How to Find the Path Function and Meeting Time for Three Moving Particles?

[Crystal037]

Homework Statement

At the corners if an equilateral triangle of side 'l', 3 particles A, B and C are located as shown. At t=0, all the particles start moving with a uniform speed v in such a manner that always velocity of A is heading towards B, B is heading towards C and C is heading towards A. Find the time after which 3 particles will meet

Relevant Equations

V(A-B) =V(A) - V(B)

Velocity of B wrt C =
(v +v*cos 60) i^ - vsin60 j^
= (3v/2)i^-((3)^(1/2)/2v)j^
But since C is also moving this initial velocity would vary. So how to find a function which defines its path and hence I can find time at which the particles meet. I was told to take rotating frame of reference that would rotate with a magnitude and direction of vsin(theta). That would cancel the (3)^(1/2)/2*v component. But I don't understand how a rotating frame which will cause some pseudo forces to cat on the system would cancel the component perpendicular to the line joining B and C. Please explain
.

 

[haruspex]

How fast is A approaching the centre?

 

[brotherbobby]

At the corners if an equilateral triangle of side 'l', 3 particles A, B and C are located as shown.

Where is the figure? Also, please write and explain your problem clearly using the mathematics tool available here, similar to $\LaTeX$. As it stands, I understand very little of the problem you just asked, though it might well be an interesting one.

 

[Crystal037]

How fast is A approaching the centre?

That is not given. Only the speed of A towards B is given.

 

[Crystal037]

Where is the figure? Also, please write and explain your problem clearly using the mathematics tool available here, similar to $\LaTeX$. As it stands, I understand very little of the problem you just asked, though it might well be an interesting one.

The figure is just an equilateral triangle and A, B and C are situated at its corners respectively.

 

[SammyS]

That is not given. Only the speed of A towards B is given.

What is the speed of A towards B ?

 

[jbriggs444]

What is the speed of A towards B ?

That's given in the problem statement:

At t=0, all the particles start moving with a uniform speed v...

Moreover, we also know the direction (relative to the rotating frame). That gives us pretty much everything we need to solve the problem.

 

[Crystal037]

That's given in the problem statement:

Moreover, we also know the direction (relative to the rotating frame). That gives us pretty much everything we need to solve the problem.

What equations are you taking into considerations. Please elaborate.

 

[jbriggs444]

What equations are you taking into considerations. Please elaborate.

The one that @haruspex is trying to get you to think about in #2.

 

[Crystal037]

The one that @haruspex is trying to get you to think about in #2.

How will I get that information. I only know how fast its approaching the other particles.

 

[jbriggs444]

How will I get that information. I only know how fast its approaching the other particles.

What angle is it moving at? Measure that relative to a line pointed at the center of the triangle.

Edit: If you had listened to @brotherbobby in #3, you'd have a diagram ready to help.

 

[Crystal037]

What angle is it moving at? Measure that relative to a line pointed at the center of the triangle.

Edit: If you had listened to @brotherbobby in #3, you'd have a diagram ready to help.

 

[haruspex]

View attachment 259896

Mark the centre of the triangle.
Do you know how to resolve vectors into components?

 

[SammyS]

Rotate that image.

So, that's a lower case L representing the initial length of each side. That $l$ doesn't show well in the default PF sans-serif font. The UNICODE character for the script version, ℓ , does look better, but can be hard to find. (See the last letter in my signature.)

Find the center of the triangle, as @haruspex suggested, then, for any vertex, find the components of the velocity relative to the center.

In my opinion, it will also be helpful to use polar coordinates. For this, vertex A is simpler to work with. Get expressions for both the position and the velocity with respect to the center.

 

[jbriggs444]

Personally, I find that starting with some simple observations about the symmetry of the system and how it must evolve over time eliminates the need for coordinates. But if you must use them, then polar coordinates are definitely nice here.