Calculations of speed and power

The obsession for speed is not a new thing in amateur cycling, the obsession for power is, since devices to measure power output have been available on the market for just over a decade. This brief and not at all comprehensive article hopefully will clarify some mysteries around power and speed and how to use your power wisely to get more speed.

There are three forces working against you in your attempt to reach a relativistically important speed (that is a speed in the ballpark of the speed of light, one that Einstein would be interested in): gravity, friction and drag. Gravity can be calculated in Newtons as roughly ten times what we call “weight” and physics calls mass. The friction coefficient can be measured, although you need a set up and roughly speaking it is a constant, which generates a relatively small “power loss” which increases linearly with speed. Aerodynamic drag is a more complicated matter and requires sophisticated equipment to be measured or it can be modelled with software packages. The equation that describes drag is:

FD = 1/2(d v2 CD A)

Where FD is the drag as a force in Newton, d is the air density, v is the relative velocity in one dimension, CD is the aerodynamic coefficient and A is the frontal area. If we assume CD to be a constant, which makes life easier and calculations marginally less accurate, basically we can say that drag increases linearly with the frontal area and with barometric pressure and exponentially with speed, or if you prefer linearly with squared speed. If you then consider that drag is a force and translates into power as P = FD v, then you see that power increases with the cube of speed. In simple currency it means that if you want to double your speed, you need eight times more power. This is a monumental waste of power, if you consider that gravity gives you back speed linearly: double the power, double the speed.

THE ZERO WIND MODEL

In the absence of external wind, it is straightforward to calculate the power needed to travel at a given speed as the sum of the three components: drag + friction + gravity. The data for an 80Kg person (including clothes and bicycle)  with an average aerodynamic coefficient and frontal area are reported in the table below:

v km/h aero drag /W friction  W 0% /W 2.5% /W 5% /W 7.5% /W 10% /W
10 3.125 5 0 55.55555 111.1111 166.6667 222.2222
15 10.546875 7.5 0 83.33333 166.6667 250 333.3333
20 25 10 0 111.1111 222.2222 333.3333 444.4444
25 48.828125 12.5 0 138.8889 277.7778 416.6666 555.5555
30 84.375 15 0 166.6667 333.3333 500 666.6666
40 200 20 0 222.2222 444.4444 666.6666 888.8888
50 390.625 25 0 277.7778 555.5555 833.3333 1111.111

So, for instance if you want to travel at 20 km/h up a 5% slope, you need 25 +10 + 222.2 = 257.2 Watt, around 86% of them are needed to overcome gravity, 10% to overcome drag and 4% to win over road friction. However, if you want to travel at 40 km/h on an absolutely flat road, then of the total 220 Watt needed over 90% are down to drag only.

Now, imagine a time trial of 20 Km, 15 are absolutely flat and then there are 5 km at 5% to get to the finish. By drawing a graph of total power output vs speed in the flat and in the 5% slope cases and extrapolating for two different approaches:

Rider A averages 250 watt on the flat and 200 Watt uphill, rider B averages 200 Watt on the flat and 250 watt uphill.

Rider A will take 21:25 to cover the flat part and 18:45 to climb the hill with a finish time of 40 minutes and 10 seconds and an average output of 227 Watt.

Rider B will take 23:40 to cover the flat part and 15:23 to climb the hill with a finish time of 39:03, and an average output of 220 Watt.

RIder B is the clear winner: over a minute faster, despite using less power than rider A!

There is an important lesson to learn here: on average power is more rewarding if used against gravity than when used against air resistance, so if you have to choose where to maximise your effort, make it a hill!

HOW ABOUT AERO WHEELS?

Aero wheels manufacturers’ wildest claims account for around 20 Watt of drag saved at 50 km/h. Often these figures are obtained in a laboratory controlled environment at unrealistic yaw angles and have little bearing on real world conditions, but let’s assume they are relevant. How much extra speed do 20 Watt buy you if you are travelling at 50 km/h? In order to travel at 51 km/h, according to the power proportional to the cube of speed you will need an extra 8% power on top of the 415 needed to travel at 50 km/h, hence 33 Watt. My calculations suggest 20 Watt should result in an increase in speed of roughly 0.75 km/h or just under half a mile. It doesn’t seem much but it is significant in a time trial, after all it does mean 18 seconds in a classic 10 mile time trial for someone posting a sub 20 minute finish.

The good news is that for those travelling slower, the speed increase is still the same, but the time saving is greater, as it takes longer to cover the same length.

The bad news is that manufacturers’ claims are typically exaggerated.

AERO WHEELS OR LIGHT WHEELS?

Again, let’s look at rider A and rider B, both churning out a constant 200 Watt over the same time trial course describe above. This time rider A chooses a set of aero wheels that in the real world save 10 Watt at 40 km/h and have negligible saving at a speed of under 20 km/h, while rider B chooses a set of super light wheels, 500 grams lighter than those of rider A.

Rider A will cover the 15 km of flat in 23:04 and the climb to the finish in 18:45 and stop the clock at 41:49.

Rider B will cover the 15 km of flat in 23:40 and the climb in 18:38, stopping the clock at 42:18.

This time rider A is the winner, indicating even on a mixed terrain aerodynamics trump weight. It is fair to say that both rider A and B did worse than when they dug in a bit deeper in the previous time trial, indicating that there is no substitute for hard work!

 

 

 

 

 

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