What-If 0154: "Coast-to-Coast Coasting"

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What-If 0154: "Coast-to-Coast Coasting"

Postby timrem » Thu Feb 09, 2017 6:02 pm UTC

Coast-to-Coast Coasting

Brandon Rooks wrote:What if the entire continental US was on a decreasing slope from West to East. How steep would the slope have to be to sustain the momentum needed to ride a bicycle the entire distance without pedaling?


Image


Follow-up question: how far could you ride on a bike made of soft clay?

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby dtgriscom » Thu Feb 09, 2017 7:04 pm UTC

Typo (or thinko):

Not only is there no way to build a slope that tall, but ice isn't even stable at those low temperatures, so there'd be nothing to slide on.


should say "pressures", rather than "temperatures".

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Soupspoon » Thu Feb 09, 2017 7:38 pm UTC

I am dissapoint that the video search link for "bicycle into water" didn't (for me, just now) give a sign of the one from The Monkees intro footage (and/or the full clip), although whilst I'm on mobile data I wasn't going to 'partake', just look at the screen-shot summary thumbnail piccies.

Also, interesting bike anatomy. I can accept the curvy dropped 'top-tube' ("ladies bike" style), and the handlebars aren't so prominant in profile as to reliably characterise (assumed near-straight), but the front fork rake looks to be totally wrong in all but the unreliable (angle of perspective) first 'at leisure' image, the (invisible, undrawn) hub being seemingly the wrong side of the steering axis for traditional stability. I'm a bit worried about the bottom bracket position, too, but that could be more of a drawing artefact.

:P

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby The Synologist » Thu Feb 09, 2017 8:27 pm UTC

dtgriscom wrote:Typo (or thinko):

Not only is there no way to build a slope that tall, but ice isn't even stable at those low temperatures, so there'd be nothing to slide on.


should say "pressures", rather than "temperatures".

This part jumped out at me too, but because I wasn't aware that ice would get less stable in a space-like environment. Does the low pressure outweigh the temperature of 0 degrees and actually cause it to boil away or something?

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby gmalivuk » Thu Feb 09, 2017 8:47 pm UTC

The Synologist wrote:Does the low pressure outweigh the temperature of 0 degrees and actually cause it to boil away or something?
It sublimates.
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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby speising » Thu Feb 09, 2017 8:48 pm UTC

Not boil, sublimate. But yes, the lower the pressure, the lower the point where water transitions to the gaseous state. After a certain point, it'll sublimate even near 0K.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Soupspoon » Thu Feb 09, 2017 8:50 pm UTC

The Synologist wrote:This part jumped out at me too, but because I wasn't aware that ice would get less stable in a space-like environment. Does the low pressure outweigh the temperature of 0 degrees and actually cause it to boil away or something?
In the following picture...
Spoiler:
Image

...the lower-left limb is the line to cross. Low pressures can make 'ice' drift straight over to 'vapour' state (sublimate, like "dry ice" does) at far below zero (celsius, but also fahrenheit if you want). But pressure from evaporation, in an enclosed container, could equilibrealise the drift. In open space, that wouldn't happen so much, and it would rely upon the general 'pressure of the vacuum' in the exact environment (within/outwith the heliopause? LEO? Bottom of a lunar crater with trace gas still hanging around?).

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Moose Anus » Thu Feb 09, 2017 9:33 pm UTC

Huh, I guess that's why they don't show puddles of water coming off of CG comets.
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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby rmsgrey » Fri Feb 10, 2017 1:43 am UTC

Is a constant slope the best option? Can you combine a steep section and a shallow section to cover a horizontal distance with less change in elevation than you'd need to just barely move on a constant slope?

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Soupspoon » Fri Feb 10, 2017 2:09 am UTC

A 'starter ramp', to gain speed over a metre or three of horizontal travel, then (dependant on formulated altitude) a very gradual slope whilst air-resistance isn't a big factor in the rolling resistance, then further increase of gradient as more air neefs to be pushed through. Seems like (apart from the tacking on of the start-ramp, which could as easily be a push off with the foot or a pusher-offerer official, as per time-trialling) a simple integral, after juggling the resistance factor function per altitude.

