Hi, I was hoping for some help with the following. This is the first question of this type I have attempted so, any help either with answering the parts of the question I could not do or if you notice any mistakes with the parts I have answered would be appreciated.

A proton of mass m moves in one dimension. Its potential energy function is: U(x)=a/x^2 - b/x where a and b are positive constants. The proton is released from rest at x0=a/b.

I also showed in earlier questions that that U(x) can be written in the following way:

U(x)=(a/(x0)^2)[(x0/x)^2-(x0/x)] and that v(x) the speed of the proton as a function of position can be written as:

V(x)=sqrt((2a/m(x0)^2)[(x0/x)-(x0/x)^2]) with x>x0

a. Give a short qualitative description of the motion in terms of the classification of Kepler orbits.

I don't know what a kepler orbit is and my textbook (or the Internet) was not much help so, I could not do this part. What is a Kepler orbit? How are they classified?

b. Show that at x=2x0 the speed of the proton is maximum and comute that speed.

My attempt:

Differentiating the first expression of U(x) with respect to x and setting the derivative equal to zero gives that:

2a/x=b therefore x=2a/b =2x0 as required.

Subbing this into v(x) gives v=sqrt(a/2m(x0)^2)

c. What is the force on the proton at the point x=2x0?

My answer: F=0.

d. Instead let the proton be released (from rest) at x1=3a/b. Derive an expression of v(x) for the new release point in terms of a,b and m, also give a qualitative description of the orbital motion.

My attempt: Initially U(x)=U(3a/b)= -b^2/9a = Total energy

So at any time: kinetic energy + U(x) = -b^2/9a

Hence 0.5mv(x)^2 +a/x^2 – b/x = -b^2/9a

This is as far as I have got for this part of the question; I do not know how to get rid of the x’s. Any ideas? I also don’t know what it means by description of the orbital motion.

Was it ok for me to use the same potential energy function for a different initial condition?

e. for each release point, (x0 and x1) find the maximum and minimum x values reached during the motion.

I did not even know how to start this part of the question, any guidance would be welcome.