# Relationship between speed and cadence?

I am having a disagreement with a friend. It stems around the scenario of 2 identical bikes on the same gear ratio side by side. If the rate of pedaling is same for both do both bikes travel at the same speed? Or does the rider's weight play a factor?

• The only variable in this scenario is the effective rear wheel diameter, determined by air pressure and weight on the rear wheel. Theoretically if tires and pressure are identical, there will be a difference and a heavier rider will travel less distance. Practically, good luck measuring it. Commented Sep 5, 2019 at 21:41
• @mattnz - Yep, there will be an effect, but it will be minuscule in most circumstances. Commented Sep 5, 2019 at 21:52
• Sorry, but wouldn't "Relationship between mass and speed at constant cadence" be a more accurate title? Commented Sep 6, 2019 at 6:10

Speed is determined only by gearing, of which crank, cassette, wheel, and tire are components

Any 2 bikes using, say, 42t cranks with a rear 32t cog and 25c tires on 700c wheels at 90rpm will be going the same speed. Bike type and size, rider/bike weight, and even front wheel/tire size and crank length don’t matter. These other factors only effect how much power it will take to keep up that cadence with that gear combination.

• There's also an assumption that neither is freewheeling at all. Commented Sep 7, 2019 at 21:50

The mechanical factors which translate pedaling rate to overall speed are:

• Gear ratio
• Size of wheels

The weight of the rider is not relevant for this question.

The weight of the rider would be relevant if you were asking about the power needed to keep the bike going, especially up any kind of hill.

• On a non inclined surface rider and bike mass has no bearing on the power needed to maintain a constant speed (ok, perhaps greater mass causes greater bearing friction, tire deformation losses, but those are small). Commented Sep 5, 2019 at 21:44
• @ArgentiApparatus surface area is (generally) proportional to mass, so you'd (generally) expect different drag at the same speed (but you also wouldn't be surprised if two riders of the same mass had different drag) Commented Sep 5, 2019 at 22:22
• @RobertLee Muscle mass is also the thing that produces the power for acceleration. Track sprints are the cycling discipline that require most acceleration and if anything, track specialists tend to be bigger than other cyclists.
– ojs
Commented Sep 6, 2019 at 8:19
• @PaulH Surface area isn't proportional to mass. It's roughly proportional to mass^(2/3), since mass scales with volume, which is the cube of a linear dimension, whereas area is the square. Commented Sep 6, 2019 at 9:58
• @PaulH "Proportional" is a much more specific claim than "one tends to increase as the other does." Commented Sep 6, 2019 at 15:22

There would likely be a small difference. If you have, eg, a bike with a rear tire that has a 68cm diameter, the effective tire diameter is reduced by the amount it compresses when the bike is carrying a rider.

Let's say that the heavier rider is heaver by 80kg, and this causes the tire to compress an additional 1cm vs the diameter with the lighter rider. The effective radius is 33cm vs 34cm, and the effective circumference is 207.34cm vs 213.63cm.

So he heavier rider would travel about 207.34/213.63 or 0.97 kilometer for each kilometer the lighter rider travels.

(In practice the difference would likely be somewhat less due to the dynamics of the tire. The lateral stiffness of the tread would tend to make the heavier bike "scoot ahead" relative to the above numbers, perhaps halving the disadvantage. This would depend on the specific tire/tread/pressure.)

• In the scenario you suggest, the heavier rider would probably cover a much shorter distance, due to having to stop because of a pinch puncture. 😉 (1cm seems like a lot of compression, to me, but I'm used to road bikes with relatively high pressure tyres; maybe it's not so crazy for a mountain bike?) Commented Sep 6, 2019 at 10:01
• If there's a difference of 80 kg between Bike 1 and Bike 2, I suspect the "heavier" cyclist may have better performance by getting rid of the second person clinging to his back. Commented Sep 6, 2019 at 17:24

If the weight difference is significant and the mountain bike tyres are inflated to low pressure, yes the ligther biker can be slightly faster than the heavier one.

The cadence (same) is directly bound via gear ratio (same) to the wheel's angular speed (rad per second). The velocity is proportional to wheel's angular speed (same) and the radius, which must be measured in the point of contact.

If you want to quantify the difference, sit on the bike and masure the distance of the rear wheel axis to the ground for both riders. Divide those numbers and you'll get the ratio how much faster/slower the other bike will ride.

• "sit on the bike and the distance of the rear wheel axis to the ground." - I think you're missing a couple of words here? Commented Sep 7, 2019 at 13:57
• @npostavs Thanks, I was rewording the sentences way too many times... Commented Sep 10, 2019 at 10:21

Unless the weight was of significant delta, it would be near on impossible to see one bike travel faster or slower.

There are also too many other factors to consider: Rider position Rider height (longer legs, etc) Rider muscle composition (stronger legs) Cadence Bike maintenance Road surface etc

• I've no idea what you mean by "of significant delta". And none of the factors you mention will alter the distance travelled at a specific cadence and gear ratio. They would certainly affect the rider's ability to maintain that cadence at that gear ratio, but that's not what the question is about. Commented Sep 6, 2019 at 10:03
• @DavidRicherby "delta" is the mathematical representation of "difference". "Delta from reference" is the same as "difference from reference", whatever that reference point may be. Therefore, "significant delta" means "significant difference", for example a heavy adult vs a small child. Agree with the rest of your comments. Commented Sep 6, 2019 at 12:08
• The only delta or difference will be the effort and time taken to reach the max speed for the cadence and gear used. The top speed will be the same regardless Commented Sep 6, 2019 at 13:15
• @FreeMan Except that mathematicians say "difference"... Commented Sep 6, 2019 at 13:26
• @DavidRicherby Race car engineers use "delta", especially when referring to lap time. ;) Commented Sep 6, 2019 at 14:57