h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

A cyclist needs to ride up a slope, which for the purposes of this question is an infinitely wide plane. It is too steep to ride straight up, but he can choose whatever angle he wants to get to the top. What sort of angle should he choose to minimise his effort?

I live on a hill and face this situation every day. I can go up a steep road which is tiring but quick, or a more gentle gradient which takes longer but is much easier. When I'm tired I go up the gentler way, but I'm not sure I'm any less knackered at the top.

I'm aware that theoretically, the work done is the same in each case, but this is more of a real-world question, and the cyclist gets tired, and can suffer from pulled muscles...

There isn't enough information to answer the question. How much power can the cyclist exert at maximum? How much for comfort?

Incidentally the amount of potential energy gained going from bottom to op remains the same outcome but not the overall work done. If you star zig zagging then you travel a much longer path and so must exert work to travel along too (or is this a theoretical cyclist in a vacuum on a frictionless surface?).

Are you already using the lowest available gear on the direct ascent?

It is easier to take the longer and easier slope - I know from experience however the more direct and challenging route is more satisfying in the end and it will get easier over time.

I found this site via hootoo but I can't remember which thread - http://www.velominati.com/blog/the-rules/ . Rule 5 and Rule 10 apply in this case.

t.

##What sort of angle should he choose to minimise his effort? ##

straight down and stop at the pub

You're dead right there Taff. Best plan any time.

t.

>>It is easier to take the longer and easier slope ../

But is it? The further you go, the longer it takes. Less work but for more time. You wouldn't travel infinity far, it would kill you, and you can't go directly up, as stated in the OP. So which slope should you take?

i suppose a 45' angle would be optimum

half the slope, twice the distance

>>45' angle/../half the slope, twice the distance

Are you are saying that it's twice the distance up a 45° slope compared to a 90° ascent to the same height? It isn't.

no

your right

for every 1 Km at 90' you go 1.4142 Km at 45'

it's late i have been up all night

This is an interesting one, in that there are two answers, one objective and factual, the other one subjective and a bit wooly.

On the one hand, if you have gone from height H0 to height H1, you have increased your potential energy from mgH0 to mgH1, regardless of the way you've accomplished it. So if you just consider your energy change, it doesn't matter.

But if you consider the WORK you have to do to achieve it... that's the force you exert times the distance you move. Since you're working against gravity, the shortest path is straight up, but that's the maximum force requirement. As you move away from the vertical, the force requirement reduces, but the path length increases. OK...

You can accomplish that in an infinite number of ways. The two variables you have are your path length (i.e. how steep a path do you choose?) and your speed (i.e. how quickly do you choose to complete that path?)

In prinicple, you can choose an infinite number of paths, from vertical to almost horizontal, and you can choose a practically infinite range of speeds, from near zero to lightspeed.

In practice, your speed choices are quite limited. Unless you're Lance Armstrong, you'd do well to pedal up even a mild slope at much more than 20mph, and anything less than 2mph and you'd probably be better off walking, to say nothing of probably falling off.

Similarly, your path choices may be limited to a straight assault or a particular alternative at a fixed angle.

What you omit to mention is whether time is a factor - is it OK for the easier route to take longer? Or do you need to get somewhere against the clock?

My experience is that my perception of how "hard" a particular physical task is is NOT governed by the energy required or the work done, but by the RATE at which I'm required to work, i.e. the POWER expended. I do not perceive walking to the shop to be in any way strenuous. I do very much perceive jogging to the shop to be strenuous. I perceive sprinting to the shop to be impossible. The energy change and the path length are exactly the same, and my feelings are entirely about how fast I'm expected to expend energy.

On that basis, if you're like me, and time is not a factor, it's always going to be the longer route at a leisurely pace. But this is obvious.

The crucial question is - if time IS a factor, which is "easier"?

My gut feel says neither - my gut feel says that the longer, "easier" route must be traversed faster. So while it's less effort per turn of the pedals, you have to turn the pedals more times per second...

I'll do the sums when I get home.

(there's a vaguely related problem in one of my recreational maths books, about the optimum path from somewhere some way inland on a beach to someone a short distance out in the water, given than one can run at 15mph across the beach but only move at 1mph in the water. The boundary conditions are a straight line (i.e. direct path from point A on the beach to point B at the swimmer), which minimises distance travelled, but has a maximum amount of the path at the 1mph, and a direct run to the point on the waterline where the line to the swimmer is perpendicular to the waterline, which minimises the swim length but is the longest possible run. There is a single, objectively correct optimum here, however, unlike (I think) the bike one as described).

Above the lower speed limit set by balance issues, or pedal-turning being too slow or force required too high even in the lowest gear, on many cycles, gearing will result in the rider being able to choose a fairly constant pedal speed/force combination on a whole range of slopes, at least if using wide (non-racing) gears.

That said, the steeper slope can be more psychologically slow, not merely because the road distance covered is lower even if the height gain per unit time might be the same as a shallower climb, but also because of the lack of momentum/kinetic energy making even brief drops in effort cause immediate drastic slowdown.

Ignoring air resistance, compared to a steep road, on a road half as steep in terms of height gain per metre travelled, the same power input will give twice the speed and 4x the kinetic energy, which would carry a coasting bike 4x higher and so 8x further along the road.

Personally, I'd probably choose a hill I could ride up at 10mph over one I could ride up at 5mph, since air resistance is still pretty low at the higher speed, but balance is much easier.