# Coast-to-Coast Coasting

What if the entire continental US was on a decreasing slope from West to East. How steep would the slope have to be to sustain the momentum needed to ride a bicycle the entire distance without pedaling?

—Brandon Rooks

Too steep to actually build, sadly. But for the next best thing, I suggest a vacation to the Hawaiian island of Maui.

First, the physics. Bikes coast downhill. On a long enough slope, a bike will reach a certain steady coasting speed. On a steep hill, their coasting speed will be faster, and on a gentle slope, they coast more slowly. If the slope is small enough, the bike will slow down and stop.

The shallowest slope at which a bike will still roll steadily forward is determined by the bike's *coefficient of rolling resistance*. In fact, the formula for this minimum slope—measured in terms of vertical drop over horizontal distance—is incredibly simple:

\[ \text{Minimum slope} = \text{Coefficient of rolling resistance} \]

"Slope equals coefficient of friction"[1]Sliding friction and rolling resistance work in different ways, but the coefficients are equivalent in these types of problems. If you want to be precise, you could use the phrase "coefficient of resistance" for all of them, but "coefficient of friction" is the more common term. is a handy general rule in physics: The coefficient of friction between an object on a surface is just the shallowest slope at which the object slides.[2]The **coefficient of static friction** is the slope at which the object starts sliding. The (usually lower) **coefficient of dynamic friction** is the minimum slope at which it keeps sliding once you give it a nudge.

For a nice bike under good conditions, the coefficient of rolling resistance can get as low as 0.002, or 1/500.[3]You can browse some test data here. That means that to travel 500 miles horizontally, you'll need a vertical drop of at *least* 1 mile. To travel the roughly 2,500 miles from New York to LA, you'd need to start off at *least* 5 miles up, higher than North America's highest mountain. I suggest bringing oxygen tanks.

But be warned—the trip could take a while.

A bike's rolling resistance mainly comes from the way the tire[4]And the ground, if you're riding on dirt.[5]And the spokes and frame, if your bike is made of soft clay or something.[6]Why do you have a bike made entirely of soft clay? deforms as it rolls, and it doesn't depend that much on how fast you're going. Air resistance, on the other hand, increases as you speed up, and under most conditions is the main drag force acting on a moving bike. To figure out how fast a bike will coast on a downhill slope, you need to calculate the point at which air resistance balances out the forward pull from gravity. At that point, the bike will stop accelerating. We can do that by using the formula for air resistance:

\[ \text{Forward pull from gravity} = \text{Rolling resistance} + \text{Drag force} \]

\[ m g \sin(\theta) = g \cos(\theta) C_r m + \tfrac{1}{2} C_d \rho A V^2 \]

\[ V = \sqrt{\frac{m g \sin(\theta) – g \cos(\theta) C_r m}{ \tfrac{1}{2} C_d \rho A}} \]

(*V* is the speed of the bike, *C _{r}* and

*C*are the coefficients of rolling resistance and air drag,

_{d}*θ*is the slope angle,

*g*is the acceleration of gravity,

*m*is the mass of the bike and rider,

*A*is the frontal area of the bike and rider, and

*ρ*is the density of air.)

For a very shallow slope of 0.2° or 0.3°, the bike would barely roll, and its top speed would be slower than a walking pace. You would need to add an extra few tenths of a degree to get the speed high enough to balance comfortably, and this would make the LA end of the slope even *higher* than the already implausible five miles.

But still, bicycles are pretty impressive coasting machines.[7]Trains have steel wheels which roll on smooth rails, so they should have very little rolling resistance. You can work out their coefficients by looking over technical specs or calculating from first principles, but a cleverer way is by watching train-pulling athletic events. Then, with a little calculation involving the limits of human strength and/or direct measurement, you can work out the coefficient from the other end. It turns out that train cars—at least, the kind used in strongman events—have coefficients of rolling resistance barely equal to that of a good bicycle. Skis, which are pretty good at sliding, actually have a coefficient of friction about 10 times higher than a bike's rolling resistance.

To ski from LA to New York, a skiier would need to start off 10 times higher than a bike to make the same trip. Instead of the top of a mountain, they would need to start from near the edge of space. Not only is there no way to build a slope that tall, but ice isn't even stable at those low temperatures, so there'd be nothing to slide on.

In practice, the longest horizontal distance you could travel on a bike with an ideal ramp is probably not more than a couple hundred miles, and that would require ideal conditions. In the real world, the longest such trip might[8]It's billed as the longest, but I wonder if there's a longer one in some random stretch of gently-sloping downhill road in rural Mongolia or something. be the Haleakala downhill bike ride, which allows you to take a 35-mile trip from near the 10,000-foot summit all the way down to sea level with virtually no pedaling required.

(And if you can't make it to Maui yourself, you can at least enjoy the video search results for bicycle into water.)

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