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Imagine you cycle 100km. On one route there is a mountain 100meter high on an other route is not a mountain but a straight way.

Usually it takes longer on the route wiht a mountain. One reason for that is that the time you spend going up the mountain is far more longer than going down. So at average you will go slower.

But how is it possible that with a moutain where the advantage of going downwards is equal to the disadvantage of going upward, at least regarding the objective obstacle.

Perhaps has something to do with the human body, but I can't see how this works.

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    Er ... 100m is a hill, not a mountain :-) – andy256 Apr 26 '16 at 21:37
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    Short answer - you spend more time and energy going up than you save/recover by going down. If you don't see that, do some empirical testing for yourself. find a hill and time your round trip ride up and down, and then do the same distance on the flat out and back. On the flip side, hills and 3D roads make a trip more interesting. – Criggie Apr 26 '16 at 22:54
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Say you can keep a 30 kph pace on a flat straightaway. On a hill, that can drop to 10 kph or even less depending on the grade you have to climb. Call it 1/3 of your flat pace. Say that half your run is uphill and the other half downhill, and let's also say that the total run is 30 km.

If you run on the flat at your regular pace, it takes one hour to do that run. But if you do 15 km of the run at 10 kph, you'll spend 1.5 hours just going uphill. There's no way you can make up the time, because you're already over by half an hour.

Now let's say you go uphill for only 10 km and downhill for 20. You still have only reached the summit when the hour's up.

There's a point at which you can do a short climb and still compensate by speeding downhill, but often you would have to attempt dangerous downhill speeds to make up your lost time. It's just not worth it. What is the obsession with time, anyway? In a race, everyone runs the same course.

Take comfort in the fact that the climbs you do are working your body much harder than running on the flats (or, obviously, downhill).

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Climbing up the mountain you are doing extra work relative to the flat route. At the top you will have accumulated potential energy. In order to break even with the flat route you will have to convert all that potential energy into motion, without any loss. This is not possible in the current universe due to the third law of thermodynamics. On the descent some of the potential energy you saved up will be lost to heat as you try to convert it to motion. This will put you at a loss relative to the flat route.

Some sources of energy lost are:

  • Wind resistance: Your drag increases as a square of your velocity. Therefore on the descent (which will experience higher speeds than the flat route) a lot of the potential energy you stored up will now be used to overcome the drag experienced at the higher descent velocities. Some of that hard work you did to climb will now be converted to heat, increasing the universes entropy and taking us one little step closer towards the heat death of the universe. If you decided to "cheat" physics and use your brakes, now you are simply turning some of that potential energy into heat via the brakes. There is no escaping the cruel mistress that is entropy!
  • Road Navigation: Even in a perfect vacuum (i.e., no wind resistance) you will still have to navigate the road safely. This means slowing for turns, thereby turning more of that precious potential energy into heat via braking.
  • Rolling resistance: While both routes will have to contend with rolling resistance, at higher speeds of the descent bearings will heat up reducing their respective inefficiencies. Therefore, even in a perfect vacuum (deep thought: does this exist?) with a perfectly straight-line descent (i.e., no turning) you will still come out at a loss relative to the flat route which will have a more consistent speed.
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    Good answer for all practical purposes, but readers should note that even in a system with no sources of friction or energy dissipation and perfect conversion of work to motion, it will always take longer to complete a route in which there are sections above z=0 and no sections below z=0, than it will to complete the course along the z=0 line, when the same work (as a function of position) is input in both situations. – Penguino Apr 27 '16 at 1:48
  • I seriously doubt you can significantly heat bicycle bearings on descent. The point about potential vs kinetic energy is true, conversely you can get a small time benefit by taking a low route, for example through underpass. – ojs Apr 27 '16 at 16:42
  • not only the drag, but the rolling resistance also goes with squared speed. (it is basically the same equation) – njzk2 Apr 28 '16 at 18:59

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