On Hills, Wind, and Cycling Power
This post is intended to get a little “under the hood” in contemplating why we ride the bike the way we do by explaining an approximation of the physics of bike riding as it relates to the application of power with the extra features of hills and wind. I start with a little introduction of the units and vocabulary to get everyone on the same page. If this is old hat, feel free to skip down. J
Getting our Units Straight
Let’s distinguish between acceleration, force, power, and energy.
Acceleration is the change of velocity (speed) with time and has units of m/s2. Technically, we can speak of “deceleration” as acceleration in the opposite direction of travel. Gravity provides an acceleration field of 9.8 m/s2.
Force can be experienced as being like “total pressure” against an object. When you press on the pedals, you apply force. (Torque is the angular analog of force.) We can write the standard equation
F = ma, which means Force = (mass) x (acceleration)
to express, for example your weight in a field of gravity, or (taken another way) how much acceleration is given by applying a certain amount of force to an object of a certain mass.
“Weight” is the force an object exerts under a gravitational field. So when we express our “weight” in kg, what we are really expressing is our mass, not our weight. (And when we express our weight in pounds, that IS a force, and yes you do weigh less on the moon, although your mass doesn’t change….bonus points for anyone who knows the English/Imperial unit of mass! See the answer at the bottom!) The lumped unit in the metric system for force is the Newton, abbreviated N. It corresponds to the English unit of the pound.
Energy can be expressed in many ways, but the most relevant one here is the application of force across a distance. When you lift an object, you apply force (to overcome gravity) across a distance. If you lift the object 1 meter, it doesn’t matter whether it takes you 1 second or 1 hour, the same amount of energy has been used. In this context, “work” is a synonym for energy.
In the context of gearing, the amount of energy expended is independent of what gear you are in when you roll a bicycle forward a certain distance. In a high (hard) gear, you will have applied a higher force to the pedals, but your pedals will not have had to go very far around. In a low gear, you will have applied a lower force to the pedals, but will have had to move the pedals much farther.
Energy = Force x distance
The units of energy/work are m2kg/s2. However, the lumped unit is Joule (abbreviated J) or 1000 of them is a kilojoule (kJ). It is related to the calorie by a factor of 4.184.
Your power meter can keep track of the amount of power you are applying at every second and add this up over time. Thus you can get the total energy spent in kJ.
Power is the rate at which energy is expended. If you raise a brick up one meter, that requires a certain number of Joules. If you do it fast, you did it at a high work rate or high power. If you do it more slowly, you did it at lower power.
Power = Energy/time = m2kg/s3 = Joules/sec = Watts
When we use a power meter on a bike, then, what we are measuring is our ability to spend a certain amount of energy per time. If your FTP is 200 W, that means your physiology allows you to apply 200 joules per second for an hour. As you get stronger, you can apply more force to the pedals across a certain distance in a shorter amount of time, i.e., press harder in a higher gear at a higher cadence.
Common Experience Vocabulary vs. Scientific units
It is central to understand that thermodynamic “work” and “energy” and “power” are different than our common experience. Two examples will suffice to make this point:
· Pick up a heavy object 1 inch from its resting spot and hold it there for a long time. As a person you will experience this as work. The same is true if you were to hold yourself up in a pull-up position. However, since no distance is crossed against the force (of gravity), no thermodynamic “work” has occurred. But we need a word to express this real human expenditure – I’ll use the general term “exertion”.
· From a physics point of view, power is power and energy is energy. However, physiologically, we know that it costs more “human exertion” to expend the same amount of energy in a shorter time (i.e., at a higher power rate). This is the basis of “normalized power” on the bike; the normalized power over a given period of time is always higher than the plain-old-average power, because WKO+ makes an approximation that “exertion” is proportional to the fourth power of actual power. This means that your high power moments “cost” you much more in terms of “exertion” than your steady moments near your average power.
Resistance on a Bicycle
In bike riding, we face various “resisting forces” that we must overcome by the application of force to the pedals over time, i.e., applying a counter-force.
- Mechanical inefficiency (mostly frictional losses, but also flexion of the bike, etc.)
- Rolling resistance (basically the losses caused by deformation of the tire at the contact point and the interaction with the road surface)
- Gravitational resistance (which is zero on a flat road, works in your favor going downhill, and against you going uphill)
- Wind resistance (basically the friction of you and your bike having to push the air out of the way – aerodynamics are all about reducing this one!)
Of these, only wind resistance depends in any significant way on the rider’s velocity.
Mechanical Efficiency
Mechanical inefficiency is a constant and the losses are only a few percent. Things like cross gearing and not keeping your bike in good alignment and well lubed can make your mechanical efficiency lower. It’s a shame to throw away even 1% of your power, so you should keep your bike well maintained, but we can ignore this term for the remainder of the discussion.
Rolling Resistance
Rolling resistance is generally expressed as a force this way:
Rolling resistance force = Crr x N,
where N is the “weight” of the bike and rider and Crr is a lumped constant that contains information about the tire width, tire pressure, road quality, etc. (This is where using good tires, latex tubes, and proper tire pressure helps you.) The important thing is that rolling resistance does not vary much with velocity .
