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experimentX
 2 years ago
Three snails at the point of equilateral triangle start to move at each other. What is the shape of curve?
experimentX
 2 years ago
Three snails at the point of equilateral triangle start to move at each other. What is the shape of curve?

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psi9epsilon
 2 years ago
Best ResponseYou've already chosen the best response.0assuming they start from vertices, to meet at centre then there are different possibilites they can meet at incentres also or at centroid too

psi9epsilon
 2 years ago
Best ResponseYou've already chosen the best response.0in either case it would be along a line or a ray

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0it can't be a ray ... they aren't moving to center. they are moving to each other ( the other snail) dw:1347897411584:dw

FweaBoyLive
 2 years ago
Best ResponseYou've already chosen the best response.0they could meet by the vertices and be moving clockwise to each other

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0nop ... they all in the direction of another snail.

Agent47
 2 years ago
Best ResponseYou've already chosen the best response.0is it going to be some polar graph?

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0don't know ..haven't solved it yet. i tried to do this couple of months ago ... couldn't do it.

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1How can one move in two direction at once

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0they are moving in cycle.

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2Great riddle let us think, thanks

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2Well I think I know the beginning:

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1I guess there will always be an equilateral triangle between them

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1Something like this:dw:1347902736343:dw

across
 2 years ago
Best ResponseYou've already chosen the best response.1The equilateral triangle shrinks and rotates at a speed proportional to the snails' pace.

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2dw:1347902612412:dw One writes the differential equation Remembering v*dt = dx

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2d theta = sin theta * dx

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2\[\frac{ d \theta }{ dx } = \sin \theta\]

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2This is integrable easily

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0sorry ... had been away ... carry on

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2This is 1st try one needs to improve the geometry

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1347903100408:dw

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1347903134187:dw

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1I think there is a possibility of that also

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0might be ... let's assume unit velocity for now ... we will generalize it later.

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2by the way my soln is a piece of spiral

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1347903217601:dw yep .. it should be spiral ...

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1ya did it that way @experimentX

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1@Mikael do u know the equation of a spiral........ Its beyond my knowledge

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0this is diverging spiral.

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0also note that I don't the answer of this Q.

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1This may be funny but this will depend upon the length of the snail

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1Because when they meet there has to be an equilateral triangle with length of the side equal to the length of the snail

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1347903644328:dw kinda seems this way.

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0@sauravshakya you have a knack for making problems difficult ... well i agree that this gives insight into things ... but for this case. try to take the size of snail that would simplify the problem. take it as a point size snail.

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2Well I think I have got it  by geberal consideration

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2Here 1) After constant time Delat t ( constant) the pictire undergoes similarity transformation by A) rotating, B) Shrinking by CONSTANT factor Z(DElta t) The only spiral that is self similar is Archimedian Spiral where r is a LInear function of t

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0well ... i didn't know that. let's say we know the answer. any method arising from calculus?

psi9epsilon
 2 years ago
Best ResponseYou've already chosen the best response.0@experimentX , the question can be solved in numerous ways, and is open to interpretation you can also use graph theory to find the shortest path which infact will define the curve other solutions like rotation of triangle are justified if sufficient and necessary conditions are provided

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0the problem is supposed to be calculus problem.

psi9epsilon
 2 years ago
Best ResponseYou've already chosen the best response.0well that should have been indicated well in advance before requesting help, makes life easy than scratching heads : )

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2Very highbrow @psi9epsilon yet I have offered a definite solution (right or wrong) but precise. Either disprove it or show my argument deficient.

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0well ... sorry my bad :/

psi9epsilon
 2 years ago
Best ResponseYou've already chosen the best response.0@Mikael, do not approve or disapprove your solution as it MAY NOT be the ONLY solution

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2I CLAIM that the only spiral that has in 0.002 timeunits A) constant rotation B) Constant zoomdown factor Z Is the above

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0it's a spiral ... at least we know that.

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.20.002 is simply to underscore the constancy of\[\Delta t\]

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1Can any one show me how the equation of a spiral looks

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2R(t) = exp(Kt) theta(t) = kt

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0let's solve this Q for particular case. assume side is 5 and speed is 1

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0@sauravshakya there are numerous way in which you can have a spiral ... try to look for polar equation or parametric equation ... this is same a circle except the radius changes.

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2\[R(\theta) = R_0e^{k*\theta}\]

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1Have no clue on polar equation or parametric equation

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0well .. that's one hell of one spiral ..

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2http://www.wolframalpha.com/input/?i=polar+plot+r%28theta%29+%3D+1*exp%280.3*theta%29

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1Well guys I have to go now.

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2AAND I HAVE PROOF ! Here it is\[R(t + \Delta T) = R(t) * e^{k \Delta T}  constant factor\]

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0http://www.wolframalpha.com/input/?i=plot+r%28theta%29+%3D+theta

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2\[\theta(t + \Delta T) = \theta(t) + k*\Delta T > constantrotation\]

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2So @experimentX @across @siddhantsharan and whoever want  check this solution !

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2Please CHECK this solution @estudier @mukushla @hartnn

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0another spiral http://www.wolframalpha.com/input/?i=parametric+plot+x%28t%29+%3D+t+cos%28t%29%2C+y%28t%29+%3D+t+sin%28t%29+from+t%3D0+to+t%3D2+pi

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2@experimentX Please accept the proof  it is rock solid. Do not evade  this is THE only Only solution

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2@mathslover please check my solution of exponential spiral

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0hold on ... I'm hunting for spirals

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2By the way THIS IS A GENERAL SOLUTION FORM FOR ANY EQUILATERAL POLYGON WITH NSNAILS IN EACH VERTEX !!!!

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2Hunting is good, but hunting for WRONG is just funny

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2THIS IS A GENERAL SOLUTION FORM FOR ANY EQUILATERAL POLYGON WITH NSNAILS IN EACH VERTEX !!!!

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0no ... just for fun. man fun is good.

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2If I am right  PLEASE say so. Check  this is justified because you asked the. And I solved it (or not  but check) question

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0i thought this would work out .. lol http://www.wolframalpha.com/input/?i=parametric+plot+x%28t%29+%3D+e^t+cos%28t%29%2C+y%28t%29+%3D+e^t+sin%28t%29+from+t%3D0+to+t%3D2+pi

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0all right ... i'll check.

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2\[R( \Theta) = R_0 e^{k \Theta}\]

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2This GIVES THE SHAPE OF THE PATHS FOR ANY REGULAR POLYGON

Mikael
 2 years ago
Best ResponseYou've already chosen the best response.2Square, Pentagon, Hexagon, Triangle whatever your n is

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1347905702311:dw yeah this is definitely a r = k e^(theta)

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1347906117851:dw

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0let's try to solve this problem using calculus method some other day.
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