Three Points at Vertices of Equilateral Triangle

[vibha_ganji]

Homework Statement

Three small snails are each at a vertex of an equilateral triangle of side 60 cm. The first sets out towards the second, the second towards the third and the third towards the first, with a uniform speed of 5 cm per minute. During their motion each of them always heads towards its respective target snail. How much time has elapsed, and what distance do the snails cover, before they meet? What is the equation of their paths? If the snails are considered as point-masses, how many times does each circle their ultimate meeting point?

Relevant Equations

v= initial velocity + at
x= initial position + initial velocity *t + at^/2

I’m not sure of how to begin solving this problem. I attempted to draw a diagram and finding the components velocity of each initial velocity vector but this did not lead anywhere. Could so please have a hint?

 

[Lnewqban]

The velocity vector originating at each vertex is not perpendicular to the centroid of the triangle.
Therefore, the triangle simultaneously rotates and shrinks.
Try to visualize what an observer standing on the centroid of the triangle, while rotating with it would see.

 

[kuruman]

Also look at the component of the velocity along the line from a snail to the centroid.

 

[vibha_ganji]

The velocity vector originating at each vertex is not perpendicular to the centroid of the triangle.
Therefore, the triangle simultaneously rotates and shrinks.
Try to visualize what an observer standing on the centroid of the triangle, while rotating with it would see.

Would the observer see each of the snails moving towards the centroid since the triangle between subsequent locations of the snails gets smaller and smaller?

 

[Lnewqban]

Would the observer see each of the snails moving towards the centroid since the triangle between subsequent locations of the snails gets smaller and smaller?

Yes.
Each snail moves toward a moving target; therefore, its vector velocity changes direction (not magnitude) respect to the ground at each instant.

Respect to its moving target, each snail sees the distance that separates it from the target reduced at each instant, until both collide.

 

[vibha_ganji]

Yes.
Each snail moves toward a moving target; therefore, its vector velocity changes direction (not magnitude) respect to the ground at each instant.

Respect to its moving target, each snail sees the distance that separates it from the target reduced at each instant, until both collide.

Oh ok thank you! That makes sense. I’m still stuck on how to find the time to collide. Could I have a hint on how to do that?

 

[PeroK]

Oh ok thank you! That makes sense. I’m still stuck on how to find the time to collide. Could I have a hint on how to do that?

Try looking at how the distance between the snails reduces over some small time $\Delta t$.

Use geometry and trigonometry.

 

[kuruman]

I’m still stuck on how to find the time to collide. Could I have a hint on how to do that?

Here are two hints:
Consider any of the three snails.
1. What is its initial distance to the center of the triangle?
2. What is its component of velocity along the snail-to-center line?

 

[PeroK]

Here are two hints:
Consider any of the three snails.
1. What is its initial distance to the center of the triangle?
2. What is its component of velocity along the snail-to-center line?

This is definitely the best approach.

 

[Orodruin]

This is definitely the best approach.

I am not so certain I would call it the "best". There are several approaches that make the problem quite simple and "best" is therefore a matter of taste. An approach that I find equally simple is to consider the component of the relative velocity between any two snails along their separation direction. Due to the angles involved this becomes very simple, no irrational numbers anywhere.

Of course, the end result is the same.

 

[PeroK]

I am not so certain I would call it the "best". There are several approaches that make the problem quite simple and "best" is therefore a matter of taste. An approach that I find equally simple is to consider the component of the relative velocity between any two snails along their separation direction. Due to the angles involved this becomes very simple, no irrational numbers anywhere.

Of course, the end result is the same.

That's the way I did it, but it was tricky to get the angular velocity about the centroid. And we need the equation of their paths.

 

[Orodruin]

That's the way I did it, but it was tricky to get the angular velocity about the centroid. And we need the equation of their paths.

I don't see how that is trickier than when you get the time by taking the radial component? Anyway, we can discuss this once OP has solved the problem.

 

[Delta2]

I am thinking of what I call the full generic approach to this problem:
For each snail define their position vector $$\vec{r_i}(t)=x_i(t)\hat x+y_i(t)\hat y$$ (motion in a plane hence two components x,y for each snail)
Given the statement of the problem we can form a system of three first order ODE's with unknowns the $\vec{r_i(t)}$ or totally 6 first order ODE's with unknown the three $x_i(t)$ and the three $y_i(t)$

I think the 6 equations are first order but not linear.

