## 0085: "Paths"

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jgf
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### 0085: "Paths"

I didn't find a "Paths" thread, so pardon if this is a repeat.

The rainy season has officially started in Berkeley, which has me wondering about a related efficiency problem every year: Assuming rain is falling at rate R with wind blowing such that it falls with incident angle Theta relative to the path of a bicycle, at what speed V should the bicycle go to minimize the amount of rain that hits it.

I'm hoping the answer isn't the speed of light, because I can't peddle that fast.

Oh yeah, I calculate the efficiency of my path too. I enter campus just to cut corners on the way to lab.

Torn Apart By Dingos
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Let's start with the most intuitive case: wind falling sideways! Then theta=0. Obviously the best thing to do is to cycle with speed R away from the rain.

Now let's assume the cyclist is shaped like a straight vertical line. We can assume the cyclist is standing still, and instead modify the rain's horizontal speed. The velocity of the rain is then r=(Rcos(theta)-V, Rsin(theta)).

The flux of rain through the cyclist is the proportional to r . n = Rcos(theta)-V (. is the dot product and n=(1,0) is the cyclist's normal vector). The minimal value is attained for V=Rcos(theta).

Note that he now gets no rain at all on him, but that's only because he's infinitely thin.

I too care about paths, and it's not as silly as it may seem. A gain factor of 1.4 (the exact value of sqrt(2), as everyone knows) can be pretty significant.

Andy
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The path that messes me up the most is one at uni.

I couldn't quite describe it in words... but there are at least 5 paths which I've experimented with, and I still don't know which one is best. Having steps in the world makes things so much tougher because if they are the right size, then they are fine, but if they are wrong, then your steps have to be rounded to their length, which slows you down a lot. Then, going down the stairs at different angles will change these lengths, so a straight line across some stairs might not be as good as one which takes the steps at a better angle. The ramp is nice, because you don't have to think about step length, but it is very much not the straight line path.

Cynic
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Andy, is that the university of NSW ?

edit: No. Looks similar but isn't it. =/

Andy
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Cynic wrote:Andy, is that the university of NSW ?

edit: No. Looks similar but isn't it. =/

Yeah it is... I can't draw

I was really curious as to whether anyone would be able to work out what uni I was at. Now, I think I've left enough bread crumbs for you to steal my identity)

I'll try and find some photos:

On the left of this is the ramp from the picture. Point B is the building on the right.
http://www.mech.unsw.edu.au/images/SMMEFrontGraphic.gif

These two are taken from near point A: (looking down, with point B on the left)
http://www.flickr.com/photos/auswy/157904474/
http://www.flickr.com/photos/benneb/162986599/

You win a prize

Rebelfish
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### Re: "Paths" discussion

As you can see in the photo of my mystery campus, I had many paths of varying curvatures to choose from (some covered from rain, some not).

With regards to the optimum speed, I remember hearing that Mythbusters did a thing on that (with people running rather than biking). The had a simulated rain shower that someone walked through and someone ran through, and found by weighing the clothes before and after that they were the same. Seems scientific enough, but I don' really trust anything they prove on that show.

Moo
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### Re:

Andy wrote:

Well the obvious solution is to take a slide-type apparatus with you and slide down the grass. A water-spraying frontal attachment could be considered for when it hasn't rained to reduce the coefficient of friction?
Other points to ponder could be a horn to warn unsuspecting students in your way; a breaking system (if you are a scaredy-cat!) and making it compact enough to carry around with you all day.

Of course this is a one-way solution; getting uphill is a different matter. Rudimentary propulsion?

EDIT: I see the trees problem. May require a re-landscaping (cutting them down) or going down the ramp instead.
Last edited by Moo on Fri Oct 05, 2007 3:15 pm UTC, edited 1 time in total.
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Aviatrix
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### Re: "Paths" discussion

I sure wish there'd been, say, a link to the comic in the basenote.

Cytoplasm
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### Re: "Paths" discussion

I always wonder the same thing, if my path of travel is efficient, when walking home from school. I have yet to truly time the possible paths out.
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lowbart
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### Re: "Paths" discussion

Take a piece of cardboard and slide down the ramp.
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Random832
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### Re: "Paths" discussion

Rebelfish wrote:With regards to the optimum speed, I remember hearing that Mythbusters did a thing on that (with people running rather than biking). The had a simulated rain shower that someone walked through and someone ran through, and found by weighing the clothes before and after that they were the same. Seems scientific enough, but I don' really trust anything they prove on that show.

IIRC the problem with the Mythbusters experiment was that they allowed the clothes to get saturated.

http://www.straightdope.com/classics/a3_395.html for a better treatment (the result, by the way, is that running is better. It doesn't examine if there is an optimal speed or if you should go arbitrarily fast)

Oh, and also, OP fails.

The original question is not sufficiently interesting because everyone knows that the fastest way between two points is a straight line. So let's make it more interesting - assume that the speed on pavement is vp and the speed on grass is vg. Let k = vg/vp.

