# How does biking speed correlate with the required effort?

The simple answer is that the faster you go the more effort/energy it takes but I would like to understand this in more detail, especially at relatively low speeds. Assume a flat smooth road and fix the bicycle and the person doing the riding. How does going at say 10, 15 or 20 km/h compare in terms of effort both per time and per distance?

The simplest physical model gives that the energy is proportional to the square of the speed but in that model maintaining a constant speed requires zero energy. To maintain constant speed in practice one needs to overcome friction of the bike and with the ground, the wind resistance and also invest some effort to keep the bike balanced and avoid falling over. How does friction scale with speed? Staying balanced gets easier with higher speed but are their formulas for that?

From personal experience (from riding with small kids) I have the impression that 15 or 20 km/h comes to about the same, so the extra speed doesn't seem to require extra effort. Going at only 10km/h seems to be more exhausting. I would like to understand whether there is some physical reasons for that or whether this is purely my subjective experience.

• You can play around with this calculator: kreuzotter.de/english/espeed.htm Commented Apr 26, 2022 at 13:22
• Riding at a speed that you feel requires concentration to stay upright while simultaneously monitoring the safety of children sounds very exhausting! You could try practicing riding slowly in a lower gear with a modest cadence if you find you end up in too high a gear doing big forceful strokes then coasting at low speed which a lot of people do.
– Affe
Commented Apr 26, 2022 at 15:49
• I think there is a psychological factor in the time it takes to complete the ride/route. Maybe exerting a little more effort for a shorter time, "feels" less exhausting. I some times feel very tired after a given ride accompanying slower riders, and the same ride at my own pace "feels" less tiring. And even riding along faster riders feels I feel just fine immediately after the ride, but a few hours later, I get extremely tired. Commented Apr 26, 2022 at 23:07
• @Jahaziel riding with slower riders, particularly if some of them are in front of you (as tends to be the case when I'm riding with kids so I can deal with the traffic) takes extra concentration. For example if my daughter descends on the brakes because freewheeling feels scary, she lets them off later than I would, so gets to the bottom then suddenly decelerates. I then have to brake sharply especially if I was looking over my shoulder at the wrong moment. The extra physical effort from that is minimal but it's mentally draining. Commented Apr 27, 2022 at 10:37
• Technically friction with the ground isn't something that slows you down and you need to overcome. Tyres are designed to have very high friction with the ground, aka traction, which is what you need too accelerate, brake, and avoid falling over. The thing that slows you down is rolling resistance.
– bdsl
Commented Apr 27, 2022 at 12:41

The simplest physical model gives that the energy is proportional to the square of the speed but in that model maintaining a constant speed requires zero energy.

This is a slight misstatement. If you examine the equations in Michael's link, the short version is that the power required to overcome aerodynamic drag is proportional to the cube of the speed. That is, the power to maintain a modeled forward speed depends on the aerodynamic drag.

Slightly complicating reality: you also need to overcome rolling resistance and drivetrain friction, and if not on flat ground you need to overcome gravity. For the first two sources of drag, I believe they are proportional to speed.

From personal experience (from riding with small kids) I have the impression that 15 or 20 km/h comes to about the same...

Here's one possible explanation. If you do the math, you must use more energy to travel at 20 km/h than at 15 km/h. That's not disputable. However, the way we perceive physical efforts is not linear with the actual power produced. You will start to notice the efforts much more as increase your power - especially when you have to use significant amounts of anaerobic power to maintain your speed.

• @RChung made a (now deleted?) comment advising that anaerobic threshold is no longer a preferred term among exercise physiologists. The answer previously used that term, but I modified it. Commented Apr 26, 2022 at 20:43
• I don't agree with everything in this video but I saw it recently and I mostly agree with the discussion of thresholds. Commented Apr 27, 2022 at 7:00
• Since we're nitpicking, energy over fixed distance is proportional to square of speed. Power is proportional to cube, but the time spent is proportional to inverse of distance.
– ojs
Commented Apr 27, 2022 at 7:52
• "This is a slight misstatement." I don't think it is a misstatement - the OP seems to be referring to the kinetic energy formula, K.E. = ½ mv². That works in the model where we ignore everything like gravity on a slope, rolling resistance and air resistance. May be reasonably accurate for high acceleration over very short distances.
– bdsl
Commented Apr 27, 2022 at 12:35
• @Arthur I think monotonic is the correct term. It's a very technical term in this context, and it means (of a mathematical function) that it never decreases. Commented Apr 27, 2022 at 16:07

I have also noticed that sometimes it feels harder to ride slow (or run or walk slow) than to ride faster.

The cause I have attributed to the impression is that I have a pace that's comfortable. Riding slower than the comfortable pace feels like it takes more energy than riding at the faster comfortable pace.

Some of the factors that come into play are momentum, getting enough wind to stay cool etc.

