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Integration by parts and approximation by power series

  1. Sep 15, 2016 #1
    1. The problem statement, all variables and given/known data

    An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants.
    a) Find v(t) and x(t).
    b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3.
    c) Find the object’s terminal velocity.
    2. Relevant equations


    3. The attempt at a solution
    I showed my work for finding v(t) and getting the approximation for v(t) for small t. but Im missing the 1/2 at the front and I cant seem to find where it comes from.
     

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  3. Sep 15, 2016 #2

    BvU

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    Perhaps you need to further develop ##e^{-\lambda t}## ? The term linear in ##t## cancels, so you need to develop up to ##t^2##
     
    Last edited: Sep 15, 2016
  4. Sep 15, 2016 #3
    but if I include a t^2 term then I end up with a t^3 term that doesnt cancel
     
  5. Sep 15, 2016 #4

    BvU

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    Yes. So the outcome will be correct up to order ##t^2##. .
     
  6. Sep 15, 2016 #5

    BvU

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    You can also do the development before the integration: ## t\;e^{-\lambda t}= t -\lambda t^2 ## and ignore higher orders
     
  7. Sep 15, 2016 #6
    Thanks for your help, i got the right answer. But can you help me understand something, why is it enough to include only up to the t^1 term before the integration, but after the integration we have to include the t^2 term?
     
  8. Sep 15, 2016 #7

    BvU

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    "Integration increases order of ##t## by one" is the answer that comes to mind. But I agree with you that it makes a weird impression.
     
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