head and tailwinds

Today a friend said he was having a difficult time concentrating - ".. like running in circles, but with headwinds."   After hanging up I found myself thinking about running or cycling on a closed course in a wind.  Just how much does a wind cost if you have an equal amount of head and tailwind?   A bit of high school physics is required.

 

Here's a simple model.  Consider a square closed course with the wind blowing at a constant speed w from the left..

 

Let's make it easy.  Rather than thinking about what speed you can run on each leg, imagine a constant speed - say v - on each leg of the course and calculate the amount of power you need to generate to overcome the aerodynamic drag force. Everything else you do running should be the same.  

 

Power is force times velocity. Since your mass isn't changing, just look at what power is proportional to.  Tricks like these are common in physics - just leave out terms that stay constant where we only care about the comparison rather than a number.   The very quick answer would be since the power increases as the cube of the effective wind speed you will never gain as much from a trailing wind as you'll loose going into the wind.   Calling wind speed s

 

sorting it out:

 

for leg 1   required power proportional to (v + s)³

for leg 2   required power proportional to  (v - s)³

 

the crosswind terms are the same:  v(v² + as²)  where a is some constant showing how much extra force you need to use to deal with the crosswind.  It could be very low or high depending..  For now consider the simple case where a is 0 and expand the expressions.

 

leg 1: v³ + 3s²v + 3sv² + s³

leg 3:  v³ + 3s²v - 3sv² - s³

leg 2 = leg 4: v³

 

So if we add these up (we're just doing a comparison) we get 4v³ + 6s²v  (you can get away with this as the running speed v is constant).

 

With no wind the power around the course is proportional to 4v³, so having a wind always requires you to generate more power.  What you gain from the tailwind never compensates for what you lose from the headwind and letting the crosswind term a be greater than zero (it will be) only makes things worse.

 

Perhaps a simpler way to think about it is to consider just the drag force from air resistance.  Let's say you're running at 10 mph and have a 3 mph wind.  Going into the wind you're feeling 13 mph of effective windspeed so the drag force will scale as the square of that or 169. With the tailwind you're feeling 7 mph so the drag force scales as 49.  With zero windspeed the force scales like 100.    You lose 69 going into the wind and pick up 51 running with the wind.  

 

So you want to draft behind other racers going into the wind and try to make sure no one is right behind you when you have tailwind. And if you're worried about record setting it's best not to have a wind on a closed course.