# Bicycle performance

Bradley Wiggins in the yellow jersey, finishing the 2011 Critérium du Dauphiné.
A heavy duty freight bicycle made by SCO, Denmark can carry more than 100 kilograms (220 lb).

A bicycle's performance, in both biological and mechanical terms, is extraordinarily efficient. In terms of the amount of energy a person must expend to travel a given distance, investigators have calculated it to be the most efficient self-powered means of transportation.[1] From a mechanical viewpoint, up to 99% [2] of the energy delivered by the rider into the pedals is transmitted to the wheels, although the use of gearing mechanisms may reduce this by 10–15%.[3][4] In terms of the ratio of cargo weight a bicycle can carry to total weight, it is also a most efficient means of cargo transportation.

## Energy efficiency

A human being traveling on a bicycle at 16–24 km/h (10–15 mph), using only the power required to walk, is the most energy-efficient means of transport generally available.[5] Air drag, which increases with the square of speed,[6] requires increasingly higher power outputs relative to speed, power increasing with the cube of speed as power equals force times velocity. A bicycle in which the rider lies in a supine position is referred to as a recumbent bicycle or, if covered in an aerodynamic fairing to achieve very low air drag, as a streamliner.

Racing bicycles are light in weight, allow for free motion of the legs, keep the rider in a comfortably aerodynamic position, and feature high gear ratios and low rolling resistance.

On firm, flat ground, a 70 kg (150 lb) person requires about 60 watts [7] to walk at 5 km/h (3.1 mph). That same person on a bicycle, on the same ground, with the same power output, can average 15 km/h (9.3 mph), so energy expenditure in terms of kcal/(kg·km) is roughly one-third as much. Generally used figures are

• 1.62 kJ/(km∙kg) or 0.28 kcal/(mi∙lb) for cycling,
• 3.78 kJ/(km∙kg) or 0.653 kcal/(mi∙lb) for walking/running,
• 16.96 kJ/(km∙kg) or 2.93 kcal/(mi∙lb) for swimming.

Amateur bicycle racers can typically produce 3 watts/kg for more than an hour (e.g., around 210 watts for a 70 kg rider), with top amateurs producing 5 W/kg and elite athletes achieving 6 W/kg for similar lengths of time.[citation needed] Elite track sprinters are able to attain an instantaneous maximum output of around 2,000 watts, or in excess of 25 W/kg;[citation needed] elite road cyclists may produce 1,600 to 1,700 watts as an instantaneous maximum in their burst to the finish line at the end of a five-hour long road race.[citation needed] Even at moderate speeds, most power is spent in overcoming the aerodynamic drag force, which increases with the square of speed.[6] Thus, the power required to overcome drag increases with the cube of the speed.

## Typical speeds

In an urban environment, there are no typical speeds for a person riding a bicycle; an elderly person on a sit-up-and-beg style roadster might do less than 10 km/h (6.2 mph) while a fitter, younger person could easily do twice that on the same bicycle. For cyclists in Copenhagen, the average cycling speed is 15.5 km/h (9.6 mph).[8]

On a fast racing bicycle, a reasonably fit rider can ride at 50 km/h (31 mph) on flat ground for short periods.[citation needed]

### Cycling speed records

The highest speed officially recorded for any human-powered vehicle (HPV) on level ground and with calm winds and without external aids (such as motor pacing and wind-blocks) is 133.78 km/h (83.13 mph) set in 2013 by Sebastiaan Bowier in the VeloX3, a streamlined recumbent bicycle.[9] In the 1989 Race Across America, a group of HPVs crossed the United States in just 5 days.[10][11][12][13] The highest speed officially recorded for a bicycle ridden in a conventional upright position under fully faired conditions was 82.52 km/h (51.28 mph) over 200m.[14][15] That record was set in 1986 by Jim Glover on a Moulton AM7 at the 3rd international HPV scientific symposium at Vancouver.

