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

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

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

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

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

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

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

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

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

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

experimentX Group TitleBest ResponseYou've already chosen the best response.0
A>B >C>A
 2 years ago

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

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

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

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

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

sauravshakya Group TitleBest ResponseYou've already chosen the best response.1
Yep.. @across
 2 years ago

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

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

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

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

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

Mikael Group TitleBest ResponseYou've already chosen the best response.2
\[\theta(x) = \cos(x)\]
 2 years ago

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

experimentX Group TitleBest 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.
 2 years ago

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

Mikael Group TitleBest ResponseYou've already chosen the best response.2
Here 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
 2 years ago

Mikael Group TitleBest ResponseYou've already chosen the best response.2
http://www.wolframalpha.com/input/?i=Archimedes+spiral
 2 years ago

Mikael Group TitleBest ResponseYou've already chosen the best response.2
Sorry r Linear in theta
 2 years ago

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

psi9epsilon Group TitleBest 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
 2 years ago

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

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

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

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

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

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

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

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

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

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

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

experimentX Group TitleBest 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.
 2 years ago

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

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

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

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

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

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

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

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

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

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

experimentX Group TitleBest ResponseYou've already chosen the best response.0
another 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
 2 years ago

Mikael Group TitleBest 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
 2 years ago

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

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

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

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

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

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

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

experimentX Group TitleBest ResponseYou've already chosen the best response.0
i 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
 2 years ago

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

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

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

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

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

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

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