 15 mph? (an OK pace for leisure cycling) 0.0165 153 2.09 15 87W 600 W
2012 Olympic champion Bradley Wiggins averaged 32 mph during the time trial What was his average power output? Power The power P W for a cyclist to maintain a speed v mph on level ground, using an average adult + bike of 90kg, is given by the formula P 0.0165v3 2.09v
1. Use the formula to find the power 3. Mark Cavendish sprints at 45mph needed to maintain a speed of: when winning stages of the Tour de France. How many 60W light bulbs a) 5 mph could he power with this effort? 3 v 45P 1597.6125 0.0165 5 2.095 13W b) 20 mph 1597.612560 26.62... 0.0165203 2.0920 174W about 26 light bulbs! 2. A cyclist has a maximum power 4. The coefficient of v in the power output of 800W. To the nearest formula is due to mass and friction. mph, what speed can he achieve, What would happen to this coefficient when using his maximum output? in the following scenarios?
v 35P 781W a) The cyclist were heavier v 36P 845W it would increase b) The cyclist switched from a closest is 35 mph mountain bike to a racer? it would decrease Sir Chris Hoy reaches a maximum speed of 49 mph during the Track Sprint
How much power does this require? P 0.0165v3 2.09v
2000W That’s enough power to run a house and ten times more than the average person can achieve! Many Tour de France riders only use around 100 W of power to maintain 25mph. The graph tells us that you need 300W to achieve this speed, so how is this possible?
Riding in the pelaton (main group) of cyclists saves a lot of energy by slip-streaming! Air resistance Moving objects experience a force known as air resistance or drag The faster an object is moving, the greater the air resistance, hence the rapid increase in power required to reach higher speeds Elite road & track cyclists try to minimise air resistance in a number of ways: leaning forward in a race ‘position’ slip-streaming behind other cyclists
press press me! me!
this reduces drag by around 10% this reduces drag by around 40% Technology can be used to further improve aerodynamics: In 1993 an amateur cyclist called Graeme Obree shattered the World Record for the greatest distance cycled in 1 hour on a track, without any slip-streaming allowed. He came up with his own distinctive riding position that reduced drag by 20% more than the position used by other riders. This allowed him to go faster and with less effort than his rivals, provided he could endure the painful position for an hour! However the governing body for cycling decided to ban Obree’s unorthodox riding position and strip him of his World Records, as they felt his technique was ‘ugly’ and didn’t befit the tradition of the sport.
His main rival at the time – Chris Boardman – was riding a prototype bike that cost half a million pounds
Obree’s bike was home-made from parts including bits of his washing machine! Air resistance 1. The graph shows how the power/velocity function changes, depending on the position of a cyclist. Use the graph to answer the following questions: a) A cyclist has a maximum power output of 600 watts. How fast can he go in each of the three positions, in mph? Racer position
Racer ___33 Obree ___36 Slipstreaming ___37½ Obree position b) Another cyclist is travelling at 30 mph. How much more power does he need to do this in a Slip-streaming racer position than when slip-streaming? 460 – 330 = 130W c) How much further can Graeme Obree travel in 1 hour than someone in a normal racer position, assuming they both sustain a power output of 400 watts? roughly 32.5 – 28.5 = 4 miles 3. The coefficient of v3 in the power formula P 0 . 0165 v 3 2 . 09 v is due to air resistance. What would happen to this coefficient in the following scenarios? a) The cyclist sat upright b) The cyclist is on a heavier bike it would increase no change! If there wasn’t any air resistance, cyclists could go a lot, lot faster...
