Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Deriving Acceleration from Potential Energy?

  1. Aug 3, 2004 #1


    User Avatar

    I have a known potential energy, V, expression:

    V(x,y,z) = α·x + β·y2 + γ·z3

    I'm given: @(0,0,0), v = v0 and then asked to find v at (1,1,1).

    I can determine v from Conservation of Energy:

    v2 = v02 - (2/m)·(α + β + γ)2

    In general, what is the expression for the accelerations ax, ay, az?

    Do I find F from -∇V?

    If so, what's next (as far as finding the acceleration's x, y and z-components)?

  2. jcsd
  3. Aug 3, 2004 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    1.Why have you squared the potential energy term??

    2. Yes, and divide F by m to find the accelerations.
  4. Aug 3, 2004 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor

    arildno sorta told you how to start it off. You should have F (and a) from the gradient of V. However, you will notice that "a" has a dependence on x, y, and z. If a is a function of t, then it is trivial to find v. But you don't have that here.

    So what you need to do to find v is to use some calculus gymnastics by invoking the chain rule, i.e.

    a = dv/dt = (dv_x/dx * dx/dt)i^ + (dv_y/dy * dy/dt)j^ + (dv_z/dz * dz/dt)k^

    It is easier to solve this component by component, so for the x-component, you have

    a_x = dv_x/dx * v_x (since dx/dt = v_x)

    Thus, a_x dx = v_x dv_x

    I think you should be able to handle the baby integral here using the initial conditions given. Do the same thing for the other 2 components.

  5. Aug 3, 2004 #4


    User Avatar

    Thanks a lot Zz.

    When solving, for example, az, I arrive at:

    az = -(3γ/m)·z2

    In general, is this a sufficient expression for az,
    or should it be reduced or otherwise expressed differently?

  6. Aug 3, 2004 #5


    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor

    ASSUMING you did the gradient correctly, that should be a sufficient expression for the a_z to play with.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Deriving Acceleration from Potential Energy?