I love this stuff. Discovery News has details on a bike that could break 100 mph. The current human-powered bike speed record is at 82 mph and held by Sam Whittingham. This record will be challenged by Graeme Obree, who has broken speed records before.

But enough about the news, how about some physics?

Let me assume that for this bike the primary problem is the air resistance. At a constant speed, the bike would have the following forces on it.

Let me go over these forces. First, the gravitational force. Not too much to say about this except that it is pulling down. The ground also pushes up on the bike - I call this the normal force and it is represented by the F N . These two forces are in the vertical direction and since the acceleration in the vertical direction is zero, they have the same magnitude. In the horizontal direction, there is the air resistance. In one model, this force is proportional to the square of the bike's speed. It can be written as having the following magnitude.

The other parameters are the density of air (ρ), the cross sectional area of the bike (A), and the drag coefficient (C). Since the bike is traveling at a constant velocity, the net force in the horizontal direction must also be zero. What force is pushing in the opposite direction to the air resistance? Technically, it would be the frictional force from the ground on the tires. Just imagine trying to bike against a strong headwind on icy ground. It wouldn't work because the tires would just slip.

If I assume that there is enough frictional force to keep the bike at this speed, then the limiting factor will be the human. This isn't a terrible assumption since motorcycles can go 100 mph (of course they have a larger mass, which does matter in this case). Here is what I want to do. Instead of looking at the force of friction, instead let me examine the "force from the cyclist." This force doesn't show up in the diagram above since it is a diagram showing the forces on the bike with the cyclist as part of the bike. But you can't get a free ride. The cyclist still has to do something.

So, at a constant speed the force from the cyclist will have the same magnitude as the air resistance force. But force isn't what I want either. I'm never happy, am I? Let me consider a short distance that the bike moves at this constant speed. I will call this distance s. Over this distance, I can pretend like the cyclist is pushing the bike with a force. (I will call this force F c .) This force will do work on the bike-system of an amount:

Recall that the work done on (in the case of the force pushing in the same direction as the displacement) would just be the force times the displacement. But I don't really want work either. I want power. Power is defined as:

Now to remove time from this expression. How long will it take to move over this distance of s? If the bike is moving at a constant speed, then:

If I use this expression for the change in time with the expression for power, I get:

Now I have an expression for power that doesn't depend on time, just on the velocity. Next, I can put the magnitude of the air resistance in for the force of the cyclist.

Maybe now you can see the problem. Suppose I am putting all my effort into pedaling a normal bike and I can get to a speed of 15 m/s. What if I want to go twice as fast? I can't increase the power, because I already said I am giving it all I have. The only option is to decrease this other stuff. Of course, decreasing the density of the air isn't really an option. (Well, I could go to a higher altitude.) That just leaves me with the cross-sectional area and the drag coefficient. How much do I have to decrease the AC product? Let me write the power for maximum velocity (normal bike) as:

Now, if I double the speed, what would the new product of AC need to be (with the same power)?

This says the product of AC would have to be reduced by a factor of 8 just to go twice the speed. Here you can see some logic behind Obree's crazy-looking bike design. His design does two things. First, by putting the rider in a prone position the cross-sectional area is decreased. Second, by covering the bike in a plastic shell the aerodynamic shape is changed reducing the drag coefficient.

Suppose I assume a maximum power output of around 1,000 Watts. __Update: __Where does 1,000 Watts come from? This is just a ballpark estimate that I have used before for the power output of a human. The Wikipedia entry on human power claims that elite cyclists can get up to 2,000 watts, but just for a short period.

What would the maximum speed for this power be as a function of the product of cross section and drag coefficient? In order to make a plot of the max speed, I need an AC value to start with. Suppose the bike had the person completely upright with arms spread out. This would be a terrible position in terms of both area and the drag coefficient. However, I could get an estimate for it. Why? Because it is similar to the position of a sky diver. If a 70 kg skydiver is falling at a terminal speed of 120 mph (54 m/s) then at this speed the air resistance force is equal in magnitude to the gravitational force. Like this:

With an air density of 1.2 kg/m3, the skydiver would have an AC value of 0.392 m2.

This suggests that to get to a speed of 100 mph, you would need an AC value around 0.02 m2. Let me estimate a reasonable value for C. Wikipedia list some values for drag coefficients. It puts a "streamlined body" at 0.04 and a "streamlined half-body" at a value of 0.09. I guess a bike would probably be closer to the 0.09 value; let me start with an estimated C of 0.07. If I use this value for C, the cross-sectional area of the rider would have to be around 0.285 m2. If I pretend like the rider is a cylinder, the shape of this area would be a circle with a radius of 30 cm. That seems reasonable.