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Help finishing a linear differential equation. Mechanics

  1. Mar 23, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the distance which an object moves in time t if it starts from rest and has an acceleration d^2x/dt^2 = ge^-kt.
    Show that for small t the result is approx "x=(gt^2)/2" and show that for very large t, the speed is approximately constant. the constant is called the terminal speed.

    2. Relevant equations

    3. The attempt at a solution
    I ended up with v = -(ge^-kt)/k + v_0 and x = (ge^-kt)/k^2 + v_0t + x_0 however im not sure what to do next. I have tried to solve for k and also set t as zero to get x = g/k^2 but it doesn't seem to be the answer the book is looking for.

  2. jcsd
  3. Mar 23, 2015 #2


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    For small t, look at the Taylor expansion for the exponential.
    For large t, take the difference between ##\int_0^{T+\Delta} ge^{-kt}\, dt-\int_0^T ge^{-kt}\, dt ## for large T, or instead of ##T+\Delta,## use ##\infty##.
  4. Mar 23, 2015 #3


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    ##k## is a given parameter. You don't want to solve for it.

    You might find it helpful to use definite integrals, e.g.,
    $$\int_{v_0}^v \,dv = \int_0^t ge^{-kt}\,dt.$$ The problem statement says the object starts from rest, so use that bit of information too.
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