How to find trajectory as a function of time with energy

[Cocoleia]

Homework Statement

I am given that an object of mass m has an attractive force F=-A/x^2 acting on it, where A is a constant and x>0. I need to find the potential energy. After i need to suppose initial conditions (x₀, v₀) such that total energy=0. I need to find the trajectory x(t) with v>0

Homework Equations

Potential energy = mgh

The Attempt at a Solution

So far I said that F=ma and then found the acceleration. I integrated the acceleration to find the speed, then I integrated that to find the position, which is A/m ln(x). I said that the potential energy = gAln(x). From this point I am stuck, I can't figure out how to find the trajectory as a function of time.

 

[mfb]

F=ma gives you the acceleration as function of distance. Converting this to acceleration as function of time (what you need for the integration steps) is not trivial.

Potential energy = mgh

That does not work. This is not a "gravity on a lab scale" setup.

then I integrated that to find the position, which is A/m ln(x)

Your position depends on the logarithm of the position?

 

[Cocoleia]

F=ma gives you the acceleration as function of distance. Converting this to acceleration as function of time (what you need for the integration steps) is not trivial.
That does not work. This is not a "gravity on a lab scale" setup.
Your position depends on the logarithm of the position?

Ok. Then how will I find the potential energy ?

 

[mfb]

The definition of potential energy directly gives a way to calculate it based on the force.