OTOH, for a quicker transit, doubly-inflected with a proper steep start whilst the drag, square of the speed but linear to pressure, is compensated sufficiently by the low latter even against the exponential former, before reverting to a still-high-altitude flatter gradient to maintain a decent velocity in still rarified air, before needing to increase the slope to not lose the velocity quite so quickly as the increasing pressure would normally do, before finally rolling over the finish line... Again, suitably deconstructed values are needed to be juggle through a composite integral, to find the optimum compound curve...

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Pfhorrest » Fri Feb 10, 2017 2:28 am UTC

Isn't what we really want a brachistochrone curved ramp?
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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Mikeski » Fri Feb 10, 2017 2:39 am UTC

Pfhorrest wrote:Isn't what we really want a brachistochrone curved ramp?


Possibly, but the question was "What if the entire continental US was on a decreasing slope from West to East. [...]", and that solution does not meet the requirement, with the low point somewhere in the middle.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Soupspoon » Fri Feb 10, 2017 9:17 am UTC

Pfhorrest wrote:Isn't what we really want a brachistochrone curved ramp?

Complicated by the fact that its a radially curved 'plane' being travelled across (which I meant to mumble about as a common lication, last message). But that's an answer to "the fastest slope", not "the slope with the lowest high end" (for which either a laser-straight (effectively curved) or a constant decrease of radius per instantaneous circumferential travel distance (actually curved but practically straight) is likely the intended solution - almost certainly the latter, as the "sublimating snow" illustration seems to suggest), and for fast-transit I'm wondering how high a (geosychronous, but not as high) starting freefall down a vertical track, outside of all atmosphere, would usefully create a velocity that in the lower atmospheric curving to sea-level acts as the shotest-time (initially hyposonic?) re-entry path that crosses the whole continent, exactly before a rolling "splash-in" at the opposing beach.

Too many scenarios! And with my high-school maths a bit rusty, so I can't even fully sanity-check my in-head imaginary-back-of-the-envelope simulations of each kind.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby speising » Fri Feb 10, 2017 9:33 am UTC

Mikeski wrote:
Pfhorrest wrote:Isn't what we really want a brachistochrone curved ramp?


Possibly, but the question was "What if the entire continental US was on a decreasing slope from West to East. [...]", and that solution does not meet the requirement, with the low point somewhere in the middle.


That's not true, a brachistochrone is always decreasing. But if it started at the same height as the straight path, it would be too shallow for the bike for much of its length, so the bike simply would stop.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Mikeski » Fri Feb 10, 2017 11:38 am UTC

speising wrote:
Mikeski wrote:
Pfhorrest wrote:Isn't what we really want a brachistochrone curved ramp?


Possibly, but the question was "What if the entire continental US was on a decreasing slope from West to East. [...]", and that solution does not meet the requirement, with the low point somewhere in the middle.


That's not true, a brachistochrone is always decreasing.


Then someone should fix the picture on the wikipedia page, which shows... not that.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby speising » Fri Feb 10, 2017 11:52 am UTC

Mikeski wrote:
speising wrote:
That's not true, a brachistochrone is always decreasing.


Then someone should fix the picture on the wikipedia page, which shows... not that.

Yes, they should.
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http://www.math.wpi.edu/Course_Material ... ject2.html

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby cellocgw » Fri Feb 10, 2017 1:59 pm UTC

Methinks Randall made a less-than-optimal assumption about the required starting altitude. I mean, gosh, just a week ago we were figuring out how to dig nice deep holes, so why not design the ramp to be at ground level in the MIdwest and somewhere below the IRT tracks in Manhattan?

