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Clarification on finding acceleration

  1. Sep 25, 2014 #1
    1. The problem statement, all variables and given/known data
    If a block slides w/o friction down a fixed, inclined plane with an angle theta of 30 degrees, what is the block's acceleration?

    (This problem is an example from my textbook, meaning the solution is available. I am confused on the method that they used. Any clarification is greatly appreciated.)

    2. Relevant equations
    Textbook clipping.png
    3. The attempt at a solution
    I understand the process on how to determine the acceleration, I also understand the that to find velocity you integrate acceleration, I don't understand the purpose of the multiplication by 2(overdot)x. It seems as though its used to manipulate the constant acceleration.
     
  2. jcsd
  3. Sep 25, 2014 #2

    BiGyElLoWhAt

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    It let's you set up a derivative product rule on the left hand side. It's a similar method to an integration factor when solving linear first order differential equations. This is the first time I've seen this done, though. Seems legit.
     
  4. Sep 26, 2014 #3

    vela

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    You're starting with ##\ddot{x} = \frac{dv}{dt} = g\sin\theta##. When you integrate, you'd get
    $$\int_{v_0}^v dv = \int_{t_0}^t g\sin\theta\,dt = (g\sin\theta)(t-t_0),$$ which is fine if you're trying to calculate how fast the block is moving after a certain time. The problem, however, is asking you to find the speed after the block has slid a distance ##x_0##. By introducing the factor of ##\dot{x}## on the righthand side, you're changing variables from ##t## to ##x##.
     
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