## Weight vs power

There has been major corporate competition to lower the weight of racing bikes. Wheels are available with comparatively lower friction bearings and other features to lower resistance, however in measured tests[which?] these components have almost no effect on cycling performance when riding on flat ground. The UCI sets a limit on the minimum weight of bicycles to be used in sanctioned races,[16] to discourage making structures so thin that they become unsafe. For these reasons recent designs have concentrated on lowering wind resistance by using aerodynamically shaped tubing, flat spokes on the wheels, and handlebars that position the rider's torso and arms for minimal drag. These changes can impact performance dramatically[quantify], cutting minutes off a time trial.[citation needed] Less weight results in larger time savings on uphill terrain.

### Kinetic energy of a rotating wheel

Consider the kinetic energy and "rotating mass" of a bicycle in order to examine the energy impacts of rotating versus non-rotating mass.

The translational kinetic energy of an object in motion is:[17]

$E = \tfrac{1}{2}mv^2$,

Where $E$ is energy in joules, $m$ is mass in kg, and $v$ is velocity in meters per second. For a rotating mass (such as a wheel), the rotational kinetic energy is given by

$E = \tfrac{1}{2}I \omega^2$,

where $I$ is the moment of inertia, $\omega$ (pronunciation: omega) is the angular velocity in radians per second. For a wheel with all its mass at the outer edge (a fair approximation for a bicycle wheel), the moment of inertia is

$I = m r^2$.

Where $r$ is the radius in meters

The angular velocity is related to the translational velocity and the radius of the tire. As long as there is no slipping,

$\omega = \frac{v}{r}$.

When a rotating mass is moving down the road, its total kinetic energy is the sum of its translational kinetic energy and its rotational kinetic energy:

$E = \tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2$

Substituting for $I$ and $\omega$, we get

$E = \tfrac{1}{2}mv^2 + \tfrac{1}{2}mr^2 \cdot \frac{v^2}{r^2}$

The $r^2$ terms cancel, and we finally get

$E = \tfrac{1}{2}mv^2 + \tfrac{1}{2}mv^2 = mv^2$.

In other words, a mass on the tire has twice the kinetic energy of a non-rotating mass on the bike. This all depends, of course, on how well a thin hoop approximates the bicycle wheel. In reality, all the mass cannot be at the radius. For comparison, the opposite extreme might be a disk wheel where the mass is distributed evenly throughout the interior. In this case $I = \tfrac{1}{2}m r^2$ and so the resulting total kinetic energy becomes $E = \tfrac{1}{2}mv^2 + \tfrac{1}{4}mv^2 = \tfrac{3}{4}mv^2$. A pound off the disk wheels = only 1.5 pounds off the frame. Most real bicycle wheels will be somewhere between these two extremes.

One other interesting point from this equation is that for a bicycle wheel that is not slipping, the kinetic energy is independent of wheel radius. In other words, the advantage of 650C or other smaller wheels is due to their lower weight (less material in a smaller circumference) rather than their smaller diameter, as is often stated. The KE for other rotating masses on the bike is tiny compared to that of the wheels. For example, pedals turn at about $\tfrac{1}{5}$ the speed of wheels, so their KE is about $\tfrac{1}{25}$ (per unit weight) that of a spinning wheel. As their center of mass turns on a smaller radius, this is further reduced.

### Convert to kilocalories

Assuming that a rotating wheel can be treated as the mass of rim and tire and 2/3 of the mass of the spokes, all at the center of the rim/tire. For a 180 lb (82 kg) rider on an 18 lb (8 kg) bike (90 kg total) at 25 mph (40 km/h ; 11.2 m/s), the KE is 5625 joules for the bike/rider plus 94 joules for a rotating wheel (combined 1.5 kg of rims/tires/spokes). Converting joules to kilocalories (multiply by 0.0002389) gives 1.4 kilocalories (nutritional calories).

