The problem of cockroach races

[ValeryOchkov]

The cockroach needs to run from start (from different positions of the start line) to finish-line in minimum time.
On the field of running there is a section in the form of a circle (see Fig.), where the running speed
differs by V1> V2 or V1 <V2.
Which route (trace) from the start line to the finish line should the cockroach use from different positions?

Better to get numerical, symbolical and plot solution (with animation!).

[ValeryOchkov]

More correct Mathcad 15 sheet for v2>v1.

My egg head cockroach is waiting the solution for v2<v1

[LucMeekes]

So, I rebuilt the system, cleaned the bread crums away from the circle, and got the silverfish to also explore the dark places:

And I also reversed the speeds, but that really confuses the silverfish:

So there's more work to be done.

Luc

[ValeryOchkov]

Fine!

The best name for your fish is Snell. You see it, sorry, her thru air, glass ans water!

See please thru your displey https://en.wikipedia.org/wiki/Refractive_index

PS I used cockroaches to catch fish in my childhood!

[ValeryOchkov]

About your second animation. It was a drunk cockroach running. It is described by Mikhail Bulgakov in the novel "The Run".

Scream in the carousel: "Janissar fails!" Gul.  (Janissar  is a cockroach-favorite)
CHARNOTA (at the cash desk): How bad? It can not be !!
                    Voice in the carousel: "Do not cry, child!"
                    Another voice: "Come on, come on, come on!" To kill Arthurka is not enough!
          The little face protrudes uneasily from the cash register. Police officers they are anxious, they look into the carousel.
Figure (running out of the carousel): Scam! Janissar is drunk with a beer!
          Arthur breaks out of the carousel. Both the coats of his coat torn off, the cylinder is turned into a flat cake, there is no collar.

           The face is covered in blood. Behind him chasing a crowd of players.
Arthur (screaming desperately): Marya Konstantinovna, call the police!

Крик в карусели: "Янычар сбоит!" Гул.
Чарнота (у кассы). Как сбоит? Быть этого не может!!
                    Голос в карусели: "Не плачь, дитя!"
                    Другой голос: "Давай, давай, давай!" Убить Артурку мало!
Личико  беспокойно  высовывается  из  кассы. Полицейские проявляют беспокойство, заглядывают в карусель.
Фигура (выбежав из карусели). Жульничество! Артурка пивом опоил Янычара!
          Артур  вырывается  из  карусели.  Обе фалды фрака у него оторваны,  цилиндр  превращен в лепешку, воротничка нет. Лицо в крови. За ним гонится толпа игроков.
Артур (кричит отчаянно). Марья Константиновна, зовите полицию!

[MJG]

Valery,

To include the possibility of v2<v1, I think you just need to add one more case (Time3):

• Travel from (x, r+s) to the tangent point on the same half of the circle (left tangent point if x<0; right tangent point if x>0; runner's choice if x=0).
• Follow the curve to (-r, 0) or (r, 0), as appropriate.
• Travel straight down to the finish line at either (-r, -r-s) or (r, -r-s)

Note that Time3 is only applicable for -r<x<r.

The minimum race time would then be min(Time1,Time2,Time3) regardless of if v1<v2 or v1>v2.

I'll leave the math up to you.

[ValeryOchkov]

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@MJG wrote:

Valery,

I'll leave the math up to you.

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Thank you very much! For my studens!

[MJG]

OK, curiosity got the bettor of me.  It was a little more complicated than I originally stated.  Note that I had to add a couple constraints to your minimize function.

Sorry for lack of notation on my new calculations.  I don't even know what your Russian notation even says.

Animation (thanks, Luc):

Mathcad attached.

[LucMeekes]

You convert the (.avi) animation to an animated .GIF picture, then insert that .gif as a picture.

Luc

[ValeryOchkov]

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@LucMeekes wrote:

You convert the (.avi) animation to an animated .GIF picture, then insert that .gif as a picture.

Luc

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[MJG]

Valery,

Your last image includes a "?"  I'm not exactly sure what your question is.

The curved line in your image is the path corresponding to my Time3 calculation.  My method of graphing this was to approximate by using 10 line segments.  With some extra effort, you could get mathcad to draw the actual curve.

Hope that helps.  If not, let me know what the question is.

[LucMeekes]

I guess Valery is wondering if, considering the small ratio in velocities v1 and v2, it would not be faster to run the straight line with the somewhat lower speed, than to run the rurved line, at the slightly higher speed.

Note that the straight line between the two points on the curve is always shorter than the curved line.

There must be some (angular) distance between the two points where the run time along the straight line gets lower than the run time along the curve. And yes there is such an angle. But i guess that angle is such that at that moment it is faster to run across the circle, past the horizontal mid-line.

You can see that ts<tc when phi > about 0.4. That's quite a large angle!

Luc

.

[ValeryOchkov]

My cockroach Pythagoras also can to read my thoughts. As Luc. I'm just going to, I just think to go to the kitchen to mingle with him (with Pythagoras not Luc) as he quickly trotted to his hole in the shortest way!

[LucMeekes]

I wasn't reading your thoughts, I was reading your picture...

Luc

[ValeryOchkov]

You can free download the chapter about cockroach races

from the site of the book

2^5 Problems for STEM Education 

See please Study-27-Divertsment-for-Website.pdf in the Support Material.