Also, I wasn't paying a lot of attention, but what happens if we get picky about GmM/R^2 and observe a decrease in the gravitational force at the top of the ramp? While the rolling friction might be reduced in part due to the lower gravity, there's still some "stickiness" that's independent of the normal force, at least to first order.
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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Flumble » Fri Feb 10, 2017 5:34 pm UTC

speising wrote:
Mikeski wrote:
Pfhorrest wrote:Isn't what we really want a brachistochrone curved ramp?


Possibly, but the question was "What if the entire continental US was on a decreasing slope from West to East. [...]", and that solution does not meet the requirement, with the low point somewhere in the middle.


That's not true, a brachistochrone curve is always decreasing. But if it started at the same height as the straight path, it would be too shallow for the bike for much of its length, so the bike simply would stop.

Except a brachistochrone curve is defined for two points on the same height too:
Image

If I understand it correctly, the curve is only fully decreasing as long as the slope of the straight line more extreme than 2/pi.

Too bad the brachistochrone curve is useless when dealing with a bit of friction (rolling or air).

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby bondsbw » Fri Feb 10, 2017 5:59 pm UTC

The endpoint of a brachistochrone curve cannot be on a positive slope, otherwise the property that starting anywhere on the curve would get you to the endpoint in the same amount of time would not hold. (For example, starting at the lowest point.)

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Flumble » Fri Feb 10, 2017 10:52 pm UTC

bondsbw wrote:The endpoint of a brachistochrone curve cannot be on a positive slope, otherwise the property that starting anywhere on the curve would get you to the endpoint in the same amount of time would not hold. (For example, starting at the lowest point.)

First of all, just think: it is possible to go from point A to point B on the same height h=0 by going down a bit and up again. See the path C to A in my previous post, for example. Now, you can't go any higher than h=0, because you have no energy at h=0 to go any higher, so ending with a downward slope at B (at h=0) is not possible. A constant slope at B will not lead to a fastest trajectory, because, even if it doesn't result in infinite time to reach B (can someone prove/disprove this?), a straight line is both shorter and has a higher average velocity than a diminishing upward slope (which is the only way to reach y'=0 at B ). So any fastest curve will have to end in a positive slope.
So either the brachistochrone curve isn't the fastest curve, or your property doesn't hold.

Spoiler: there is no property that you get to the end point at the same time regardless of starting position. If you're talking about (the at wikipedia ill-explained similarity with) the tautochrone curve: the property you're speaking of holds for the lowest point of the curve, not for an end point.
The similarity at wikipedia is described completely intelligibly, though, because after lots of reading my understanding is that a brachistochrone curve always describes a part of a horizontal (inverted) cycloid, while a tautochrone curve does the same and always includes the lowest point of that cycloid. So a brachistochrone that doesn't include a point with zero slope (i.e. Δx/Δy<π/2, with Δx and Δy the horizontal and vertical distance between A and B) is not tautochronous.
Equations (20) and (21) in Douglas' article really need to make their way into the wikipedia article by the way. They describe the constants to use for the x,y parametrization of the curve/cycloid. (something the mathworld article withholds from us too) :D


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You'd better edit your comment at the wiki page. (assuming it was you by the posting times) :P

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby speising » Fri Feb 10, 2017 11:21 pm UTC

If A and B are on the same height, that's right. But that is not the situation we have here (or in the animation on the wikipedia page)
Watch this: https://youtu.be/skvnj67YGmw?t=16m43s

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby RCT Bob » Sat Feb 11, 2017 8:48 am UTC

I'm quite sure that those curves are without friction. Friction removes energy from the system and turns it into heat. For the energy removed by rolling resistance, the most energetically favourable way is travelling in a straight line, because rolling resistance is fairly independent of speed. So that energy loss is simply the path length multiplied by the friction force, and the straight line simply has the shortest path. For energy loss due to air resistance, the system is more complicated as air resistance depends on the air speed. For low speeds (below roughly 0.3 times the speed of sound) the drag coefficient is fairly independent of speed, depending on the shape of the cyclist and its angle of incidence, etc. The energy loss is still the drag force integrated over the path, but the actual math becomes more difficult as your speed depends on the path you take too, and the drag force itself also influences the speed.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby rmsgrey » Sat Feb 11, 2017 3:26 pm UTC

speising wrote:
Mikeski wrote:
speising wrote:
That's not true, a brachistochrone is always decreasing.