Those 1.4 kilocalories are the energy necessary to accelerate from a standstill, or the heat to be dissipated by the brakes to stop the bike. These are kilocalories, so 1.4 kilocalories will heat 1 kg of water 1.4 degrees Celsius. Since aluminum's heat capacity is 21% of water, this amount of energy would heat 800 g of alloy rims 8 °C (15 °F) in a rapid stop. Rims do not get very hot from stopping on flat ground. To get the rider's energy expenditure, consider the 24% efficiency factor to get 5.8 kilocalories—accelerating a bike/rider to 25 mph (40 km/h) requires about 0.5% of the energy required to ride at 25 mph (40 km/h) for an hour. This energy expenditure would take place in about 15 seconds, at a rate of roughly 0.4 kilocalories per second, while steady state riding at 25 mph (40 km/h) requires 0.3 kilocalories per second.

The advantage of light bikes, and particularly light wheels, from a KE standpoint is that KE only comes into play when speed changes, and there are certainly two cases where lighter wheels should have an advantage: sprints, and corner jumps in a criterium.[18]

In a 250 m sprint from 36 to 47 km/h to (22 to 29 mph), a 90 kg bike/rider with 1.75 kg of rims/tires/spokes increases KE by 6,360 joules (6.4 kilocalories burned). Shaving 500 g from the rims/tires/spokes reduces this KE by 35 joules (1 kilocalorie = 1.163 watt-hour). The impact of this weight saving on speed or distance is rather difficult to calculate, and requires assumptions about rider power output and sprint distance. The Analytic Cycling web site allows this calculation, and gives a time/distance advantage of 0.16 s/188 cm for a sprinter who shaves 500 g off their wheels. If that weight went to make an aero wheel that was worth 0.03 mph (0.05 km/h) at 25 mph (40 km/h), the weight savings would be canceled by the aerodynamic advantage. For reference, the best aero bicycle wheels are worth about 0.4 mph (0.6 km/h) at 25, and so in this sprint would handily beat a set of wheels weighing 500 g less.

In a criterium race, a rider is often jumping out of every corner. If the rider has to brake entering each corner (no coasting to slow down), then the KE that is added in each jump is wasted as heat in braking. For a flat crit at 40 km/h, 1 km circuit, 4 corners per lap, 10 km/h speed loss at each corner, one hour duration, 80 kg rider/6.5 kg bike/1.75 kg rims/tires/spokes, there would be 160 corner jumps. This effort adds 387 kilocalories to the 1100 kilocalories required for the same ride at steady speed. Removing 500 g from the wheels, reduces the total body energy requirement by 4.4 kilocalories. If the extra 500 g in the wheels had resulted in a 0.3% reduction in aerodynamic drag factor (worth a 0.02 mph (0.03 km/h) speed increase at 25 mph), the caloric cost of the added weight effect would be canceled by the reduced work to overcome the wind.

Another place where light wheels are claimed to have great advantage is in climbing. Though one may hear expressions such as "these wheels were worth 1–2 mph", etc. The formula for power suggests that 1 lb saved is worth 0.06 mph (0.1 km/h) on a 7% grade, and even a 4 lb saving is worth only 0.25 mph (0.4 km/h) for a light rider. So, where is the big savings in wheel weight reduction coming from? One argument is that there is no such improvement; that it is "placebo effect". But it has been proposed that the speed variation with each pedal stroke when riding up a hill explains such an advantage. However the energy of speed variation is conserved; during the power phase of pedaling the bike speeds up slightly, which stores KE, and in the "dead spot" at the top of the pedal stroke the bike slows down, which recovers that KE. Thus increased rotating mass may slightly reduce speed variations, but it does not add energy requirement beyond that of the same non-rotating mass.

Lighter bikes are easier to get up hills, but the cost of "rotating mass" is only an issue during a rapid acceleration, and it is small even then.

## Aerodynamics vs power

Heated debates over the relative importance of weight savings and aerodynamics are a fixture in cycling. This is an attempt to at least get the equation-based parts of the debate clarified. There will always be those who argue that "experience trumps mathematics" on this issue, so this will attempt to highlight those areas where experience might disagree with the math. From this, perhaps further discussion can focus on the topics of dispute rather than questioning known physics. To be as clear as possible, this will cover 1) the power requirements for moving a bike/rider, 2) the energy cost of acceleration, and then 3) why experience and the math might disagree.