So:
• 1 = t
• 2 = t*(1+k√2)/3
• 3 = (tk√5)/3

• What value of k make these three equivalent?
• Or, if there is no such value, what values make any two equivalent?
• What is the optimal path for these values of k?
• Construct a general formula for the travel time for any given point where he leaves the pavement.
• Construct and prove a general formula for where along the NS path he should leave the pavement for optimal travel time for any given value of k.
• Compare the time saved by choosing the optimal path over the course of the number of trips in a semester with the time a first-year undergradute math student would take to work this out.
• A senior?
• A PhD student?
• An english major?
• Consider the case where he walks east and then proceeds diagonally to the destination point.
• Consider the time spent crossing the strip of pavement in the center at vp
• This depends on the width. Introduce it as a variable, then try for different angles.
• Also consider the width of the edge pieces of pavement.
• Would it be beneficial to cross the central walkway at a different angle than he was proceeding across the grass?
• Assume direction cannot be changed instantaneously. Would the optimal path become a curved path rather than connected straight-line segments?

mturyn
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### Hystere[sis|ria]

I have an additional difficulty, a psychological one: I both intensely want to optimise, and also "wear out" paths I've used too many times, so I have to add an history-dependent piece to the equation.... Fortunately, for very short paths I can just grit my teeth and bear the repetition, and longer paths allow for more piece-wise variability.

cj-maranup
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### Re: "Paths" discussion

So glad I'm not alone on this one - I spent 4 years at UWA finding the optimal routes for accessing various buildings, accounting for variables such as rain, sun, insane cyclists, bus arrivals, cricket matches & the avoidance of the Arts building... Since then I have been known to use google maps to find the optimal angle across various cities...

Andy - do you not find that the over-large steps mean you take larger steps & thus walk faster? Is speed a primary concern, or is it really about minimising the distance?

Lunch Meat
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### Re:

Andy wrote:The path that messes me up the most is one at uni.

I would tend to walk on a mostly straight-line path and just jump the larger steps. Of course, many times when I walk to class, if I'm not in a hurry I strive to make it interesting, meaning I'll go over railings, walk on retaining walls and suchlike.

My main problem is that I could pretty easily walk on a straight-line path, but then I would feel like a jerk for walking on the grass instead of all the nice concrete paths the school went to all the effort to make, even though they're not efficient.

anyone
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### Re: "Paths" discussion

I hope you don't carry credit cards, with many of your paths going through that magnetic field.

NThisStyle-10-6
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### Re: "Paths" discussion

My main problem is that I could pretty easily walk on a straight-line path, but then I would feel like a jerk for walking on the grass instead of all the nice concrete paths the school went to all the effort to make, even though they're not efficient.

See... the thing there is, you're probably paying to go to school there - even if it's public. Grass doesn't die from one person's path unless you walk it in exactly the same way every time, and it should be the school's job to put down sidewalks in the most efficient building to building manners. Of course you don't want a concrete campus, that would be ugly, but why not put down grass and when paths get worn into them, pave that? I don't know.

It might make me a vandal, but I walk across "Do Not Walk: Soft Sod" areas pretty often.

ivory_tinkler
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### Re: "Paths" discussion

I drive myself mad sometimes trying to figure out the most efficient routes to places. Don't you just hate those parks or campuses with weird wiggly paths?! When I'm not in a hurry (not very often, due to scatty nature) I will console myself with taking the 'scenic route' and trying not to worry about it.

TheKrikkitWars
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### Re: Re:

Lunch Meat wrote:
Andy wrote:The path that messes me up the most is one at uni.

I would tend to walk on a mostly straight-line path and just jump the larger steps. Of course, many times when I walk to class, if I'm not in a hurry I strive to make it interesting, meaning I'll go over railings, walk on retaining walls and suchlike.

My main problem is that I could pretty easily walk on a straight-line path, but then I would feel like a jerk for walking on the grass instead of all the nice concrete paths the school went to all the effort to make, even though they're not efficient.

Surely the solution is to purchase a bike, and add pnematically adjustable height suspension and caterpillar tracks and head down the bigger of the two steps (or listen to Song 2 by Blur and pedal really fast).
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neoliminal
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### Re: "Paths" discussion

GOOMHR. I do this constantly. I've found Google Maps doesn't really help because it's skews the distances. I even use my timer on my iPhone to measure the time to various places. I live exactly between two subway stops, with a 5 second difference to the platform, so I go to the station that is further down the path on the off chance that there is a train under me as I walk that I would have missed going to the other station.
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Mumpy
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### Re: Re:

TheKrikkitWars wrote:Surely the solution is to purchase a bike, and add pnematically adjustable height suspension and caterpillar tracks and head down the bigger of the two steps (or listen to Mars by Fakeblood and pedal really fast).

Fixed for best cycling song ever.

Grimer11
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### 85: Paths

https://xkcd.com/85/

Just re-reading through the entirety of xkcd...

As with most of the comics on the site I related to this one pretty well and just wanted to add that along with taking the shortest route time-wise there is a variable I always have to carefully consider, especially when walking a daily route - whether or not I should sprint.

It always trips me up because I'm naturally an introvert and don't want attention drawn to me and when there's several people milling about casually from one end of an open space to the other (think extensive parking garage or multiple city blocks sidewalk) the best way to draw attention to yourself is to run. It always leads me to an internal struggle between seeing the long walk ahead of myself and wanting to sprint for the sake of efficiency or just staying in line and walking with the group even though it's killing me inside cuz it's taking WAY longer than if I just ran... it takes a psychological toll on me daily.