Find the path traced out by each ant

[mahesh_2961]

Hi guys
Please help me work out this problem
The problem is like this
There is a square table. On each vertex there is an ant sitting. Now these ants (four of them) starts moving in such a way that at any instant of time the direction of motion of each ant is directed towards an ant which started from a vertex adjacent to it (say the vertex to its right side).
Now the question is:
to find the path traced out by each ant (all of them will trace similar paths).
I tried drawing the paths of each ant to solve this, from the drawings i can only understand that at any time the velocity vector of two adjacent ants are perpendicular to each other and the four of them will eventually meet at the center of the square.

regards
Mahesh

 

[dextercioby]

Hi guys
Please help me work out this problem
The problem is like this
There is a square table. On each vertex there is an ant sitting. Now these ants (four of them) starts moving in such a way that at any instant of time the direction of motion of each ant is directed towards an ant which started from a vertex adjacent to it (say the vertex to its right side).
Now the question is:
to find the path traced out by each ant (all of them will trace similar paths).
I tried drawing the paths of each ant to solve this, from the drawings i can only understand that at any time the velocity vector of two adjacent ants are perpendicular to each other and the four of them will eventually meet at the center of the square.
regards
Mahesh

Why wouldn't they simply move on a circle,the same circle,the circle circumscribed to the initial square??If they're on a circle,every ant's direction of motion (arch of a circle) is definitely towards the ant which is in front of it.While the velocity vectors remain tangent to the circle (trajectory),just as they should.If they move at the same velocity (tangent),then the distance (arch of circle) will remain distant and they could go round like that till they age and die. :tongue2:

Daniel.

 

[mahesh_2961]

But in that case the condition that " at any instant of time the direction of motion of each ant is directed towards an ant which started from a vertex adjacent to it " will not be satisfied
The path i got by drawing it is attached .. I am not very good at sketching

These paths satisfy the above condition but the thing is i am not able to find this mathematically...

regards
Mahesh

 

[dextercioby]

But in that case the condition that " at any instant of time the direction of motion of each ant is directed towards an ant which started from a vertex adjacent to it " will not be satisfied.

Are u kidding me,or are u running low with your imagination...?Read very carefully:"at any instant of time the direction of motion of each ant is directed towards an ant which started from a vertex adjacent to it "
If they move on a circle,then at any instant of time,the direction of movement of each ant will be directed towards the ant which is in front and which started to move from the adjacent vertex.On your picture the directions of movement for each ant are perpendicular for two neighboring ants,not the same.In my case,of the circle,the 2 neighboring ants walk the same path.It's actually true for all of them.

Daniel.

PS.I hope u realize.Else,i won't argue with you...

 

[BobG]

They will meet in the middle.

Geometrically, it's easy to show. All the ants are the same distance from the center. They lie on the circumference of a circle. Their path isn't along the circumference, though, so their path is not perpendicular to the radius. Their instantaneous path is along the secant of the circle, since they are headed directly towards the next ant. There's no way back to their original position. Their radius can only get smaller and smaller.

Setting up the math will take a little thought. The radius is constantly changing, getting smaller. The path of each ant should be a function of the radius and the angle necessary to head towards the adjacent ant.

Edit: Thinking about it, the angle between the radius and the secant line stays constant. It's just a square with a shrinking radius. I take it the ants' linear speed stays constant. You want to solve this as a related rate between the radius and the angular velocity.

 

[mahesh_2961]

hai,
before i can start doing it, would like to know whether my drawing was correct or not ? I am very confused right now .. please help

Thanks in advance
Mahesh

 

[BobG]

Yes, it captures the general idea.

All four follow the same path, just separated 90 degrees.

 

[Gokul43201]

The path is a logarithmic spiral that ends at the center, just as you've drawn it.

The equation I get for the spiral is :
$$r = \frac{a}{2}~e^{\theta /2}~, ~~-\infty < \theta \leq 0$$

I got this by integrating to find the arc length of the spiral starting at a/2, and equating this length to a, where a is the side of the square table.

The reason the arc length, or the total distance traveled by each ant, is a, is easy to see using relative co-ordinates. Imagine a frame that is rotating with the same angular velocity as the rotation of the square formed by the ants. I am one of the ants. The ant in front of me has no component of motion along the line joining us, so I merely have to travel a distance equal to our initial separation to meet up with it. So the total distance traveled is simply equal to the initial separation, a.

PS : This is not the rigorous way to derive the path equation. I used a short cut because I knew that the path is a logarithmic spiral.