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Linear model of air resistance

  1. Nov 15, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi,
    I have to model the velocity and displacement of different paper shapes assuming that air resistance, R is proportional to velocity, v.

    I have also conducted an experiment, and the equation of the line
    h = 1.532t - 0.366
    Where h is the height the paper shape was dropped from, if t is the time taken for it to reach the ground.

    Model:
    R = kv
    Assuming that the only forces acting on the cup are it's mg downwards and R upwards.
    And that the motion is vertical only.

    when t=0, v=0 and x=0

    I have found the equations for v and x in terms of t. The problem I have is finding k.

    k= constant in the the assumption R is prop to v.
    mg= weight of paper cup
    R= air resistance
    dv/dt = acceleration
    x= displacement
    v= velocity of the paper cup

    Downwards is taken as positive.

    I will go through what I have so you know what I am working with and the nature of the problem, but ultimately the problem I have with is k.

    2. Relevant equations

    N[II] gives : m(dv/dt) = mg -kv

    (dv/dt) + (k/m)*v = g

    Using an integrating factor: I = e^(kt/m)

    So d(v*e^[kt/m])/dt = ∫ge^(kt/m)dt

    Finishing this and using the initial conditions:

    v = [mg/k]*(1-e^[-kt/m])

    Integrating this we get displacement:

    x = [mg/k]*(t+(m/k)e^(-kt/m)+c)

    Using initial conditions:

    x= [mg/k]*(t+(m/k)*[e^(-kt/m)-1])

    3. The attempt at a solution



    I know that t -> ∞ v-> mg/k [which will be an asymptote when plotted]

    So the terminal velocity = mg/k

    I also know that the x against t graph will look like a curve with an increasing gradient, until the time at which the terminal velocity is reached. At this time the gradient will become constant.

    Ok so now we find k--- how?

    Thanks in advance.
     
  2. jcsd
  3. Nov 15, 2012 #2

    haruspex

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    Your experimental result has h = ct+d. So for large t, h ~ ct. Doesn't that give you your terminal velocity? And presumably you know m.
     
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