a bolt down the track

A pre-Olympics minipost

 

Competing in the 2009 Berlin World Championships, Usain Bolt set the 100 meter sprint world record at 9.58 seconds.  A few of us have analyzed his performance using position and speed data taken every tenth of a second by a laser tracking system. The exercise gave me a deeper appreciation for what's going on in this type of performance.

 

Physics suggests world class sprinters should be taller than world class distance runners, but at 195cm, Bolt is extremely tall even for a sprinter.  Aerodynamic drag is where most of a runner's effort goes in sprinting as the power required to overcome drag goes as the cube of your speed relative to the wind.  Part of sorting out what's going on requires knowing the effective area he presents running. The coefficient of drag (C_d) is multiplied times the cross sectional area of a body moving through a fluid (air in this case) to give an effective area - C_dA.  A very streamlined body is going to present much less drag than a brick of the same area.   A human runner has a C_d of about 1.2 ..  most cars are 0.3 to 0.4 and a bike racer is about 0.8, so you can see he's not very streamlined. And being so tall increases only makes matters worse by increasing the area he presents.  

 

But he makes up for it. 

 

Usain pushed off the starting blocks with about 830 Newtons of force, about the same as his weight, giving an initial acceleration of 9.5 meters per second. Around 0.9 seconds out he's delivering about 2600 watts of power.  That peak falls off rapidly, but the number needs to be put into perspective.  When you're riding a bike at a comfortable speed you're delivering between 50 and 150 watts.  A world class bike racer can supply about 400 watts for an hour.  A horsepower is 746 watts.  

 

From one to four seconds he's supplying a roughly constant force, but his acceleration slows as he fights drag. His speed continues to build - four seconds out he's reached a maximum speed of nearly 12.2 m/s - 27.3 mph.  The force from drag is now equal to the force he can generate.  He's reached an effective terminal velocity.  Bolt manages to maintain about 99 percent of that speed for the remainder of the race.

 

Now for the part that shocked me. The total energy that goes to motion is about 1.8 watt-hours, but since he's applying a nearly constant force you can work out the total energy he needs to run and overcome drag  - about 22.6 watt-hours. Ninety-two percent of his energy goes into overcoming air resistance!  He's almost entirely relying on anaerobic metabolism which can supply these enormous power bursts for short amounts of time (under 30 seconds) He jokes he couldn't run a mile at any speed.  Anaerobic metabolism is inefficient and the waste heat needs to go somewhere. The core body temperature of these sprinters rises quickly and would be a limiting factor (eg - they'd pass out or die) if they could maintain their pace for longer periods.

 

So why the picture?  The world record has dropped over the decades, but people have calculated the impact of better tracks (world record tracks have a special harder surface so less energy is lost during the run), better shoes, and starting blocks.  Given the same technical changes, Jessie Owens' best performance falls in the high 9.60s - about a tenth of a second difference between the two great athletes.  Of course we don't have time machines, but ...

 

Why is riding a bike so much more efficient than running?  That's for another time...