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Calculating power as a function of time

  1. Nov 19, 2013 #1
    1. The problem statement, all variables and given/known data
    A force of 5 N acts upon a body of 8 kg in the +x direction. Formulate an expression for the power generated as a function of time. The body is in the beginning at t=0 and x=0 (Sorry I'm translating this question from German, so excuse the mistakes).


    2. Relevant equations

    Work = Force * distance

    Power = ΔW/Δt

    Force = mass * acceleration

    3. The attempt at a solution

    Since the force of 5N is acting on the body, by Newton's third law, it is also exerting the same force, so that:

    F = m * a

    5 = 8*a

    a=5/8 m/s^2

    By integrating the acceleration twice I get the distance as a function of time:

    x(t) = 5/8 t^2

    Work = Force * distance = 5 * (5/8)t^2 = (25/8)t^2

    P(t) = (25/8)t

    Was my method correct?
     
  2. jcsd
  3. Nov 19, 2013 #2

    haruspex

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    You forgot something in doing that integration.
    You have divided the total work done over the distance by the total time taken. That will give you average power, but I think it is the instantaneous power at time t that is wanted.
     
  4. Nov 20, 2013 #3
    Thanks I realised my mistake in the integration part. That leads to x(t)= 5/16 t^2.

    The body starts moving at t0=0, so at any given point of time Δt = t - t0 = t

    Hence:

    P= (F*d)/t

    P(t) = 25/16 t
     
  5. Nov 20, 2013 #4

    Curious3141

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    No. You missed out a factor of half in your integration, so the correct expression for total work done at time t should be ##W(t) = \frac{25}{16}t^2##. This expression is not linear in t, so can you simply divide by t to find the power at time t? Shouldn't you be differentiating?

    Basically, your original expression was correct - simply because your errors "cancelled out". You missed a half when integrating, then divided instead of differentiating, so you didn't include a requisite factor of 2.
     
  6. Nov 20, 2013 #5
    Great, thanks for the explanation!
     
  7. Nov 20, 2013 #6
    Another way of arriving at the answer is to remember that the instantaneous power is equal to the force times the velocity. The velocity at time t is 5t/8.
     
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