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How Do You Find Velocity When Acceleration Is Not Constant?

  1. May 1, 2008 #1
    In planetary motion, how do you find change in velocity and change in time in relation to a change in distance? (a=GM/x^2)
    since you cannot simply use an equation like vf=vi+at, unless "a" is constant, how do you do it?
  2. jcsd
  3. May 1, 2008 #2


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    Staff Emeritus
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    It's called "calculus" and was indeed the mathematical problem that Newton needed to solve before he could formulate his mechanics.

    velocity is the integral of the acceleration.
  4. May 1, 2008 #3


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    Staff: Mentor

    Since planetary motion is two-dimensional (actually three-dimensional, but the orbit is confined to a plane if we're dealing with only one planet at a time, and neglecting perturbations from the other planets), you have to solve a pair of coupled differential equations:

    [tex]\frac{d^2 x}{dt^2} = \frac{GMx}{(x^2 + y^2)^{3/2}}[/tex]

    [tex]\frac{d^2 y}{dt^2} = \frac{GMy}{(x^2 + y^2)^{3/2}}[/tex]
  5. May 1, 2008 #4
    Alright, let me make sure I understand that, since I don't know much calculus. Are the equations relating acceleration at distance x and distance y? Also, how do you exactly go about solving the coupled differential equations? It might be a bit over my head right now, but thanks anyways for your help.
  6. May 1, 2008 #5


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    Do you know about vectors yet?

    When you solve differential equations, you get x(t) and y(t), that is, formulas (or tables) for x and y at time t. The first derivatives dx/dt and dy/dt give you the x and y components of the velocity at time t. The second derivatives [itex]d^2 x / dt^2[/itex] and [itex]d^2 y / dt^2[/itex] give you the x and y components of the acceleration.

    In practice, people usually solve differential equations like this using computer software. You give it the initial values of x and y at t = 0, and it calculates a table of x and y at later times. It calculates each point based on the results for the preceding point, going one step at a time. There are various methods (algorithms) for doing the calculation, with different combinations of simplicity, speed and accuracy: Euler's method, Runge-Kutta methods, etc. You typically learn the details in a numerical-methods course.
  7. May 1, 2008 #6
    Ok that makes a lot of sense, but it means that I am kind of back where I started hehe. I have a program updating calculations for a, v, x, and y in small increments, assuming that a is constant for a tiny amount of time, which seems more or less like Euler's method. I was wondering how exact my answer would be. Would evaluating the differentials produce a different result than if i simply used a=GM/r^2, vf=vi+at, and dx=vit+1/2at^2 to calculate the new values at every time increment?
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