• Walking at shopping pace compared to striding out is a particularly obvious example for me. But I wonder if the dominant effect here is a mental one related to progressing slowly enough that it feels tedious Commented Apr 27, 2022 at 10:32
• @ChrisH If you consider a automobile's most efficient speed, it's not the lowest speed possible: the overhead of the engine becomes dominant. There's likely a similar situation with our legs where the overhead of moving them starts to dominate at low speeds. There are also other factors such as friction in the axels and gears is not linearly related to the rotational speed e.g. static friction tends to be much higher than kinetic friction. Commented Apr 27, 2022 at 21:56
• @JimmyJames but a car has a much smaller range of gears, and on a bike we can much more readily freewheel, turning over very little of the drivetrain, and aerodynamics does purely increase with speed. Going very slowly on the flat it can be easiest to use not the lowest gear, and to ride one pedal stroke, wait, then the other pedal. I do this for the last 100m to the station every day, because the bike rack requires the derailleur to be in a small sprocket but you can't arrive fast. I end up doing about 6rpm. I do the same if sharing with lots of pedestrians Commented Apr 28, 2022 at 5:47
• And walking isn't rolling so much of the movement stops with every pace anyway. Commented Apr 28, 2022 at 5:47

The power (so likely the effort) is proportional to speed multiplied by force: E = F*v. This is why riding uphill faster is more difficult than riding slower (the force does not need to be larger, the speed is) and switching into lower gear does not reduce the power required (now you need faster cadence for the same speed).

A bicycle needs very little force to move if at low speed and over flat good road. The rolling resistance does not change much with speed, yet you need more power to overcome it while riding faster.

Very approximately, from 5 km/h the air drag becomes a factor and since then it is roughly proportional to speed within the speed range of the casual cyclist (up to 25 km/h). At about 12 km/h it approaches the force from the rolling resistance.

Very great formulas can be found here.

• "E = F*v" is misleading; "E" is commonly used for energy. Furthermore, the discussion of riding uphill is a distraction because the question is about resistance, e.g. friction. Commented Apr 27, 2022 at 8:53
• I found riding uphill a good explanation what the equation means. It is somewhat contra-intuitive that the power depends on the speed and is not constant. Commented Apr 27, 2022 at 13:31
• The question literally says "Assume a flat smooth road...", continues with the kinetic energy and then asks "How does friction scale with speed?". I think the OP has no trouble understanding that it is harder to go uphill fast than slow. And still: What about that "E" which usually stands for Energy, not Power (for which, intuitively, "P" is used)? Commented Apr 27, 2022 at 14:13

When riding in the plain the only thing limiting your velocity at a certain level of steady exertion is friction: Rolling and gear friction as well as drag. Interestingly, the force to overcome rolling resistance and friction in your pedals etc. is constant; if you were riding in a vacuum, your legs would need to exert the same force at higher speeds than at lower speeds (once you finished accelerating of course, which needs force as well, as good old Isaac formulated). The energy spent per distance on a bike in a vacuum does not depend on the velocity. Still: Because you are going faster, you cover more distance per time, expending more energy in the same time. That's what physicists call power, measured in Watt.

The power needed to overcome rolling and drive train friction grows linearly with the velocity.

The second kind of friction to overcome is aerodynamic drag. That force grows quadratically with the velocity: If you go twice as fast, you need four times the force. As before though: Because at higher speeds you cover more ground in the same time unit, the energy per time, or power grows with an additional factor:

The power needed to overcome drag grows with the cube of the velocity.

One consequence is that the drag is negligible for small velocities but grows quickly and quickly becomes the dominant factor restricting speed. The diagram below, from Wikipedia, shows that nicely. The velocity is given in m/s. To get km/h, multiply with 3.6, e.g. 4 m/s equals 14.4 km/h.

Note how the power expended for drag (the blue line) is about 20 W at 4 m/s but about 160 W for double the speed at 8 m/s; doubling the velocity needs eight times the power.

From Wikipedia, Theosch, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

A note to your personal observation that going 15 km/h doesn't seem to be easier than going 20. As David said in his answer, there may be a most "natural" speed for each person. I can imagine though that little children would struggle to maintain 20 km/h over larger distances. Maybe check your tachometer for accuracy? A mis-measurement of only 25% results in a mis-estimation of the drag of a factor 2.

But generally, the energy expended by our bodies and the exhaustion felt does not directly correspond well to the work done in the physics sense: For example, carrying a heavy load across a plane doesn't accelerate or lift the mass at all, and still is very exhausting. Even just holding a heavy load is exhausting. Likewise, standing on an incline and balancing a stationary bike is likely exhausting even though, as in the other example, no physical work is performed. Going slow on a bike uses a lot of muscle movement and other "ancillary efforts" to control the body to no effect. At low revolutions much of the effort simply goes into pressing the pedal without effect (like a rocket just strong enough to hover — all the fuel is wasted). Together, and with the fact that at slow-ish speeds the forces are small to begin with, that may well explain your perception.

But from a physics standpoint the case is clear: For low speeds the dominant power drains correlate slowly, linearly, but for higher speeds they rise very quickly, with the third power, which is why we mere mortals never go 40 km/s without some tailwind (and why even 5 km/h tailwind are a godsend at higher speeds).