There’s money in the bank: the physics of indoor track and “optimal” speeds

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In January, there were two indoor mile races here at the University of Michigan with two distinct outcomes.  They were the first footraces held on U of M’s new hydraulically banked indoor track. The first race was the men’s mile, and it saw Nick Willis, a multi-time Olympic medalist in the 1500m, win in 3:57.99, out-sprinting compatriot Hamish Carson, who finished in a respectable 3:58.68.  Both men left a bit disappointed though, as they came up short of the Michigan all-comers record of 3:57.89, and more importantly, the 3:55.00 qualifying standard for the upcoming IAAF World Indoor Championships. The second race on the new track, however, the women’s mile, had a similar plot but a different ending.  Like the men’s mile, it ended up being a duel, with Nicole Sifuentes pulling away from Shannon Osika for the win.  However, over this 5,280-foot footrace, both Sifuentes and Osika went under the previous Michigan all-comers record and, more impressively, the IAAF world champs standard of 4:28.50.  The ladies stole the show.

With that performance, Sifuentes put herself at the top of the world-leading list for the mile in the 2017-2018 indoor season.  Interestingly, the man with the world’s fastest mile performance of the season at that time was none other than Hamish Carson, who had run 3:57.76 at Boston University a few weeks prior.  Carson was likely more fit for the mile race in Ann Arbor, but he wasn’t able to better, or even replicate, the BU performance.

Now, the BU track is a facility famous for facilitating infamously impressive footraces, with a reputation in the running community for being probably magical, possibly short, and definitely fast.  Later in February, the world saw Edward Cheserek blitz a 3:49.44 mile there to move himself to the #2 all-time best indoors, then, a few weeks later, on a single evening, 14 men went sub-4 at the BU Last Chance meet.  The Boston University indoor track boasts, among other characteristics, an incredibly steep bank on its turns, rumored to be 18-degrees.  U of M’s new track maxes out at 10-degrees.  Obviously there are many multitudes of factors that can take or give seconds in a given distance race on a given day, but this one begged an engineer’s mind to explore a bit deeper.

The Forces of Going Around a Turn

For an object to turn while moving at a set velocity, it needs an additional force: a centripetal force.  That force is the acceleration of continuously changing the direction of the given speed as the object rotates around the turn.  When running, this centripetal force is generated from the mediolateral friction (think side-to-side) beneath the foot.  It is proportional to the speed you’re running (squared!), and inversely proportional to the turn’s radius.  That is, the faster you go, the more centripetal force you need to generate to turn, and the bigger the radius (the wider the turn), the less force you need to generate.  This is why a flat 400-meter outdoor track (turn radius of 36-37 m) is easier to run around than a flat 200-meter indoor track (turn radius of 17 m or less) – it requires less than half the centripetal force.

Screen Shot 2018-03-01 at 4.04.59 PM

Now, as NASCAR drivers, cyclists on a velodrome, and Intro to Physics students working on their rotational kinematics problem sets know, something interesting happens when you tilt that surface you’re going around.  It involves another weird force aptly named the “Normal Force”.  The normal force is simply the force that the ground applies back to an object that is in contact with it (so that when you stand, for example, you don’t just keep moving downward into the depths of the earth).  It is thus equal and opposite (Sir Isaac’s Third Law) to the force you put into the ground.  So on flat ground, the normal force shoots straight back up (“normal” (i.e. perpendicular) to the surface).  But, on a tilted surface (e.g. banked turn), it shoots back normal to that surface, so off the angle of the tilt.  Why this becomes interesting when moving around a turn lies in the components of that now-angled normal force.  One component of it is shooting straight up (against gravity), and one component is now shooting inward, the tilted direction of that banked surface (and radially inward of the turn!).  The “tilted” normal force that the object (runner) experiences is thus the sum of these two components.

Now, how does this relate to that centripetal force we talked about earlier? Well, as you increase the tilt (angle) of that bank, more and more of that total normal force is directed inward rather than upward.  If the bank was at 45°, half the normal force would be inward and half would be upward.  Remember, for a given speed, there is a centripetal force inward required to move around a turn.  So, for a given speed, there is a magical angle at which the inward component of that surface’s normal force is exactly equal to the centripetal force required to turn.  This means that just by moving around the turn at that speed, you don’t have to exert a wasteful lateral frictional force to generate centripetal force and turn.