### Power required

There is a well-known equation that gives the power required to push a bike and rider through the air and to overcome the friction of the drive train:

$P = g m V_g (K_1+s) + K_2 V_a^2 V_g$,

where $P$ is in watts, $g$ is Earth's gravity, $V_g$ is ground speed (m/s), $m$ is bike/rider mass in kg, $s$ is the grade (m/m), and $V_a$ is the rider's speed through the air (m/s). $K_1$ is a lumped constant for all frictional losses (tires, bearings, chain), and is generally reported with a value of 0.0053. $K_2$ is a lumped constant for aerodynamic drag and is generally reported with a value of 0.185 kg/m.[19] If there is no wind, $V_g=V_a$ and the result simplifies to:

$P = g m V_g (K_1+s) + K_2 V_g^3$,

which is proportional to the ground speed cubed in its leading term.

Note that the power required to overcome friction and gravity is proportional only to rider weight and ground speed. The aerodynamic drag is roughly proportional to the square of the relative velocity of the air and the bike. Being that the total power requirement to propel the bike forward is a sum of these two variables multiplied by speed, the degree of proportionality between power requirement and speed varies according to their relative magnitude, in an interval between the linear and cube: at higher speeds (riding fast on a flat road) power required will be close to being a cube function of speed, at lower speeds (climbing a steep hill) it will be close to being a linear function of speed.

The human body runs at about 24% efficiency for a relatively fit athlete, so for every kilojoule delivered to the pedals the body consumes 1 kcal (4.2 kJ) of food energy.[citation needed]

Obviously, both of the lumped constants in this equation depend on many variables, including drive train efficiency, the rider's position and drag area, aerodynamic equipment, tire pressure, and road surface. Also, recognize that air speed is not constant in speed or direction or easily measured. It is certainly reasonable that the aerodynamic lumped constant would be different in cross winds or tail winds than in direct head winds, as the profile the bike/rider presents to the wind is different in each situation. Also, wind speed as seen by the bike/rider is not uniform except in zero wind conditions. Weather report wind speed is measured at some distance above the ground in free air with no obstructing trees or buildings nearby. Yet, by definition, the wind speed is always zero right at the road surface. Assuming a single wind velocity and a single lumped drag constant are just two of the simplifying assumptions of this equation. Computational fluid dynamicists have looked at this bicycle modeling problem and found it hard to model well. In layman's terms, this means that much more sophisticated models can be developed, but they will still have simplifying assumptions.

Given this simplified equation, however, one can calculate some values of interest. For example, assuming no wind, one gets the following results for kilocalories required and power delivered to the pedals (watts):

• 175 W for a 90 kg bike + rider to go 9 m/s (32 km/h or 20 mph) on the flat (76% of effort to overcome aerodynamic drag), or 2.6 m/s (9.4 km/h or 5.8 mph) on a 7% grade (2.1% of effort to overcome aerodynamic drag).
• 300 W for a 90 kg bike + rider at 11 m/s (40 km/h or 25 mph) on the flat (83% of effort to overcome aerodynamic drag) or 4.3 m/s (15 km/h or 9.5 mph) on a 7% grade (4.2% of effort to overcome aerodynamic drag).
• 165 W for a 65 kg bike + rider to go 9 m/s (32 km/h or 20 mph) on the flat (82% of effort to overcome aerodynamic drag), or 3.3 m/s (12 km/h or 7.4 mph) on a 7% grade (3.7% of effort to overcome aerodynamic drag).
• 285 W for a 65 kg bike + rider at 11 m/s (40 km/h or 25 mph) on the flat (87% of effort to overcome aerodynamic drag) or 5.3 m/s (19 km/h or 12 mph) on a 7% grade (6.1% of effort to overcome aerodynamic drag).

