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Constraints on Acceleration Given Endpoints of Motion

  1. May 11, 2013 #1
    1. The problem statement, all variables and given/known data
    A particle of mass M moves on the X-axis as follows: It starts from rest at t = 0 from the point x = 0 and comes to rest at t = 1 at the point x = 1. No other information is available about it smotion at intermediate times (0 < t < 1). If α denotes the instantaneous acceleration of the particle, then prove that |α| must be ≥ 4 at some point in its path.

    2. Relevant equations
    [tex]
    \int_0^1{v(t)\,dt} = x(1) - x(0) = 1\\
    \int_0^1{a(t)\,dt} = v(1) - v(0) = 0\\
    [/tex]


    3. The attempt at a solution
    If constant accelerations of equal magnitudes are assumed for the period of acceleration and the period of deceleration, one obtains that the acceleration is 4 for the first 1/2 second and -4 for the last 1/2 second. I can see why the statement is true given these calculations, but could someone suggest a more rigorous way to prove this?
     
  2. jcsd
  3. May 11, 2013 #2

    mfb

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    2016 Award

    Staff: Mentor

    If |α| <=4, what is the quickest way to reach x=1? You can use symmetry to consider the acceleration part only, if you like.
    If |α| <4, can you still have the same time?
     
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