# Relationship between power and speed in a time trial

I’ve been reading a lot about aerodynamics and power – they are two of the most frequently used terms in time trial discussions.  ‘Testers’ (time trial riders) are often obsessed with them.  Seemingly for good reason.  In a flat, or flat-ish time trial, the weight of rider and bike doesn’t matter all that much.  What matters is how aerodynamic they are, and how powerful.  Without wishing to state the obvious, the former is about how easily your shape (you and the bike) slip through the air, and the latter is about how hard you’re able to press the pedals.

I decided to write this post because I found it interesting to read about how speed, power and overall time to complete a course are related.  Clearly, the more powerful a rider is, the faster they will travel, and the shorter the time they will take to finish.  What’s interesting is the varying effects of increased speed and power.  Here’s a chart I made.  I will attempt to explain below:

Speed vs Time vs Power over 10 miles

The X axis shows the speed the rider is travelling, in mph.  Most TT riders will complete 10 miles at a speed somewhere in this range.  The blue line, and the left-most Y axis shows the time taken to complete the 10 miles.  So if we take the upper left most point on the line, a rider travelling at 20mph will complete 10 miles in 30 minutes.  At the other end of the line, a rider travelling 30mph will complete the distance in 20 minutes.  Hopefully we can agree on that!?

So far, so obvious.  What I find interesting is what happens in between.  It’s tempting to think that each incremental increase in speed would result in a similar linear reduction in time.  But this is not the case. Consider rider A, who previously has averaged 20mph and so has finished in 30 minutes.  If he increased his speed by 1mph to 21mph, he will finish 1 minute and 25 seconds (1:25) quicker.  Rider B has previously averaged 24mph and has finished the course in 25 minutes.  If he increased his speed by the same 1mph increment, to 25mph, he would complete the course in 24 minutes, i.e. 1:00 quicker.  So at lower speeds, an extra 1mph is worth significantly more time than at higher speeds.  It gets worse the faster you go.  Rider C has previously averaged 29mph and has increased his speed to 30mph.  As a result, he only gains 41 seconds.  So, the faster you go, the smaller your gains per extra mph.  Increasing speed from 20mph to 21mph will saver approximately twice as much time as increasing speed from 29mph to 30mph.  Shame!

Now let’s look at power.  Firstly, it has to be said that a chart showing power and speed is dependent on aerodynamics.  A rider who is very aerodynamic can ride the distance in a set time with less power than another rider who is less aerodynamic.  The specifics of this are too complex for me to understand at the moment, beyond that simple truth.  So the green line on the chart represents the power required to ride at each speed, given some rider’s aerodynamic characteristics.  The actual numbers are generalised.  The relative differences are not, and it’s these I want to talk about.

The key point is that there is an (approximate) cube root relationship between speed and power.  That is to say, a given increase in speed requires a cubed increase in power.  So in the extreme case, in order to double your speed, you need approximately 8 times as much power.  I’ll let that sink in for a second or two!

Looking at the green line on the chart, we can see that this rider must push 142 watts in order to ride at 20mph.  Again, I must say this is due to their specific aerodynamics (and to a lesser extent, weight).  If he wants to increase his speed to 21mph, he must increase his power to approximately 160 watts.  So 18 additional watts for 1mph.  Over time he gets faster, and now he’s riding at 24mph, pushing 224 watts.  If he wants to increase his speed by the same 1mph, he must push 250 watts, i.e. an additional 26 watts.  That’s 44% more power increase required for going from 24 to 25mph, than 20 to 21mph.  Again, it gets worse the faster you go.  After a few years our rider has got so fast that he’s now travelling at 29mph.  If he wants to increase to 30mph he needs an additional 35 watts. Almost twice as many additional watts as it took him to go from 20 to 21mph.

It’s a double whammy – an extra 1mph at the top end saves you half as much time as at the lower end, but costs you twice as much additional power.  I guess this is why the faster riders care so much about optimisation.  So it seems the faster you go, the harder it gets to improve.

If you spot any mistakes, please comment.  If you don’t spot any mistakes, you’re also very welcome to comment! 🙂