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Help! I Can't Solve This!

Here is a problem I found in a book. Even after many attempts, I couldn't solve it. None of my friends could solve it! Here it is, directly taken from the book.

Six particles situated at the corners of a regular hexagon of side 'a' move at a constant speed 'v'. Each particle maintains a direction towards the particle at the next corner. Calculate the time the particles will take to meet each other.

It must be based on the concept of Relative Velocity (or Change of Frame) somehow. Can anyone help me? Thanks in advance...

Note by Ameya Salankar
3 years, 4 months ago

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I think that's from H.C. Verma, right?

Dinesh Chavan - 3 years, 4 months ago

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Yeah! Do you know how it's done?

Ameya Salankar - 3 years, 4 months ago

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It's simple, just find the component of velocity of any particle towards the center, and it won't change throughout the motion (imagine this, it's the "emotions" of physics .... Dinesh will understand the statement :P)

So, the component of the velocity towards center, you can find it easily, and then the distance between center and any particle, then just get the time= \(\dfrac{distance}{speed}\). not tough at all.....

You may want to see the solution I wrote to your problem (square one)

Note that this will happen if and only if the polygon is regular polygon and the speed of each of the particles is the same.

@Ameya Salankar , @Dinesh Chavan

Aditya Raut - 3 years, 4 months ago

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@Aditya Raut @Aditya Raut, yes, quite an easy one! Can you please elaborate your emotions of physics?

Ameya Salankar - 3 years, 3 months ago

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@Ameya Salankar emotions of physics is getting something by feeling, intuition in physics

Aditya Raut - 3 years, 3 months ago

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@Ameya Salankar Its a small Issue in our tution classes

Dinesh Chavan - 3 years, 3 months ago

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I can't help but notice the similarity with Ants on a tetrahedron

Thaddeus Abiy - 3 years, 4 months ago

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Hey friends i just tried to deduce a formula for such type of problems \(t=\dfrac{a}{2v\times \sin(\dfrac{\pi}{n})\times \sin(\dfrac{\pi}{n})}\) where \(a\) is side length and \(n\) denotes the total sides of the regular polygon

Mehul Chaturvedi - 2 years, 5 months ago

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Can you please show how did you do it ?

Shah Yug - 11 months, 3 weeks ago

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Nailed it ! NYC DONE BRO >

Atul Saswat - 1 year, 7 months ago

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The time it takes is exactly 2 times the time it takes for any particle to travel the length of one side of the hexagon. If there were 4 particles in a square, the time is the same as travelling a side. Given any regular polygon, since at any time after pursuit begins, the particles are at the corners of the same regular polygon but a shrinking one, the time it takes is the same as finding the length of logarithmic spirals, which have the property of self-similarity.

Most pursuit problems are harder to analyze.

Michael Mendrin - 3 years, 4 months ago

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But how do we prove that they will eventually collide with each other? (it seems to defy reasoning)

Ameya Salankar - 3 years, 4 months ago

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Okay, if you want to prove they will, first draw a hexagon slightly smaller the first, but inside it, such that after each particle has traveled a very short distance, they're all still headed towards each other. By property of self-similarity, this can be repeated until the innermost hexagon has shrunk to a point. The paths traced out by the particles is a logarithmic spiral, which has a finite arc length from any point on it down to the center.

A shrinking hexagon indicates that the particles are getting closer to each other, because the sides are shrinking.

As a counterexample, if each particle was constrained to move only along a circular path common to all of them, then they'll never get any closer.

Michael Mendrin - 3 years, 4 months ago

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@Michael Mendrin @Michael Mendrin, Thanks for explaining! I got it now.

Ameya Salankar - 3 years, 4 months ago

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ANSWER:
2a/v

Plz admire this image for understanding my answer... and plz consider it as a regular hexagon

img

img

**now, Initial separation between two particles = Side of hexagon = a

Final separation = 0

Therefore, Relative displacement between two particles = a

Particle B has a component v cos60 along particle BC (a side)

Therefore, relative velocity with which B and C approaches each other = v - vcos60 = v/2

Since, v is constant, thus time taken by these two balls to meet each other is given by

=(Relative displacement) / Relative velocity = a/(v/2) = 2a/v

So, the time taken by the particles to meet each other = 2a/v

In reality each particle will follow a curved path and eventually meet at the center of the hexagon.**

Ess Kp - 3 years, 2 months ago

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@ess kp , I have edited your comment so now the image is appearing instead of the link. Please see the changes I've made so that next time you can make your image appear in comments, notes and problems.

Aditya Raut - 3 years, 2 months ago

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