Energy used cycling on flat surface/hills

[clandarkfire]

Do you use more energy cycling on, say, a 10 km flat stretch, or on a stretch where you spend the first 5km cycling uphill and the second 5km coasting downhill? What if you spend the first 1km cycling up a pretty steep incline, and the remaining 9km coasting down a more gentle slope? What if you add the constraint that you have to complete both routes in equal amounts of time?

I've considered this a bit intuitively, but I'm not sure how to model it mathematically.

A thought:
1. It seems that if your bicycle is really efficient (here I just mean that it doesn't lose much energy to friction), hills might be easier. You could theoretically spend only one meter going forward, gain a little elevation, and then coast down for the remaining 10 km without using any energy. But if your bicycle is that efficient, then I guess you would also be able to go forward a long ways using little energy on flat ground.

Intuitively, I have a feeling that the same amount of energy should be expended on a flat course as on a hilly one. But I don't know how to justify it mathematically. Thoughts?

 

[berkeman]

Do you use more energy cycling on, say, a 10 km flat stretch, or on a stretch where you spend the first 5km cycling uphill and the second 5km coasting downhill? What if you spend the first 1km cycling up a pretty steep incline, and the remaining 9km coasting down a more gentle slope? What if you add the constraint that you have to complete both routes in equal amounts of time?

I've considered this a bit intuitively, but I'm not sure how to model it mathematically.

A thought:
1. It seems that if your bicycle is really efficient (here I just mean that it doesn't lose much energy to friction), hills might be easier. You could theoretically spend only one meter going forward, gain a little elevation, and then coast down for the remaining 10 km without using any energy. But if your bicycle is that efficient, then I guess you would also be able to go forward a long ways using little energy on flat ground.

Intuitively, I have a feeling that the same amount of energy should be expended on a flat course as on a hilly one. But I don't know how to justify it mathematically. Thoughts?

Does this wiki entry on basal metabolic rate change your thinking?

http://en.wikipedia.org/wiki/Basal_metabolic_rate

.

 

[clandarkfire]

A bit. The takeaways seem to be that BMR is difficult to calculate and varies quite a bit from person to person, and that a person might burn more calories if the trip takes longer. So I suppose that a trip that is mostly slow coasting might still burn a lot of energy.

What I'm more interested in calculating, I guess, is how much energy it takes for each course independent of physiological expenditures. What if the bicycle were powered by a robot that gets turned off as soon as it starts coasting?

What I might be asking for is just which course requires more work. But I'm not quite sure if that's the appropriate term.

 

[willem2]

You should really add the condition that both trips need to take the same time. Because of air-resistance, the energy you need will increase with speed. You really can't ignore air resistance when cycling,

If you need to take the same time, the hilly trip will take more energy because of two effects:

1. You will spend more time on the slower sections, and this will decrease your average speed. if you go 5 km uphill with 20 km/h and 5 km downhill with 30 km/h, the average speed is 24 km/h and not 25 km/h. If the first half of your trip is a 10% climb, your speed will be less than half your cruising speed on the flat, and you'll never catch up, even if the descent is instantaneous.2. Air resistance. Because it depends on the square of the speed, you'll get really punished on any section that is faster than average, such as on a descent. Wich is probably a good thing, because you would either go much too fast on a long descent or your brakes would overheat.

 

[dipole]

Willem has it right with the air resistance. If you want to make an absolutely fair comparison between two possible courses - one which is perfectly flat the whole time, and the other which includes a hill, and you impose that both are the same distance and must be completed in the same time, then they'd really be about equal without air resistance.

Because of the air resistance though, you expend a LOT of energy pushing yourself up the hill, but on the down slope you lose that energy at a rate proportional the square of your speed - the bigger the hill, the more energy you lose going down it. This means that to complete the trip in time, you'll have to compensate for the energy you lost due to the air resistance because your speed increased on the downhill section.

Plus, I can tell you from personal experience that biking on flat level pavement is a breeze, but throw some hills and grades in there and it becomes brutal.

 

[CWatters]

Consider what happens on a very high hill..

On a very high hill you will gain more PE than is needed to overcome the remaining air resistance on the downhill section. To complete the route in roughly the same time/speed you would therefore have to waste some of that PE as heat in the brakes to stop you going too fast (eg to keep the comparison fair). This suggests to me that a hill which gives you just enough PE overcome air resistance on the remaining part of the route would be a sort of cross over point. Any higher and you are going to have to waste heat in the brakes.

However it's not clear that a slightly lower hill offers any advantage. I suspect it's break even if you set up the problem the right way. A low hill wouldn't give you enough stored energy to overcome air resistance on the remaining part so you would have to add some more on the way down. Under the right conditions I think it should balance out. The hill is just storing energy for use later so no net gain.

 

[dipole]

Consider what happens on a very high hill..

On a very high hill you will gain more PE than is needed to overcome the remaining air resistance on the downhill section. To complete the route in roughly the same time/speed you would therefore have to waste some of that PE as heat in the brakes to stop you going too fast (eg to keep the comparison fair). This suggests to me that a hill which gives you just enough PE overcome air resistance on the remaining part of the route would be a sort of cross over point. Any higher and you are going to have to waste heat in the brakes.

However it's not clear that a slightly lower hill offers any advantage. I suspect it's break even if you set up the problem the right way. A low hill wouldn't give you enough stored energy to overcome air resistance on the remaining part so you would have to add some more on the way down. Under the right conditions I think it should balance out. The hill is just storing energy for use later so no net gain.

No, because a good chunk of that PE is lost due the air resistance when you're coasting at higher speeds on the down slope. The power required to keep the bike moving is going to be proportional to the square of your velocity (roughly). At higher speeds, it requires more power to keep the bike moving at that speed - so when you store up all this PE by pedaling up the hill, a good chunk of it is lost from the air resistance as you race back down the slope. The bigger the hill, the faster you'll coast down, but in the process expend a huge amount of energy to accelerate to those higher speeds, and quickly you'll slow down again once it levels off and that energy is gone.

 

[CWatters]

Fully understand about the velocity squared issue but I thought we had already decided that to make the comparison fair the trip had to be completed at roughly the same speed (eg no big departures from average speed or so slowly that air resistance wasn't a big factor?).