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Differentiating acceleration using constant of motion

  1. Oct 25, 2012 #1
    Let's say we have an object falling through an accelerated field from r to s. I would say a gravitational field, as it is by the same process as gravity, but I will be applying a different constant of motion within the field.

    If the constant of motion is something like (1 - b / r) = (1 - (v_r/c)^2), or (1 - b / r) / (1 - (v_r/c)^2) = K (constant for all r), where b is also constant, then the acceleration in terms of r can be found with

    b / r = (v_r/c)^2, b / s = (v_s/c)^2

    a = d(v^2) / (2 dr)

    a = (v_s^2 - v_r^2) / (2 dr), where r - s = dr

    a = c^2 (b / s - b / r) / (2 (r - s))

    a = c^2 b (r - s) / (2 r s (r - s))

    a = c^2 b / (2 r^2)

    But what I want to know is how we would find the acceleration similarly to the above example with a constant of motion within an accelerated field of (1 - b / r) = (1 - v_r / c) / (1 + v_r / c) instead?
     
  2. jcsd
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