Then someone should fix the picture on the wikipedia page, which shows... not that.

Yes, they should.
Image
http://www.math.wpi.edu/Course_Material ... ject2.html


Whether the brachistochrone is monotone decreasing or not is determined entirely by the ratio of the drop to the horizontal distance - if you want to travel more than Pi/2 times as far horizontally as vertically, assuming friction to be negligible, you're better off borrowing energy by dropping below your target altitude.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Flumble » Sat Feb 11, 2017 4:39 pm UTC

RCT Bob wrote:I'm quite sure that those curves are without friction.

Indeed they are. You can incorporate friction in the equation, but like mathworld says "the problem can also be solved analytically, although the solution is significantly messier". (though I'd say the parametrization in x and y is still quite nice, except for the part where you probably have to vary both θ and k to optimize for time)

speising wrote:If A and B are on the same height, that's right. But that is not the situation we have here (or in the animation on the wikipedia page)
Watch this: https://youtu.be/skvnj67YGmw?t=16m43s

I used the same height as the simplest counter-example to bondsbw. That doesn't mean it's the only counter-example.
What you see in that video is, once again, a slope of approximately -2/π, so the brachistochrone curve is monotonically decreasing and ends horizontally. It's a pity no one on youtube has an experiment showing that the fastest trajectory on a shallow slope goes below the end point.

And since I don't have a wood/acryllic workshop, we'll have to do some math. Assuming a gravity of g=1 and a slope of 1:20*
proofs.png

Let's have a look at the green line. It would take 10 units of time to traverse it (because the velocity at height B is 1/2mV²=mgh => 1/2V²=1*1/2 => V=1 and the horizontal distance is 10). And we can agree that the green path is faster than any path from A to B that doesn't go below B. (It's shorter than the shortest distance between A and B and it's at maximum velocity all over the 'curve'.)

Now take the red path: it takes y=2=1/2gK² => K²=2/(1/2)/g => K=2 to get down, x=10=vt=(g*2)L => L=10/2/g L=5 to get across and M=K-Z, 1/2=1/2gZ² => Z²=1 => M=K-Z=2-1 =>** M=1 to get back up. So the red path takes a total of t=K+L+M=8, which is definitely less than the green path.
And as this page explains, having tight corners takes longer than a smooth curve, so you can improve on t=8 by rounding the corners of the red path a bit (and thereby making them physically feasible, in case you don't accept corners in a counter-example).

Since the paths that stay above B are slower than the green line and there exists a path faster than the green line, the fastest path (=brachistochrone curve) must go below the end point for this shallow slope.

*because it makes both the calculations and the counter-example path trivial. Changing gravity only means you have to go deeper/less deep on the red path. Making the slope even shallower works in favour of the red path (try it out with a horizontal distance of 20). And for steeper slopes, at some point, you need paths that approximate the cycloid, rather than some horizontal and vertical lines to provide a counter-example.
**It takes 2s to get from 0m to 2m and 1s to get from 0m to 0.5m, so it must take 1s to get from 2m to 0.5m. (arbitrarily using seconds and metres as the units of time and length here)

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Pfhorrest » Sat Feb 11, 2017 7:16 pm UTC

Ok so given the requirement for monotonous descrease in elevation I overlooked and the difference now explained between a brachistochrone and a tautochrone, would a tautochrone be what we actually want to get the greatest speed boost early on and so a lower maximum elevation requirement?
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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Mikeski » Sat Feb 11, 2017 7:37 pm UTC

Pfhorrest wrote:Ok so given the requirement for monotonous descrease in elevation I overlooked and the difference now explained between a brachistochrone and a tautochrone, would a tautochrone be what we actually want to get the greatest speed boost early on and so a lower maximum elevation requirement?

