the maths of cycling

The Maths of Cycling
Bikes come in different styles, depending on their intended use
racer
mountain bike
bmx
fast and efficient
can go off-road
small & manoeuvrable
sensitive steering
comfortable and
easy to ride
less parts to go wrong
What do they
look like?
Advantages
Disadvantages
What features
help them suit
their purpose?
only low speeds possible
uncomfortable
riding position
slower and heavier
dangerous for novices
inefficient on roads
uncomfortable riding
position if seated
thin wheels and tyres
with minimal tread
to reduce friction
aggressive, hunched-over
riding position to reduce
air resistance and
maximise power output
from legs
thick tyres with deep
tread to maximise
grip and stability
much smaller frame
and wheels make
tricks easier to do
relaxed, upright riding
position to increase
comfort and allow rider
to look up more easily
low saddle so it
isn’t in the way
only one easy gear to
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
by how many teeth they have
cassette
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
cassette size
gear
ratios
chain ring size
34
50
27
1.26
1.85
24
1.42
2.08
21
1.62
2.38
18
1.89
2.78
15
2.27
3.33
12
2.83
4.12
chain ring
which combination of chain ring and cassette
gear are the lowest gear ratio (easiest to pedal)
and the highest gear ratio (hardest to pedal)?
lowest – 34/27
highest – 50/12
which different combinations of chain ring
and cassette gear are almost the same ratio?
34/18 and 50/27
for every 100 rotation of the pedals, how many
rotations do the wheels make in the highest gear? 100 x 4.12 = 412
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.
They are pedalling at 50 revolutions per minute.
What is their speed in miles per hour?
Distance travelled
by wheel in 1 hour = 2 x 50 x 60 x 4
= 24000m
using the different gears and varying
your pedalling speed enables you to
reach a range of speeds:
revolutions per minute
speeds
mph
60
(steady
pedalling)
120
(very fast
pedalling)
1
4.95
9.9
2
9.9
19.8
3
14.85
29.7
4
19.8
39.6
≈ 15 miles
of course, you can only reach the
higher speeds if have the power to
pedal very fast in a high gear...
ratio
1 mile ≈ 1600 metres
Gear ratio and speed
b) Here are the gear ratios for my new bike:
1a) Can you complete the table?
cassette size
gear
ratios
chain ring size
30
40
30
1
1.333...
20
1.5
2
16
1.875
2.5
10
3
4
Gear
1st
2nd
3rd
4th
5th
Gear ratio
1.2
1.5
2
2.5
4
i) For every 60 rotation of the pedals, how many
rotations do the wheels make in 2nd gear?
60 x 1.5 = 90
ii) For every 40 rotation of the wheels, how
many rotations do the pedals make in 5th gear?
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
1.98  120 60  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!

2  c  60  r 1600  20  c  32000
120r
eg
r  3  c  89
r  4  c  67
800
3r
The gear ratio required to
break speed records is
much higher than those
found on normal bikes
The bike used by Guy Martin when he broke the
British speed record had only a gear with an even
higher ratio of 15, achieved by a system of cogs
In 1962, Jose Meiffret broke the
motor-paced cycling World Record
on a bike with a gear ratio of 8.7!
Given that he pedalled with a cadence of 97 rpm
and his bike had wheels with a circumference of
2.07 metres, how fast did he go?
112.94 mph
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
v mph on level ground, using an average adult +
bike of 90kg, is given by the following formula:
This is a cubic relationship
between speed and power:
P  0.0165 v3  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  0.0165 v3  2.09v
1. Use the formula to find the power
needed to maintain a speed of:
a) 5 mph
0.0165 53  2.09 5  13W
b) 20 mph
0.0165 203  2.09 20  174W
2. A cyclist has a maximum power
output of 800W. To the nearest
mph, what speed can he achieve,
when using his maximum output?
v  35  P  781W
v  36  P  845W
closest is 35 mph
3. Mark Cavendish sprints at 45mph
when winning stages of the Tour de
France. How many 60W light bulbs
could he power with this effort?
v  45  P  1597.6125
1597.6125 60  26.62...
about 26 light bulbs!
4. The coefficient of v in the power
formula is due to mass and friction.
What would happen to this coefficient
in the following scenarios?
a) The cyclist were heavier
it would increase
b) The cyclist switched from a
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  0.0165 v3  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
me!
press
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
33 Obree ___
36 Slipstreaming ___
37½
Racer ___
b) Another cyclist is travelling at 30 mph. How
much more power does he need to do this in a
racer position than when slip-streaming?
Obree position
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 v3  2.09v 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 windbreak attached on the Bonneville salt flats
Riding uphill
Anyone who has cycled knows
that going downhill is a lot
easier than going uphill
because of the effect of gravity!
We saw previously that the
If the
cyclist
climbing
a slope
power
P Wisfor
a cyclist
to
with gradient
m as
a percentage,
maintain
a speed
v mph,
using an
the ruleadult
has another
average
+ bike ofelement:
90kg, is
given by the following formula:
P  0.0165 v3  2.09v  3.94vm
This means it is much harder to climb
slopes, especially at higher speeds:
20 mph
10 mph
Riding uphill
The graph shows the power needed to sustain various speeds when riding up a gradient.
1. Use the graph to estimate the
power output required to maintain a
speed of 10mph on an 8% gradient
20 mph
350W
2. Dave can sustain a maximum
power output of 400W. What
speed should he maintain if he
wants to be using his maximum
output on a gradient of 4%?
15 mph
10 mph
about 17½ mph
3. Mr Walker rides up Muswell Hill
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
5 mph
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
1a) Can you complete the table?
cassette size
gear
ratios
chain ring size
30
b) Here are the gear ratios for my new bike:
Gear
1st
2nd
3rd
4th
5th
Gear ratio
1.2
1.5
2
2.5
4
i) For every 60 rotation of the pedals, how many
rotations do the wheels make in 2nd gear?
1
2
16
2.5
3
ii) For every 40 rotation of the wheels, how
many rotations do the pedals make in 5th gear?
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  0.0165 v3  2.09v
1. Use the formula to find the power
needed to maintain a speed of:
a) 5 mph
3. Mark Cavendish sprints at 45mph
when winning stages of the Tour de
France. How many 60W light bulbs
could he power with this effort?
b) 20 mph
2. A cyclist has a maximum power
output of 800W. To the nearest
mph, what speed can he achieve,
when using his maximum output?
4. The coefficient of v in the power
formula is due to mass and friction.
What would happen to this coefficient
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 ___
b) Another cyclist is travelling at 30 mph. How
much more power does he need to do this in a
racer position than when slip-streaming?
Obree position
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 v3  2.09v 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
power output required to maintain a
speed of 10mph on an 8% gradient
2. Dave can sustain a maximum
power output of 400W. What
speed should he maintain if he
wants to be using his maximum
output on a gradient of 4%?
3. Mr Walker rides up Muswell Hill
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?
20 mph
15 mph
10 mph
5 mph