Gravitational Resistance
Gravitational force = gravity x (total mass of rider and bike) x (hill gradient)
As long as we allow that the hill gradient can be positive (uphill), zero (flats), or negative (downhill), this equation covers all situations. Again, gravitational forces do not vary with rider speed.
Wind Resistance
Wind drag = A x Cd x r x (Relative wind velocity)2
Where A = effective frontal area of the cyclist and rider; Cd is a lumped constant that reflects the effect of shape, materials, etc.; and r is the density of air (which matters, but we can take as a constant for purposes of our discussion). CdA is what you are trying to minimize by having a good aero fit, having good wheels and frame, etc.
Importantly the relative wind velocity can be generated by the rider moving in still air or by wind blowing at the rider. We will neglect the effect of the sideways component of wind resistance here. That isn’t entirely fair because side wind is “bad”, but it does not change any of the conclusions we reach.
The critical thing to see here is that wind resistance goes up with the SQUARE of the relative wind velocity. But it’s even worse than that – if the wind is still, and you are generating all the relative wind velocity by moving through the wind, it goes up as the cube of your velocity because you going faster is how you are increasing the “wind”!
How does this matter in the real world?
So which forces resist us the most on the bike? We all know if you ride at some fixed power (say, 200 W) on a flat, you don’t keep going faster and faster. Instead, what happens is that the force your wattage applies in the forward direction is eventually balanced by the resistance forces, and at that point, your velocity becomes a constant…at least until the next hill or wind gust! As you get faster, the wind resistance goes up until the sum total of all resistance balances the average force going forward (The force going forward is directly proportional to the instantaneous power you are applying at the moment.)
The dominant resistance force depends on how fast you are going and whether you are going up a hill. For now, we assume you are riding on a flat surface with no outside wind.
In still air, wind resistance is obviously zero when you are not going anywhere, and it is still small when you are going slowly. Most of us do not note wind resistance when we walk at 3-4 mph in still air.
For normal-sized people and bikes, rolling resistance is the major resistance force at low velocity, but it remains constant as the rider speeds up…while wind resistance goes up with velocity. At speeds (or relative wind velocities) less than 10 mph, rolling resistance dominates. By 20 mph, 80% or more of the resistance is from the wind.
The squared nature of the wind resistance term explains a key fact: Once you are going roughly 20 mph or more, an increase of power of 10% will NOT get you to go 10% faster! In fact, you will go more like 3% faster. The exact relationship is more complex than we show here, but very roughly, on a flat, in the neighborhood of 20 mph and higher, you get about a square root’s worth of “return” on your investment of higher power.
In judging how to apply power across a race course, you have to balance the “ROI” of additional force/power vs. the “human exertion” it costs.
Add in Hills
Unlike wind resistance, you get a linear return on investment of power against gravity. If the hill is significant enough that you are going under 10 mph, if you apply 10% more power, you will go almost 10% faster because the wind resistance is small. Obviously on a more gentle hill, the “ROI” of your power increase will be somewhere between the linear and square root figure.
How About Wind?
So, let’s say you are riding in a flat, but into the face of a 20 mph wind. Well, now even at low velocity, the wind is the most important factor slowing you down: remember that our wind resistance term is a relative velocity of you and the wind. What happens if you apply extra force? Well, you increase the already high wind velocity! So, as a result, you get that low (approximately square root) ROI on your power investment.
What about the case with the wind at your back? With the wind in your back, investing additional power has a MUCH higher ROI than when the wind is in your face. Why? Because the relative wind speed is very small! So increasing your power will increase your speed very efficiently.
How You Would Ride If You Were A Perfect Machine
Let’s say you have a fixed amount of ENERGY to expend in a race, and you are an ideal machine. In that instance, you could run a little simulation and decide how to apply your power (energy per time) with respect to hills, wind, etc., and the answer you would get would have the following components:
- HIGH power going uphill because you get a linear response to the application of that power.
- EXTRA HIGH power going uphill with the wind at your back.
- HIGH power with the wind at your back
- LOW power going into the wind because reducing your power doesn’t cost you that much velocity
- LOW power going downhill because there is already acceleration and thus a fairly high wind velocity
- “REGULAR” power at the top of a hill about to go down, because you are not yet going fast into the wind.
Bad News: You are not an Ideal Machine
The key assumption in the above is that you are an ideal machine and how much power you are applying doesn’t matter, as long as you spend the same total amount of energy. Anyone who has ridden with a power meter for any length of time recognizes that this is NOT the case! The whole purpose of NP and TSS is to estimate the physiological cost of applying power, which I’ve been calling “exertion”. In fact the NP expression suggest that there is a physiological response that scales with the fourth power of applied power.
Considering the downhill or wind-at-your-back case, there will come a bike velocity when we imperfect human beings will run out of gears to efficiently apply power, just as we run out of gears that are easy enough on steep hills. This is another “non-ideality” of your real life experience.
Now if you re-examine how to make that ride with hills and wind, you are going to increase your power a LOT less than you would if you were an ideal machine because increasing your power costs so much in terms of human exertion. That much is obvious. Moreover, we have to run after we ride, so the physiological consequence is even more complicated.
