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Apparent Bycicle Paradox

  1. Feb 7, 2010 #1

    "If you cycle up a hill and then back down with no net change in elevation, it seems as if your slower uphill speed and faster downhill speed should offset each other. But they don't. Your average speed is less than it would have been had you cycled the same distance on a level road. Similarly, cycling into a headwind for half your trip and returning home with a tailwind yields an average speed less than you would have achieved on a windless day. The faster part of the ride doesn't compensate for the slower part. It seems unjust!"

    Assuming the biker transmits always the same power to the bike's pedals (200 Watts for example) prove that the mean velocity with wind is lower than it is for a windless day.

    Please help me, thanks
  2. jcsd
  3. Feb 7, 2010 #2


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    Wind resistance is the main loss on a bike above say 10mph, so if you slowly climb up a steep hill you are probably only going to freewheel down at 20mph because of wind resistance not the 30-40mph you would have to do to cancel out the 'cost' of going uphill.

    The wind always being in your face in both directions is a real effect - if the wind is coming at you from the side (even slightly form the side) your velocity adds to the wind's to give an extra drag force - it's one of bernouilli's equations.
  4. Feb 7, 2010 #3
    Thanks for the swift reply but how do you prove mathematically the paradox?

  5. Feb 7, 2010 #4


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    Just write the equations for the power consumed as a function of speed, incline etc. You get a term in V^2 so the energy used to go a distance isn't independant of speed.
    You then just need to assume that going up/down hill isn't the same speed as going on the flat.
  6. Feb 7, 2010 #5


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    The issue with the headwind and tailwind round trip is due to rolling resistance in the bike which increases approximately linearly with road speed. Your relative wind speed will be higher going into the wind than with the wind because of the difference in rolling resistance due to the difference in speed.

    The uphill and downhill sequence is due both to wind speed and rolling resistance.
  7. Feb 8, 2010 #6
    AFAIK rolling resistance is approximately independent of speed. The power that it takes to overcome it increases approximately linearly with road speed, but the work you have to do to overcome it only depends on the length of your trip (and your weight, road serface, tire pressure etc.) and does not depend on the speed of the bicycle or the wind speed.

    There are actually two reasons nearly any trip with uneven speed is slower.

    The first reason has little to do with bicycles or even with physics. If you compare a trip of distance d with an average speed of v with a trip where you do the first d/2 with speed (v-w) and the second d/2 with speed (v+w) than your average speed is going to be

    [tex] \frac {v^2 -w^2 } {v} [/tex]

    wich is less than v.
    so you need to go faster than v+w to make up for the time lost in the first half.
    If your speed on the first half is less than half of v (likely if your trip is a mountain climb plus descent) your average is going to be <v even if the second half of the trip is at infinite speed and takes no time at all.

    This first reason is valid for almost any kind of activity where time is involved: v could be m^2/hour painted, licenceplates/hour produced etc.

    The second reason is air resistance. The force is proportinal to the square of velocity. If you
    do half of the trip with speed (v-w) and the other half with speed (v+w) the work spent will be
    cd(v^2+w^2) instead of cdv^2

    On a round trip the increase in air resistance will be bigger upwind, than the decrease downwind, and even a side wind adds to the air resistance. The first reason is also still present.
  8. Feb 8, 2010 #7
    Last edited by a moderator: Apr 24, 2017
  9. Feb 8, 2010 #8
    Everything I could find about rolling resistance has F = C N, where C is the coefficient of rolling friction, and N the normal force.

    Your article is about trucks and I also doubt it because of this:

    Heat production goes up if only the speed goes up, and the load and the force of the rolling resistance remain constant.
    Last edited by a moderator: Apr 24, 2017
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