The “Optimal” Speed to Get Around a Bank

So how does this relate to indoor track? It means that every banked track has an “optimal” speed at which a runner can run so he or she does not have to exert any lateral force to turn.  Check out the free-body diagrams below:


On the left, you have the flat track case, where the runner uses friction to turn.  In the middle, you have a strange case where the runner is running on a banked surface, but below the “optimal” speed for that bank and that turn.  The runner actually has to exert a force in the opposite direction just to stay upright.  Then, on the right, we have the case where the runner is running around the banked turn at the “optimal” speed for that turn and that bank.  No lateral force, inward or outward, is required to move around the turn. Rearranging the variables in the equation above, we get the relation at the bottom of the figure:

Screen Shot 2018-03-07 at 3.45.17 PM

So using this relationship and assuming a turn radius of 17.5 m (IAAF facilities recommendation), we can get an optimal angle for a given speed!

Optimal bank angle for various speeds around an indoor track with a turn radius of 17.5 meters

Or, in a more helpful form, the optimal bank as a function of 200-meter lap times:

Optimal bank angle for a given 200 meter split

If you’re running at Boston University, the optimal speed for running around their track is about 27 seconds per lap, whereas at Michigan, it’s 36 seconds.  A huge difference! Again, this is assuming a constant turn radius: if BU has tighter turns, that speed would go down a bit.  For example, if the radius was 16 m instead of 17.5 m, the optimal lap speed would be about 28 seconds.  Regardless, it’s especially intriguing considering the lightening fast men’s races you see at Boston, and the incredible women’s races we’ve seen at Michigan. When Sifuentes and Osika ran 4:27 and 4:28 for the mile at Michigan, they were moving just a bit faster than the “optimal” speed for that facility, and thus had to generate very little lateral force to turn.  When Erin Finn ran 8:58 for the 3000m the next weekend, she was running at nearly the exact speed best suited for the track!

Running Slower than the Optimal Speed?

For Edward Cheserek’s 3:49 mile at BU, he was running close to that optimal speed for the track, but still slightly below it (and perhaps closer than the above graph indicates if the turns are tighter).  That means that even he along with the dozens of guys who ran under 4 minutes there this season may have still had to generate a bit of frictional force outwards just to stay upright.  This is an interesting situation, as it represents a muscular challenge that’s very distinct from that which a runner is habituated: generating inward force to turn left on a flat track.  This got me thinking: does a steeper bank cause a runner to increase his or her pace?  Does a runner subconsciously speed up a bit to lessen or attempt to mitigate the vestibular challenge and foreign muscular demand of staying upright?  That is to say, does the bank neuromuscularly “egg-on” the runner to go a bit faster?  I don’t think there’s an answer (yet), but a cool study would be “self-selected running speeds around turns with various bank angles”.  A good study for a facility with a hydraulic, variably banked track…

Another important consideration on the opposite end of this spectrum would be the detrimental effects of running slower around a steep bank.  Those strange lateral forces and stresses may create a new injury risk for runners spending a lot of time on that steeper bank.  This probably isn’t an issue for the isolated cases of racing, but if a team is doing a lot of intervals on a bank and jogging slowly around the curves during recovery, perhaps it may become an issue? Again, not sure there’s an answer here, but something to consider and something to explore.


Are banks regulated by the IAAF or the NCAA?  While there are conversions from the NCAA for 200m flat-to-banked and 300m flat-to-banked, it treats the bank as binary (index of facilities).  Curiously, they do have a regulation on banks, with the upper limit being 18-degrees.  Per the NCAA Rulebook, Rule 1, Article 1, Section 1e: “An indoor track may be banked. The angle of banking should not be more than 18 degrees for a 200-meter track.” Interesting. Furthermore, the IAAF follows the same vein, where it mandates that official competition venues be banked, but doesn’t specify what the requirement is.  They’ve published a “facilities recommendation ” document that suggests a bank of 10-degrees and a turn radius of 17.5 meters.

