Convergence Time of Moving Points on an Equilateral Triangle

[Saitama]

Homework Statement

Three points are located at the vertices of an equilateral triangle whose sides equals a. They all start moving simultaneously with velocity v constant in modulus, with the first point heading continually for the second, the second for the third, and the third for the first. How soon will the points converge?

Homework Equations

The Attempt at a Solution

I suppose this one is an easy problem, but i am still not sure if i did it the correct way. I found the original distance of a point from the centroid (where the points converge) which is a/√3 and the component of velocity along the line joining the centroid and a point is vcos(30°), therefore time taken by the points to converge is vcos(30°)/(a/√3)=2a/3v. This matches the answer given in the answer key but i don't understand why this method works here? What if it was not an equilateral triangle? Any explanation on this would help.

Thanks!

 

[I like Serena]

Hi Pranav!

Suppose all 3 points have moved a little.
What does the resulting figure look like?
And what is the radial speed then?

 

[Saitama]

Hello ILS!

Suppose all 3 points have moved a little.
What does the resulting figure look like?

It will be still an equilateral triangle.

And what is the radial speed then?

Um..i have no idea on this. :uhh:

 

[I like Serena]

Hello ILS!

It will be still an equilateral triangle.

Yep!

Um..i have no idea on this.

No? So how did you get vcos(30°)?

 

[Saitama]

No? So how did you get vcos(30°)?

I was confused when you said "radial speed". Sorry about that!

 

[I like Serena]

I did not understand what you meant by radial speed. Sorry about that!

Ah yes, radial speed is the amount that the distance to the centroid gets shorter per unit of time.

So... does that answer your question?

 

[Saitama]

Ah yes, radial speed is the amount that the distance to the centroid gets shorter per unit of time.

So... does that answer your question?

Thanks, but what about my second question? What would happen if the sides were unequal?

 

[I like Serena]

Thanks, but what about my second question? What would happen if the sides were unequal?

Oh yes. Completely missed that. Sorry.

Well, then it becomes a bit more complex doesn't it?
Then you can't use the symmetry anymore.

So what I'd do is draw a non-equilateral triangle - any triangle will do.
Say, a triangle with a base of 2 cm and 10 cm high?
Then take a step of a couple of millimeters in the proper directions and draw the new triangle.
Then repeat a couple of times.
That should give you a clue about what's happening...

Alternatively, you can set up equations and try and solve those.
But that is way more work, and you don't like a lot of work do you?

So I'd first try to draw a couple of steps and see if you can find a pattern.
At worst you'll have a pretty figure!

 

[Saitama]

Well, then it becomes a bit more complex doesn't it?
Then you can't use the symmetry anymore.

I was expecting that.

So what I'd do is draw a non-equilateral triangle - any triangle will do.
Then take a step of a couple of millimeters in the proper directions and draw the new triangle.
Then repeat a couple of times.
That should give you a clue about what's happening...

Alternatively, you can set up equations and try and solve those.

Thanks for the tips, i would be sure to use your hints if i encounter any question like this. For now, i am off to bed, getting late here.

But that is way more work, and you don't like a lot of work do you?

Yes, i hate it. :tongue2:

 

[I like Serena]

Yes, i hate it.

Ah, but you do like pretty pictures, don't you?

 

[Saitama]

Ah, but you do like pretty pictures, don't you?

Yes.

 

[I like Serena]

Yes.

Oh well, sleep tight and dream of rotating triangles that becomes smaller and smaller. :zzz: