Optimal speed on an indoor track

Consider yourself running on an indoor running track surface. The turn has a banked incline, which ostensibly helps making fast turns easier. How fast does one need to run around a banked curve to balance the forces of gravity and centripetal motion?

When running slow, say a light jog, a banked turn can feel awkward (you feel yourself being pulled off the track). Run too fast and you'll feel a pull outwards (though less than if there was no bank at all.)

The following diagram outlines the physics involved.

Wikipedia diagram of a banked turn; balancing the forces.

The specific forces we wish to balance are gravity and centripetal, at angle theta:

Assuming no friction, the velocity v for angle theta and radius r is

The IAAF gives a detailed plan for building an indoor 200m track here. Curves are banked at 10 degrees (18% grade) and the internal radius r = 17.5 m. Each additional lane (e.g. 2, 3..6) adds an extra 0.9 meters to the total curved radius.

The optimal speed v for the first lane of a track is therefore 5.5 m/s, or 19.8 km/h. The optical speed for the 6th lane is 22.2 km/h. One should feel the most "balanced" racing a banked turn at approximately 20 km/h. To place this in perspective, a 10 minute 3000m race is 18 km/h; at 9 minute it is run at 20 km/h, and 8 minutes is 22.5 km/h. 20 km/h is around the speed of a competitive women's 1500m-3000m university race.

Of course one can run much faster than 20-something km/h on an actual course since cleated footwear is high in fractional forces. Friction, mu, prevents sideways movement hence adds to the maximum speed.

Assuming a static friction coefficient of mu = 1, you'd stay glued to a 10-degree banked surface up 15.6 m/s or 56 km/h, enough for even the fastest sprinter. What if we increased the banking? If mu = 0 (no friction) but banked at 45 degree like a Velodrome, you could run up to 13 m/s. The optimal speed for a given banking is given the table below:

If there is no banking at all (theta = 0) then all centripetal motion is countered with frictional forces. Assuming mu = 1, the maximum speed around a non-banked curve is 13.1 m/s, still fast enough to support any runner. Even a tiny flat track of radius r = 5 m could support a speed of 7 m/s (25 km/h), fast enough for a 3:36 1500m run. This is why we never see people fly off the outside of a track.

Returning to our official track of r = 17.5 and theta = 10 degree, then running the inside lane at 20 km/h, or 36 seconds a lap, is ideal if interested in minimizing sideways forces.