My first thought was, “How ridiculous! By suggesting facilities bank at 10 degrees, they’re losing an opportunity for awesomely fast times, both in distance races and sprints!”  Then I considered the case of running slower than the optimal speed for that bank, and that banking tracks steeper would detrimentally affect some of the women’s distance races at the international level, and all sorts of races at the collegiate or high-school level.  Thus, a 10-degree bank ensures that just about all the elite races, male and female, sprints to long distance, will be running faster than a speed that would cause them to have to generate that weird force to stay upright.  I’m not sure if that was the logic or calculation of the IAAF, but 10-degrees seems like a justifiable all-purpose high-performance  bank to me.

Here’s where hydraulic banks would be a huge opportunity: they can be adjusted to several levels. This is the case at U of M, but it maxes out at 10 degrees!  This is the same with other hydraulic facilities in the country, and Mondo, one of the major companies that manufactures and installs these surfaces and systems, specs their hydraulic offerings at a 10-degree max.  This is what really bums me out – the ideal case would be a system where you can tune the track to the race being run!  Some guys want to go for sub-4? Set the bank to 15-degrees! Somebody wants to go after the world record in the 400? Crank it to 25!  Some ladies are trying run a fast 5000m? Bring it back to 10! I’m not sure if the companies that make the hydraulic systems max them out at 10 per the IAAF recommendation, or if maybe there are some engineering challenges associated with designing a hydraulic structure across those ranges of angles (i.e. would designing and building a system that can go from 0 to 15 degrees instead of 0 to 10 be an exponentially greater structural and mechanical design problem?).  I’m not sure what the rationale is, but as it stands, that might be a huge opportunity lost right now.


  • For athletes and coaches: Plan ahead and pick your target races based on the facility best suited to the speed you’ll be racing.  As discussed, more is not necessarily better when it comes to a banked track.  The BU track may be great for guys at the high end of the performance spectrum, but a shallower bank is going to be much better for even some of those guys racing a longer distance.  For ladies, a 10 degree bank may be incredible for some distance races, but some girls, a lighter bank will be optimal.
  • For coaches: if you’re training on a banked track, be aware of the physical challenges of running slowly around a steep bank.  Some coaches are aware of the danger of running a lot of unidirectional laps around tight indoor turns, but the forces in the opposite direction on the steep bank could be a hidden devil.  Change directions, or better yet, if you’re jogging slowly on recovery between reps around a bank, jog on the flat interior.
  • For the NCAA (because they definitely will be reading this): Document and publish the bank angles and turn radii of your member institutions’ indoor facilities! That way athletes and coaches can plan and target races accordingly!

Final Thought

After running two heart-breaking miles in 4:00 and 4:01 on back-to-back weekends at U of M’s new facility, former NCAA Steeplechase champ and Michigan harrier Mason Ferlic jokingly asked me if I had any biomechanical advice for getting under that mystical four-minute barrier.  I sarcastically replied, “increase the magnitude of the instantaneous time-derivative of your center-of-mass’s horizontal displacement vector!” (or rather, “Run faster!”).  Now, I have some better biomechanical advice: buy a plane ticket to Boston.

5 thoughts on “There’s money in the bank: the physics of indoor track and “optimal” speeds

  1. Fantastic article! Your Classical Mechanics professor would be proud. The NCAA provides a conversion chart for qualifications standards that is given to 5 significant figures. This is static for all ‘banked and oversized’ tracks. I always thought those conversions looked a little too precise. Sounds like that level of precision would only make sense if bank angle, turn radius, and whole host of other factors were considered.


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  3. Awesome article Thank you for posting. Beynon Sports built the track surface at BU and the hydraulic track system at U of M. The hydraulic track at U of M does have the ability to adjust the degree of the banked turns.. Your comment on adjusting the degree of the banks by event to optimize times would be great except that this is not allowed by the IAAF (now World Athletics). I also believe the NCAA does not allow for adjusting the degree of the banked turns during events.


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