@Orodruin, @PeroK ,@kuruman is it solvable that way? Yes I understand this approach is sort of too generic or blind or brute force maybe...

EDIT: On second thought I think it might be better if work in polar coordinate system instead of cartesian system.

 

[Orodruin]

I am thinking of what I call the full generic approach to this problem:
For each snail define their position vector $$\vec{r_i}(t)=x_i(t)\hat x+y_i(t)\hat y$$ (motion in a plane hence two components x,y for each snail)
Given the statement of the problem we can form a system of three first order ODE's with unknowns the $\vec{r_i(t)}$ or totally 6 first order ODE's with unknown the three $x_i(t)$ and the three $y_i(t)$

I think the 6 equations are first order but not linear.

@Orodruin, @PeroK ,@kuruman is it solvable that way? Yes I understand this approach is sort of too generic or blind or brute force maybe...

I mean, yes, you can do it that way. But regardless what I said about a "best" solution being a dubious statement, I don't think that this particular solution would objectively not qualify for that title

You would have (identifying indices modulo 3):
$$
\dot{\vec r}_i = \vec v_i = v_0 \frac{\vec r_{i+1}-\vec r_i}{\sqrt{(\vec r_{i+1}-\vec r_i)^2}}
$$
with initial conditions
$$
\vec r_i (0) = \ell R_{2\pi/3}^{i-1} \hat x
$$
where $R_{2\pi/3}$ is a rotation by $2\pi/3$. You are right that this is highly non-linear.

Now, using symmetry and some variable substitutions and rewritings, you can of course reduce this to one of the simpler cases above, but why bother?

Edit: Actually, the most important rewriting would be to change independent variable from $t$ to $s$ according to
$$
\frac{ds}{dt} = \frac{1}{\sqrt{(\vec r_{i+1}-\vec r_i)^2}}
$$
(because of the symmetry it does not matter what $i$ you pick). This turns the differential equation into the linear differential equation:
$$
\frac{d\vec r_i}{ds} = v_0 (\vec r_{i+1} - \vec r_i).
$$
Later you will have to solve the differential equation for $s$ to get $t$ back.

 

[PeroK]

I think we need to hear something from the OP.

 

[Orodruin]

I think we need to hear something from the OP.

I agree. And please note that I would not recommend the above in any way or form.

 

[Delta2]

Thanks @Orodruin, I don't think I would 've thought of changing the independent variable...

 

[Orodruin]

I agree. And please note that I would not recommend the above in any way or form.

Ok, so now having actually done it this way by explicitly solving the differential equation after applying symmetry arguments - I can confirm that it is indeed possible, gives the same answer (phew!), and that it is not something I recommend.

 

[Delta2]

Ok, so now having actually done it this way by explicitly solving the differential equation after applying symmetry arguments - I can confirm that it is indeed possible, gives the same answer (phew!), and that it is not something I recommend.

You mean the differential equation regarding the change of variable , that is between s and t,is hard?

 

[Orodruin]

You mean the differential equation regarding the change of variable , that is between s and t,is hard?

No, the entire system of differential equations. Of course that also involves finding the relation between s and t, which probably was the easiest part once the rest of the solution is known.

 

[kuruman]

This is definitely the best approach.

I think so. It can easily be generalized to snails on a regular $N$-gon and one can show that in the limit $N\rightarrow \infty$, the snails will go around in a circle. I first saw this question with mice instead of snails. The solution converged faster $\dots$

 

[Orodruin]

I think so. It can easily be generalized to snails on a regular $N$-gon and one can show that in the limit $N\rightarrow \infty$, the snails will go around in a circle. I first saw this question with mice instead of snails. The solution converged faster $\dots$

I’d say ass snails except one are red herrings to be honest. The only thing they do is to ensure the same angle between the radial and the velocity. The more general generalisation would be any constant angle spiral.

I still think that the relative velocity approach is less cumbersome with the information given in this case (i.e., side length rather than radius - the relative velocity solution then involves a single trigonometric consideration whereas the radial one requires two - even if they involve the same cosine).