Technical FAQ: Weight, gravity, and coasting downhill

Lennard Zinn addressed a follow-up question regarding a recent column about coasting downhill and the forces that come into play.

More on coasting vs. pedaling downhill

Dear Lennard,
In your May 1 column “Coasting vs pedaling downhill“, you wrote, “Assuming you are rolling faster, then the next contributions are your weight and frontal area. If you have more weight and/or less frontal area, you will also run them down.”

I’m not sure that’s correct. I have this argument with people occasionally, but my recollection of high school physics says that weight should have nothing to do with it. More specifically, all objects accelerate because of gravity at the same rate: 9.8 m/s^2. What that means in real life is that while a heavier person will have a larger force of gravity (mass x acceleration, aka weight), that force is divided by each person’s mass to get back to the same acceleration. By the same token, a heavier person will have more grip on the ground due to the force of friction, but will also need that same friction to turn effectively. So mass cancels out.

However, you’re right, air friction is a function of Cd and velocity, which is why you have terminal velocity. I’ve passed people on downhills on my non-aero, fairly light bike, and my personal explanation is lower hub bearing friction, but that’s just a guess.
— George

Dear George,
You are correct about acceleration due to gravity being constant. Galileo is credited with figuring this out, although his may have been a thought experiment rather than the actual dropping of two spheres of different mass off of the Leaning Tower of Pisa he is credited with. However, two masses dropped simultaneously from the same height hitting the ground at the same time only applies in the absence of air friction.

Galileo’s possibly fictitious experiment and Jan Cornets de Groots’ actual experiment of dropping two lead balls off of a church tower in Delft, Netherlands, work only because the wind resistance of the two balls is similar enough that the difference in their speeds is not visible. If Galileo or De Groots had attempted on Earth the experiment that U.S. astronaut David Scott conducted on the moon — dropping a hammer and a feather from the same height — we all know that, unlike what happened in Scott’s experiment on the moon, the hammer would hit the ground a long time before the feather did. [related title=”More Tech FAQ” align=”right” tag=”Technical-FAQ”]

That’s analogous to a heavy rider out-coasting a lighter rider on a descent. If they were riding in the absence of wind resistance, there would be no difference in speed. But any time riders in the real world are going faster than 20 mph or so, the biggest force they have to overcome is that of air friction.

Think about the fact that the aerodynamic drag coefficient of each rider and his or her bike is proportional to their frontal area and that their mass is proportional to their volume. And think about the fact that, both being human, the two riders will have, for the sake of argument, equal density. To simplify this to make it easier to visualize, let’s say that both of these riders are spherical — so they are like two balls of human tissue of the same density sitting atop the saddles of their bicycles. The heavier rider will be a bigger ball, of course, and his or her bike frame will also likely be taller. We know that there will be greater aerodynamic drag on the bigger human ball sitting on the saddle, as well as on the taller head tube and seat tube of the bigger bike.

Here’s the key to this whole thing: the gravitational force on each rider and bike is proportional to their combined mass, and the aerodynamic drag force slowing each one down is proportional to the frontal area of bike and rider. (The drag force is also proportional to the cube of the velocity, meaning that it ramps up quickly with speed and effectively limits speed as you mention, yet it remains also proportional to the frontal area.)

Since the drag force on each rider is proportional to his or her frontal area, that would mean it is proportional to the square of the radius of the human sphere representing him or her. But the force pulling the rider down the hill is proportional to the rider’s mass, which in turn is proportional to the rider’s volume, and the rider’s volume is proportional to the cube of the radius of the human sphere representing him or her.

This means that the mass difference will decide this race. If the force pulling a rider down the hill goes up as the cube of their radius, and the force holding them back only goes up as the square of their radius, then the heavier rider will roll down faster. Add to this the fact that part of the aerodynamic drag they are facing is the drag force on the bike, and that on similar bikes the drag on the wheels and on most of the bike will be similar for both riders. So the extra air drag on the heavier rider and bike will not be big enough to counteract the extra gravitational force pulling him or her down the hill enough to let the lighter rider roll as fast.

This same reasoning applies when considering Tom Dumoulin time trialing against one of the little climbers in the Giro. Yes, the little climbers could have a bit less air drag because their aerodynamic drag will be proportional to their surface area, while their bike will have similar air drag to Dumoulin’s. Most importantly, the power they produce will be proportional to their volume, and, unfortunately for them, Dumoulin’s power will also be proportional to his volume. His greater volume and power will trump the slightly increased aerodynamic drag he faces. The cube vs. square argument explains not only why heavier riders roll downhill faster but also why bigger riders generally can beat otherwise closely matched smaller riders in time trials.
― Lennard