## The Motion of a Cyclist

Why is the "new" UCI hour record using the normal drop position roughly 6 km/h slower than the superman position ? Is the Obree-position really so much faster ? How much faster will I ride if I use the aero position instead of the normal drop position ? What happens if I use disc wheels? On this page you can calculate the effect.

First some basics:(we need some math to understand the effect...)
The total resistance opposing the motion of a cyclist on a flat terrain is the sum of the aerodynamic drag of air  FAir and the rolling resistance  FRoll.
The wind resistance FAir (in Newton) depends on the aerodynamic drag coefficient cw, the projected frontal area A of bike and rider (in square meters), the air density r (about 1.225 kg/m³ at sea level for 15° C), and the square of the air speed v (in m/sec).
The air density is important if one rides at different altitudes, for a detailed view of this topic look at my hour record at altitude page.
Since it is difficult to quantify the area A precisely, one associates cw and A  to the effective frontal area cwA.
FAir = 0.5 ×cw×A×r×v² =  0.5 ×cwA×r×
The power needed to overcome the air resistance is FAir multiplied with the velocity v:
PAir = FAir×v = 0.5 ×cwA×r×
Typical values for cwA are around 0.25 m² (see table below). Example: For a cwA value of 0.25 m2 you need 92 Watts to overcome the air resistance at 30 km/h, that's quite easy... But if you double the speed to 60 km/h, you would need a factor of 2³ = 8 more, that's 736 Watts!

The rolling resistance  FRoll depends on on the rolling coefficient cr, the mass m of the rider in kg, and the acceleration due to gravity g (9.81 m/s²). It is independent of the velocity v.
FRoll = cr×m×g
Values for cr are around 0.003-0.006 for racing bikes on a track, giving 3-5 Newton as rolling resistance for a 70 kg rider plus 10 kg bicycle.
Of course the power needed to overcome the rolling resistance depends on the velocity v:
PRoll = FRoll×v= cr×m×g×v
Putting in some values you'll see that at low speed the rolling resistance is not negligible compared to the aerodynamic drag. But when you go faster, the rolling resistance gets less and less important compared to the aerodynamic drag.

The total power needed is then:   Ptot = PAir + PRoll

Though the efficiency Eff of the bicycle is quite high (around 95-98%), but not equal to 100%  due to losses in the drivetrain and hubs, the power output the rider has to deliver is:  PRider = Ptot / Eff

Using those equations we can caclulate how much power we need for a given input. Most important at high speeds is the value for cwA. Here is a table with different measurements. One can see that the differences are quite large.

Values for cwA
TOUR 9/96 Different Wheels (1) German Magazines (2) Scientific Publications (3)
Wheel Type cwa Reference cwa Reference cwa
Disc 0.2328 Drop Position TOUR 9/94 0.252  Grappe 1997 Drop Position 0.276
Shamal HPW12 0.2423 Extreme Drop Positon TOUR 0.235 Grappe 1997 Aero Position 0.262
Cosmic 0.2439 Obree TOUR 9/94 0.210 Grappe 1997 Obree Position 0.216
Citec 12 spokes 0.2446 Boardman VELO 9/94 0.22 Davies 1980 Drop Position 0.280
Spinergy 0.2462 Obree VELO 9/94 0.18  Kyle 1975 Drop Position 0.272
HED Jet 2510 Moser 1984 VELO 9/94 0.30 Menard 1990 Aero Position 0.260
Rigida DP 18 0.2510
Spengle Tri-Spoke 0.2518
Standard 36 spokes 0.2731
(For some more recent wheel tests see below)

Now let's calculate the power! (Efficiency of the chain = 98%) (Click on button "Power")
You can calculate the power for two different setups (change cwA, cr, or the mass of the rider plus bike). If the values on the right side are smaller than on the left side, the power will be smaller. If you click on the button "Calculate Improvement Table", a window will open which shows how much faster the rider will be. Clicking "Plot sec per km" will create a plot showing how many seconds per kilometer one gains for a given speed, clicking "Plot km/h per km" will do a plot showing the gain in km/h.
 speed (km/h) total weight (bike + rider) (kg) cw*A value cr value power due to air resistance power due to rolling resistance power (air + roll) total power (due to 98% efficiency)
(uses Java applet tChart.)

(1)Bike with aerobar setup. Tested were different wheels on a track using a SRM powermeter at 45 km/h. cwA is calculated with FRoll = 4,5 Newton and 98% Efficiency.
(2)Wind tunnel measurement by german magazines "Tour" and "Velo"
(3) Grappe 1997: Ergonomics, 1997, Vol 40, No 12, 1299
Menard 1990: Contribution a l'amelioration des performances du coureur cycliste, Institute Aerotechnique de Saint- Cyr,288
Kyle 1979: Ergonomics, 1979, Vol 22, 387

Wheel Tests: (wind tunnel test of aerodynamic wheels)

In these tests the single wheel is tested in a wind tunnel, and one gets the power in Watts absorbed at a specific speed. These values can be used in the calculator above. Since the rear wheel is influenced by the frame and the rider the results are in principle only valid for the front wheel.