More About Friction

This item from Fair and Unbalanced knocked a stray thought loose that's been bouncing around for a while. Don't ask me how I got from Fred Flintstone to the Flash, because for the life of me I can't remember. But I do remember the basic question: what's the theoretical maximum speed a human could run?

We'll have to consider that two different ways: with and without cleats.

It's easy enough in principle to find the maximum speed of anything that has to move through the air. When the force pushing you forward is equal to the drag force exerted by the air surrounding you, you're not going to get any faster. So, we need to find the drag force:

Drag = 0.5 * (rho) * (v) * (v) * (CdS)

where rho is the density of air, v is the velocity, and CdS is the drag coefficient (Cd) times the surface area (S). It's wickedly hard to measure them for something like a running person, so we leave them lumped together. For that matter, how do we even find it? We can make an educated guess, by comparing a running dude to a falling one ... not a perfect analogy, but I know where I can look up terminal velocity. We can set the falling person's weight equal to the drag force, and then solve for CdS.

Drag = 180 lbs.

rho = 0.002377 slugs/ft^3

v = 175 ft/s

Solving for CdS gives 4.8893 ft^2.

(Incidentally: slugs are the English system's unit for mass. One slug of steel held in your hand would weigh 32.174 pounds, give or take.)

This gets us almost to the point where we can figure out the maximum speed. Now we need to figure out how a runner exerts force upon the ground. Which is why we need to split it into cases with and without cleats: with cleats, you get the added force from jabbing metal stakes into the ground, otherwise you're just relying on good old friction.

Without Cleats: The coefficient of friction between rubber and dry concrete is 0.85. This means that a 180-pound runner can only exert 153 pounds of force sideways upon the ground before he will begin to skid. Now, that lets us set up the drag equation again, and solve for v:

Drag = 153 lbs.

rho = 0.002377 slugs/ft^3

CdS = 4.8893 ft^2

Solving for v gives us 162.26 ft/s, or 110.63 miles per hour.

With Cleats: Here, the limiting factor isn't friction, it's the point where the cleats will snap off. We're going to assume "ideal" cleats here, which is to say that the cleats stick into the ground without tearing the surface. It has the dual virtue of both giving us the most beneficial possible conditions, as well as simplifying the problem. That's because now, all we have to worry about is steel's strength in shear. That's 50,000 pounds per square inch. Now, let's assume six spikes per shoe, with the spikes being 1/20 inch thick; that gives a cross-sectional area of 0.0118 square inches. In order to cause the spikes to fail in shear, you'd have to apply 589 pounds of force. This would be the maximum drag force. We can put that in the above equation, and solve for v:

Drag = 589 lbs.

rho = 0.002377 slugs/ft^3

CdS = 4.8893 ft^2

Solving for v gives us 318.38 ft/s, or 217 miles per hour.

Some interesting conclusions follow:

1) Since we know the Flash can run much, much faster than that, the obvious implication is that he's not using friction to keep his feet on the ground ... or whatever surface he's running across. (I tried to find a picture of Flash running up the side of a building. I know I've seen it. But when you Google "Flash running up the side of a building", you ... well, don't do that. Or if you do, don't come crying to me trying to unsee what you've seen.)

2) Steve Austin running 60 miles per hour? Totally possible, provided you have atomic-powered artificial legs.

3) Track and field world records have a long, long way to go. We'll probably reach human bio-mechanical limits before we even get close to the theoretical ones.

And with that, I'm gonna lace up my running shoes. I've got some work to do.