Particle subject to position dependent force

[jimz]

Homework Statement

A particle with total energy E and mass m is subject to a force $$F(x)=\xi x^4$$. Find the velocity v of the particle as a function of the position x, and sketch a phase diagram for the motion.

Homework Equations

$$T=\frac{1}{2}m\dot{x}^2$$

$$U=constant$$

$$F=m\ddot{x}$$

The Attempt at a Solution

$$x=\sqrt[4]{\xi m \ddot{x}}$$

Not sure where to go from here, or what the phase diagram axes should be. Do I just take the time derivative of x and that's my velocity?

 

[ehild]

Use the work-energy theorem.

 

[jimz]

Use the work-energy theorem.

"The net work done by all the forces acting on a body equals the change in its kinetic energy."
Not seeing it...

 

[ehild]

Assume one-dimensional motion along x. You need the velocity of the particle as function of the position: v(x). At t=0 let x=0 and the kinetic energy=E. During some time period t, the displacement of the particle is x(t) and the change of KE is:

$$\Delta E = 1/2 mv(x)^2-E$$

The particle is subjected to a force of form

$$F(x) = \xi x^4$$.

The work done by this force while the particle moves from position x=0 to some x(t) is

$$W=\int_0^{x(t)}{F(x)dx}=\int_0^{x(t)}{\xi x^4dx}$$

Calculate the integral, make it equal to the change of KE, express v(x), sketch v(x) as function of x.