So what is the answer? How should you ride a course in the wind and hills?
Rich, Patrick, and others have accumulated a lot of real life data that point to the strategy we are all acquainted with by now: at the IM distance, the smart play is to more-or-less ignore the hills and ride very steady. We apply only a modest increase of power going up the hills, and continue to apply power across the top of the hill until we are going so fast that application is impractical and inefficient. This contrasts to spending too much exertion going up the hill (even if mathematically efficient if you were a perfect machine…which you are not) and then dying at the top of the hill (which is a bad idea if you are a perfect machine)
Ride steady up the hill so you are able to apply power in the favorable case of downhill slope but low velocity (i.e., the top of a hill!) instead of bonking. Ride steady to maximize the average power you can apply while minimizing the physiological cost.
But also consider the modifications given for hills and not given for wind.
Our normal advice is to ride 5-10% harder uphill than target watts. Now we see that this is an efficient application of the extra power. Beyond that, some of the hills require us to apply (at least) that power just to stay riding on the bike.
We are advised to stop pedaling in the mid 30s mph. There are two reasons for this. One is physiological – it is hard to efficiently apply power at that speed with “normal” bike gearing. But furthermore, unless the wind is at your back, it is not an efficient time to apply the power because your wind resistance is so high.
There is no guidance to increase power into the wind, even though it “feels” like a hill.
There are two good reasons for this. The first is that most of the hills we ride are fairly short, so only last several minutes. Thus, the increased power is for a limited time and a limited physiological cost. However, a headwind can last for an hour and can get worse and worse. This is fairly unpredictable and you would be committing to a potentially much larger “investment” of your TSS points.
The second reason is that it is pretty stupid to pedal into the wind for the basic physical reasons we have laid out here – you just don’t get much return! Going up a steep hill, you will get a more or less linear response for that expenditure. Going into the wind, you get only the square root of it (or worse!). Low ROI. Very bad idea!
The temptation, in fact, should be to ride with higher power when the wind is at your back and everyone else is cruising.
However, like the headwind case, you don’t know how long the wind will last. Moreover, if the wind is strong, you may already be in that mid-30s mph range where you start to spin out your gears anyway. So it’s not clear that you should always do this either. However, what IS clear is that riding with the wind at your back is NOT the time to ride easy unless you are already going so fast you are spinning out your gears.
At the HIM distance, where it can be rational to spend some extra effort going uphill, it seems likely that it is also rational to spend some extra effort in a modest tailwind. Unfortunately, that assertion is not backed up by any current data. Regardless, what we can take away from this discussion is that
- Small increases in power going uphill are worthwhile
- In wind-neutral conditions, resting some at high speed makes a lot of sense
- With the wind in your face, resist temptation to power through itWith the wind at your back, resist temptation to coast easily unless you spin out your gears.
One more Postscript on Not Being a Perfect Machine – Calculating Food Calories expended on a ride from your power meter
Unfortunately, we express energy in food content using a different unit than the Joule. Instead we use the Calorie, which is based on fundamental heat measurements, rather than fundamental work measurements. The conversion is that 1 Cal = 4.184 J.
This means that you could take the number of kJ from your power meter and divide it by 4.184 to precisely determine how many “food Calories” (which are actually kilocalories) you spent on your ride. However, your physiology is not perfect. In fact, the conversion factor of 4.2 for Cal to Joules is pretty wel balanced by the physiological efficiency of producing work. The latter is that you only get 20-25% of the “work” available in a Calorie when you use muscles. As a result, you can get a reasonable approximation of the number of food calories expended in a ride by just adopting the number of kJ expended at the power meter. Yes, there is individual variation, but it’s as good as most of us will probably ever get, and certainly better than a heart-rate based method of calculating calories expended. It’s not perfect, but it ain’t bad either.
Pop quiz answer: The unit of mass in Imperial/English units is the slug! One slug weighs 32 pounds on earth.
Comments
This is such a well written analytical analyisis of why we ride the way we do (or are taught to). I thought I understood why I needed to follow the EN bike pace protocol, this simplified and clarified things in a big way for me. Enough so I don't want to be spiking watts - especially on the downhills like I have a habit of doing.
This definitely need to be in the Wiki as part of the Racing with Power.
Thanks for shareing Professer!
The follow up question I have what would you do differently if there was no run after the bike? In otherwords, in an open TT where total time to cover a distance is all that matters, would it be better to follow the "ideal machine" strategy or still stick to the steady approach?
I know this is not really a long course triathlon question but you got me thinking about when the ideal machine approach is appropriate, if ever.
What about riding a hill with a strong headwind?
I am not much of a mathematician or programmer, but I see on inherent reason that you could not do the simulation as using TSS points instead of total energy. If you really buy that the relationship between NP and "exertion" is correct, this would be a pretty straightforward way of figuring out how to mete out effort as a "human". It also might serve as a reality check on NP/TSS :-)
What a great post. Thanks for your time and effort!
Thanks, WJ!
Article has been wikified here!