What is the Trajectory of Four Rockets in a Square Formation?

[onsem]

Hi,

first I want to apologize for my english. I'm not from an English speaking country and I have to practise.

I have a quite interesting physical assignment for you. I was trying but I still can't solve it.
If you would have an idea or if you would know how to solve it please let me know. I will be very grateful.

So, how the assignment sounds:

We have four rockets in each corner of the square one. Each rocket aims on the rocket in front of it. Length of square side is a. All the rockets starts at the same moment. They have constant speed (no acceleration). I want to work out their trajectories and how long they will fly until they explode?

Picture (before the start):
picture is on this site:
http://forum.matweb.cz/viewtopic.php?id=7793

(00 - rocket pointed on the other one)

Picture (general trajectories:
picture is on this site:
http://forum.matweb.cz/viewtopic.php?id=7793

r - a,b,c,d - rockets
a,b,c,d - trajectories
P.S.: I apologize for my painters and language abbilities

 

[CompuChip]

If they are not accelerating, why then are they not moving in straight lines but in spirals toward the center?

Do you mean that the rockets keep turning so their instantaneous speed is always pointing to the position of the next rocket, while its magnitude stays the same?

 

[onsem]

If they are not accelerating, why then are they not moving in straight lines but in spirals toward the center?

Do you mean that the rockets keep turning so their instantaneous speed is always pointing to the position of the next rocket, while its magnitude stays the same?

Each rocket is heading on the rocket in front of it (in every moment) and each one has the same speed v as the other three rockets (they also started at the same moment so they have to have the same trajectory turning into the centre of the square).

I hope that I have ansanswered on your question.

 

[tiny-tim]

Welcome to PF!

Hi onsem! Welcome to PF!

Call the positions r₁ r₂ r₃ and r₄ …

what is the differential equation which says that each one has constant speed v towards the next one?

and what is the equation which says that each one (because of symmetry) is 90º round from the next one?

 

[rcgldr]

The speed of the rockets doesn't matter, just the direction. Relative to a radial from the center of the paths, the direction is always 45 degrees inwards (or outwards if going backwards), of perpendicular to that radial, which creates a spiral. Somehow you're supposed to get the insight that the (outwards) direction of curve, ${tan}((r\ d\theta)\ /\ dr) \ =\ \pi/4$ (or any constant angle for this type of spiral), and continue from there.