How to derive the velocity's direction from a potential

[PhysicsPrincess]

Hi, I need some help with a problem in my Classical Mechanics undergrad course.
We were assigned a problem in which we were given a one dimensional potential (some sort of a parabola).

We were asked to determine the velocity's direction at different point, and even though it is somewhat "intuitive", I can't find any kind of general connection between the potential at a certain place and the velocity's direction.

I'd really appreciate your help, thanks.

 

[Dale]

There isn't such a connection. Consider a mass attached to a spring oscillating horizontally on a frictionless surface. The potential is as you describe, but the direction of the velocity goes either way at each location in the potential.

 

[mfb]

You can find the direction of acceleration (due to this potential), but the direction of velocity is not fixed.

 

[HallsofIvy]

Did you mean acceleration? Force is the negative of the gradient of the potential function and acceleration is force divided by mass. The acceleration is in the same direction as the force, the opposite of the direction of the gradient of the potential function.

 

[PhysicsPrincess]

Thanks for the answers :)
So I understand that I didn't miss any "formula" connecting the potential and the direction of the velocity... So it thing must be something specific in the potential given in the problem:

v(x) = k(x^2-a^2) where |x|<a
0 where |x|>a

(k>0)

(we were asked the velocity and its direction in different points x for different values of E_total).

What do you think?

 

[HallsofIvy]

Well, "e_total" is the sum of potential energy and kinetic energy so for a fixed value of e_total and known potential energy you can calculate the kinetic energy. However, kinetic energy depends upon the square of velocity and will not give a direction. As DaleSpam said in his first response, consider the case of a spring. At a given position, the potential energy and kinetic energy are the same whether the mass is going up or down. The information you give simply is not sufficient to determine the direction of motion.

 

[PhysicsPrincess]

Alright, so they probably have a mistake there :/

 

[Dale]

Thanks for the answers :)
So I understand that I didn't miss any "formula" connecting the potential and the direction of the velocity... So it thing must be something specific in the potential given in the problem:

v(x) = k(x^2-a^2) where |x|<a
0 where |x|>a

(k>0)

(we were asked the velocity and its direction in different points x for different values of E_total).

What do you think?

This is the potential for a simple harmonic oscillator. So it behaves just like a mass on a spring as described above.

 

[Dale]

Hold on, I misread the potential. Let me look at it again.

 

[Dale]

OK, for total energy < 0 this system is a simple harmonic oscillator. For total energy > 0 it is mostly a free particle. But either way the direction of the velocity is not determined.

 

[PhysicsPrincess]

Okay, thanks! It has to be a mistake then.

 

[PhysicsPrincess]

What if I were given the initial conditions (initial velocity and direction, etc)?
Can I determine the direction?

 

[Dale]

Yes. Sort of. If the total energy is positive then it will never change direction. Otherwise it will oscillate.