Fred Rompelberg holds the World Record for speed on a bike whilst slip-streaming
He reached 167 mph whilst slip-streaming a drag car with a specially designed wind- break attached on the Bonneville salt flats Riding uphill Anyone who has cycled knows This means it is much harder to climb that going downhill is a lot slopes, especially at higher speeds: easier than going uphill because of the effect of gravity! 20 mph
10 mph We saw previously that the If thepower cyclist P W is for climbing a cyclist a slopeto maintainwith gradient a speed m as v mph,a percentage, using an averagethe rule adult has another + bike of element: 90kg, is given by the following formula: P 0.0165v3 2.09v 3.94vm Riding uphill The graph shows the power needed to sustain various speeds when riding up a gradient. 1. Use the graph to estimate the 20 mph power output required to maintain a speed of 10mph on an 8% gradient 350W
2. Dave can sustain a maximum 15 mph power output of 400W. What speed should he maintain if he wants to be using his maximum output on a gradient of 4%? 10 mph
about 17½ mph
3. Mr Walker rides up Muswell Hill 5 mph on his way to work. It is ½ mile long and has a gradient of 10%. It takes him 6 minutes to complete, what is his average power output?
speed is 5 mph, so power required is 200W British cyclist Chris Froome won the most prestigious bike race of all in 2013 – the Tour de France One of his most impressive wins came on the longest stage – 150 miles including a brutal 13 mile climb to the summit of Mont Ventoux... His average speed on the hour-long climb was 14 mph, requiring a power output of over 500W! Summary
We have seen that a number of factors which affect a cyclist’s speed: The gear ratio and cadence (pedalling speed) The amount of power the cyclist can generate in his legs The air resistance of the cyclist and bike Whether there is an incline (gradient) All of these factors can be understood by looking at the Mathematics behind them
Elite cyclists will be well-aware of these factors and in fact the remarkable success of British cycling in recent times is in part due to a team of experts who understand the Mathematics involved. Gear ratio and speed b) Here are the gear ratios for my new bike: 1a) Can you complete the table? Gear 1st 2nd 3rd 4th 5th gear chain ring size Gear ratio 1.2 1.5 2 2.5 4 ratios 30 i) For every 60 rotation of the pedals, how many 1 rotations do the wheels make in 2nd gear? 2 16 2.5 ii) For every 40 rotation of the wheels, how many rotations do the pedals make in 5th gear? cassette size cassette 3
2a) Chris Hoy’s bike has a chain ring with 51 teeth and a cassette with 14 teeth. How many rotations do his wheels make for every 70 rotation of the pedals?
b) The wheels of Chris Hoy’s bike have a circumference of 1.98m. If he is pedalling at 120 revolutions per minute, what is his speed, in miles per hour? Power The power P W for a cyclist to maintain a speed v mph on level ground, using an average adult + bike of 90kg, is given by the formula P 0.0165v3 2.09v
1. Use the formula to find the power 3. Mark Cavendish sprints at 45mph needed to maintain a speed of: when winning stages of the Tour de France. How many 60W light bulbs a) 5 mph could he power with this effort?
b) 20 mph
2. A cyclist has a maximum power 4. The coefficient of v in the power output of 800W. To the nearest formula is due to mass and friction. mph, what speed can he achieve, What would happen to this coefficient when using his maximum output? in the following scenarios?
a) The cyclist were heavier
b) The cyclist switched from a mountain bike to a racer? Air resistance 1. The graph shows how the power/velocity function changes, depending on the position of a cyclist. Use the graph to answer the following questions: a) A cyclist has a maximum power output of 600 watts. How fast can he go in each of the three positions, in mph? Racer position
Racer ___ Obree ___ Slipstreaming ___ Obree position b) Another cyclist is travelling at 30 mph. How much more power does he need to do this in a Slip-streaming racer position than when slip-streaming? c) How much further can Graeme Obree travel in 1 hour than someone in a normal racer position, assuming they both sustain a power output of 400 watts?
3. The coefficient of v3 in the power formula P 0 . 0165 v 3 2 . 09 v is due to air resistance. What would happen to this coefficient in the following scenarios? a) The cyclist sat upright b) The cyclist is on a heavier bike Riding uphill The graph shows the power needed to sustain various speeds when riding up a gradient. 1. Use the graph to estimate the 20 mph power output required to maintain a speed of 10mph on an 8% gradient
2. Dave can sustain a maximum 15 mph power output of 400W. What speed should he maintain if he wants to be using his maximum output on a gradient of 4%? 10 mph
3. Mr Walker rides up Muswell Hill 5 mph on his way to work. It is ½ mile long and has a gradient of 10%. It takes him 6 minutes to complete, what is his average power output?