The two-body problem with pushing force, what ll be their speed in t=infinity?

[malrf]

Hi!

I want to solve the following: I have 2 particles p1, p2, with given m1,m2,v1 , p2 stands still in the origin(x,y,z=0).
Between them there is only 1 force F=1/(d^2), the force is pushing the particles away from each other.
I start p1 from infinite distance with given speed, and i know it's equation of motion without the pushing force, and i want to know what ll be it's speed after infinite time. (p1 is closing to p2 and p1's line of motion without the force is not crossing p2)

After searching google i found that there might be a closed form expression solving the 2-body problem, but those expression only true if the course of the particles is an ellipse, so i don't know how to solve my problem.

Thanks for helping!

 

[eljose79]

Hi there!

Thank you for reaching out about your problem. It sounds like you are dealing with a classic 2-body problem in physics. This type of problem involves two particles interacting with each other through a force, in this case the force is inversely proportional to the distance between the particles squared.

First, it is important to note that the motion of the particles in this system will not form a perfect ellipse, as you mentioned, so the closed form expressions for the 2-body problem will not be applicable here. However, there are still ways to solve for the motion of the particles and determine their speeds after an infinite amount of time.

One approach is to use numerical methods, such as computer simulations, to track the motion of the particles over time. This can give you a good approximation of their speeds after infinite time.

Another approach is to use energy conservation principles. In this case, the total energy of the system (kinetic energy + potential energy) will remain constant as the particles move. This means that you can set the initial energy of the system (which you can calculate from the given masses and speeds) equal to the energy at any point in time, and solve for the speed of the particle at that time. Repeating this process for infinite time will give you an approximate value for the speed after infinite time.

I hope this helps! Let me know if you have any further questions. Good luck with your problem!