Shaving 1 kg off the weight of the bike/rider would increase speed by 0.01 m/s at 9 m/s on the flat (5 seconds in a 32 km/h (20 mph), 40-kilometre (25 mile) TT). Losing 1 kg on a 7% grade would be worth 0.04 m/s (90 kg bike + rider) to 0.07 m/s (65 kg bike + rider). If one climbed for 1 hour, saving 1 lb would gain between 69 metres (225 ft) and 110 m (350 ft) – less effect for the heavier bike + rider combination (e.g., 0.06 km/h (0.04 mph) * 1 h * 1,600 m (5,200 ft)/mi = 69 m (226 ft)). For reference, the big climbs in the Tour de France have the following average grades:

The equation can be separated into level ground power

$P_{level} = g m V_g K_1 + K_2 V_a^2 V_g$,

and vertical climbing power given by

$P_{climbing} = mg(h/t) \approx gm (V_g s)$.

### Energy cost of acceleration

$P_{accelerating} = m*a*V$

## References

1. ^ S.S. Wilson (March 1973). "Bicycle Technology". Scientific American.
2. ^ Wilson, David Gordon; Jim Papadopoulos (2004). Bicycling Science (Third ed.). Massachusetts Institute of Technology. p. 343. ISBN 0-262-23111-5.
3. ^ Phil Sneiderman Homewood (August 30, 1999). "Pedal Power Probe Shows Bicycles Waste Little Energy". Johns Hopkins Gazette. Archived from the original on 1 February 2010. Retrieved 2010-02-21.
4. ^ Wilson, David Gordon; Jim Papadopoulos (2004). Bicycling Science (Third ed.). The MIT Press. p. 318. ISBN 0-262-73154-1. "When new, clean, and well-lubricated, and when sprockets with a minimum of 21 teeth are used, a chain transmission is highly efficient (at a level of maybe 98.5 percent or even higher)."
5. ^ MacKay, David J C (2008). Sustanible Engergy (First ed.). UIT Cambridge Ltd. p. 128.
6. ^ a b Wilson, David Gordon; Jim Papadopoulos (2004). Bicycling Science (Third ed.). The MIT Press. p. 126. ISBN 0-262-73154-1. "aerodynamic drag force is proportional to the square of the velocity"
7. ^ Kohsuke Shimomura, et al (2009-11-10). "A study of passive weight-bearing lower limb exercise effects on local muscles and whole body oxidative metabolism: a comparison with simulated horse riding, bicycle, and walking exercise". Retrieved 2014-07-26.
8. ^ "Bicycle statistics". City of Copenhagen website. City of Copenhagen. 13 June 2013. Retrieved 12 December 2013.
9. ^ http://www.hptdelft.nl/en/index.php?option=com_content&view=article&id=508:hightech-recumbent-from-delft-breaks-world-record&catid=15:blog&Itemid=58
10. ^ Wired.com (2008-09-25). "World's Fastest Cyclist Hits 82.3 MPH". Archived from the original on 26 September 2008. Retrieved 2008-09-26.
11. ^ "International Human Powered Vehicle Association Official Speed Records". Archived from the original on 12 April 2008. Retrieved 2008-03-04.
12. ^ "Fastest Human Powered Lists". Archived from the original on 8 March 2008. Retrieved 2008-03-04.
13. ^ "HPV And Bicycle Speed Records Men – Single Rider". Archived from the original on 12 April 2008. Retrieved 2008-03-04.
14. ^ "Moulton Bicycle Company: Records and Racing". Archived from the original on 12 April 2010. Retrieved 2010-02-26.
15. ^ "Aerodynamic research using the Moulton small-wheeled bicycle". Retrieved 2010-02-26.
16. ^ uci.ch. "UCI Rules". Retrieved 2010-07-27.
17. ^ Ruina, Andy; Rudra Pratap (2002). Introduction to Statics and Dynamics (PDF). Oxford University Press. p. 397. Archived from the original on 12 September 2006. Retrieved 2006-08-04.
18. ^ "Technical Q&A with Lennard Zinn: The great rotating-weight debate". Archived from the original on 2006-10-17. Retrieved 2007-02-03.
19. ^ Corresponding to a surface area of 0.4m^2 with a drag coefficient of 0.7: Drag (physics)#Power
20. ^ Sastre wins the 2008 L'Alpe d'Huez stage. July 23, 2008. p. Velo News. Archived from the original on 19 February 2009. Retrieved 2009-01-14.