Because of rolling resistance and wind resistance, you'll always want a measurable amount of downhillness, and I don't think fancy curves and faster/slower sections will help a lot. Cute cycloid curves work because of negligible losses for the distance covered.

We're covering thousands of miles/km, but a bike will coast for, at most, a couple hundred yards/meters on level or near-level ground.

Wind resistance increases as v2, so building extra speed early just means you're wasting speed (height) to warm up the cyclist until the atmosphere slows him back down. You won't have the benefit of extra speed gained in California and Nevada when you reach Ohio. Or even Nebraska.

At 5miles/8km high, a 100kg bike+rider will be converting 7.8MJ of energy into heat (and a hundred or so J of leftover kinetic energy) over the length of the ride. So rather than negligible losses, we're all losses with negligible gains.

(In theory, theory and practice are the same. In practice, they're not.)

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Thibaw » Sun Feb 12, 2017 10:01 pm UTC

Someone please do the math for iron wheels with high-tec ball bearings on iron track. Thank you a lot.
If it is feasible we should start a kickstarter or something.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby mosc » Mon Feb 13, 2017 1:42 pm UTC

So what would the ratio be (distance over drop) to keep a bike going fast enough where it could self-stabilize? 500/1 gets you moving but it does not keep you from falling over. Is it 250/1? Less?
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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Tribunzio » Wed Feb 15, 2017 2:15 am UTC

Wind resistance can be a boon... Prevailing winds are generally from West to East and could significantly help, particularly if the bicycle is equipped with a suitable kite.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby hermitian » Wed Feb 15, 2017 5:58 pm UTC

What is all this tecno-babble in here?

I came to boast that I have done the bike ride down Mount Haleakala. It was fantastic. The only thing really breaking up your coasting is stopping to take photos of the beautiful scenery. I never expected to see it in a What if.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Mikeski » Wed Feb 15, 2017 11:20 pm UTC

Tribunzio wrote:Wind resistance can be a boon... Prevailing winds are generally from West to East and could significantly help,

True.
particularly if the bicycle is equipped with a suitable kite.

I'm not sure intentionally utilizing wind power would count as "coasting". All the ways a kitesailed bike could go wrong are certainly in the spirit of What-If, though.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Flumble » Thu Feb 16, 2017 1:57 am UTC

Mikeski wrote:
particularly if the bicycle is equipped with a suitable kite.

I'm not sure intentionally utilizing wind power would count as "coasting". All the ways a kitesailed bike could go wrong are certainly in the spirit of What-If, though.

Then again, it was Randall who turned "without pedaling" into "coasting".
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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby HES » Thu Feb 16, 2017 1:40 pm UTC

Flumble wrote:Then again, it was Randall who turned "without pedaling" into "coasting".

I mean, if we're relaxing that one there are plenty of motorised bicycles (which are a different thing to motorcycles) that would suffice.
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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby gcgcgcgc » Fri Feb 24, 2017 12:56 pm UTC

Questioner: "What if the entire continental US was on a decreasing slope from West to East."
Randall: "[...] To travel the roughly 2,500 miles from New York to LA,"

I was about to post that the North-South element of the vector would make the slope effectively shallower, then I noticed that he was going uphill the wrong way anyway. At least the signs point in the right direction so I assume this was just a typo.

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Re: What-If 0154: "Coast-to-Coast Coasting"

Postby Cougar Allen » Wed Mar 01, 2017 4:24 am UTC

mosc wrote:So what would the ratio be (distance over drop) to keep a bike going fast enough where it could self-stabilize? 500/1 gets you moving but it does not keep you from falling over. Is it 250/1? Less?


With a little practice you can learn to balance a bicycle even if you're holding still.
https://www.youtube.com/watch?v=nRR4paQnUsQ


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