A couple of months ago, I was running with a friend and we were talking about running on hilly courses. He made the comment that the most efficient way to run is at constant effort, that is, slowing on the way up and speeding on the way down as gravity helps. I tried to explain that that wasn't necessarily the best way to do it and used as an example when I ran the Afton 50K with Eve Rembleski (she dropped me about 10 miles in); she ran exactly the same pace regardless of terrain, while I slowed going uphill and sped up going down. She set the course record for women and I fell apart. Admittedly, I often fall apart in races and one instance does not make a rule, but there is more to the situation than mere physics. For one thing, it's very hard to judge just how much slower one should go uphill and how much faster downhill - I frequently barrel downhill, causing myself to use additional energy to brake.
If you run the absolute fastest you can at a steady rate on a hilly course, you could run faster still by running at a steady effort. That much is simple physics. However, to do that, you have to know ahead of time what the absolute fastest actually is. It is easier to train oneself to run even pace and there may be a saving of "psychological energy" in so doing.
If you look at cars, the fastest are the least fuel efficient: six miles to the gallon at 200 miles per hour is doing pretty good. You have to sacrifice efficiency once you go a certain speed. This is true in running as well.
When I started running ultramarathons, I gradually shifted my gait to a bit of a shuffling stride (though, being a forefoot striker, I never made the transition complete). The shuffle stride, first made famous by Derek Clayton and then by Alberto Salazar, is extremely efficient and is seen almost to the exclusion of others in 100 mile races. And while it's possible to run as fast as 4:15 per mile in this stride (as Clayton and Salazar frequently did), to run their absolute top speed, they would've had to change to the sprinter's stride, which is less economical. There are no shuffling sprinters.
I've been re-teaching myself how to run. Only this week have I started to turn the corner. I'd started running faster - but much less - and had done this for a while by increasing stride rate, but that had hit a limit. I now can sense, at about 6:30-7:00/mile, that I'm raising my knees higher, pulling my heel further up in the back kick, swinging the trailing leg through more forcefully and hitting the ground with my foot with greater elasticity and more of a pawing ankle flexion.
Unfortunately, it also makes my damaged achilles tendons ache. If I can get used to this stride and build up the distance I can run that way, I may just have some speed left.
Going up the country
1 day ago