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Free fall velocity with air resistance/drag

  1. Jan 31, 2016 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data
    I am analyzing free fall motion in my computer code class.
    We haven't really discussed much about air resistance and it is a bit of a foreign topic for me.
    I am searching the internet as much as I can for information on it, but would really appreciate talking to someone here regarding it.

    What I've been giving is
    F_d = -bv where b is a constant and v is velocity.
    Teacher said to use our decay problem as an example code for this. Suggesting that I may use something of the form Ae^...

    I'm just not sure how this all relates.


    I search online and it seems like there are many many different types/examples of air resistance equations/examples.
     
  2. jcsd
  3. Jan 31, 2016 #2

    RJLiberator

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    So I'm observing
    v(t) = mg/b-(e^(-bt))*mg/b
    But, in my problem there is no mass.
     
  4. Jan 31, 2016 #3

    RJLiberator

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    Sorry to bump, this will be the last time.

    I have to code this guy into a free fall problem, problem is, I haven't really worked with air resistance.

    Many of the sites (youtube videos and the like) all have the drag force dealing with mass and other such things that I do not consider in my coding problem.

    The teacher only gave us this problem:

    dv/dt = a - bv

    I tried solving this with my limited knowledge of differential equations and found
    a = e^(-bt)+bv
    But this doesn't make much sense to me as, how can a be a part of the velocity graph.
    If acceleration mean the change in velocity maybe I could look at it as a time step? But then e^(-bt) is dependent on time anyway.
     
  5. Jan 31, 2016 #4
    Show us how you solved the differential equation, please.
     
  6. Jan 31, 2016 #5

    RJLiberator

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    Will do.

    dv/dt = a - bv

    Divide across by dv

    1/dt = (a-bv)/dv
    flip everything (inverse)
    dt = dv/(a-bv)

    integrate
    t = -log(a-bv)/b

    Multiply both sides by -b
    -bt = log(a-bv)

    exponentiate
    e^(-bt) = a-bv

    a = e^(-bt)+bv
     
  7. Jan 31, 2016 #6
    You left out the constant of integration, required to satisfy the initial condition at t = 0.
     
  8. Jan 31, 2016 #7

    RJLiberator

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    At time = 0 there would be an initial velocity?

    So would it just become a = e^(-bt)+bv-v_i
     
  9. Jan 31, 2016 #8
    The initial velocity is zero, and your equation doesn't satisfy that. The correct solution should be:

    $$v=a\frac{(1-e^{-bt})}{b}$$
     
  10. Jan 31, 2016 #9

    RJLiberator

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    v = (a-ae^(-bt))/b

    So this equation is not dependent on mass, which is good.
    But, the problem I am having with this is it is dependent on acceleration and velocity.
    I have the duty of coding a graph of the equation of velocity from a free falling object.
    I have successfully made the graph for the case without air resistance.

    So I figure the graph with air resistance shouldn't be too hard. It should be a simple extension.
    When I look at this equation, I am left wondering how to incorporate it into my code.

    v is what we are looking for.

    1) b, I have no idea what b is. I know it's a constant that is dependent on the fluid it traverses, but how am I suppose to give it numerical value? Should I make it a user input and suggest some values for b?
    b is determined by vterm = mg/b
    But I don't have a mass, and what would be my terminal velocity? Don't have that either.

    I suppose I can let the user input values for b, but I am struggling even finding realistic values to suggest.

    2) And then there is acceleration. Can I represent that by a small change in time of the velocity? OR could that Just be gravity = 9.81?
     
  11. Jan 31, 2016 #10
    I don't know how you want to handle b. It depends on how technically correct you want to be. Is this for a physics course or a math course?
    You should have a = g in your equations. As far as terminal velocity is concerned, it is the value of v when t is infinite. What does your equation say?
     
  12. Jan 31, 2016 #11

    RJLiberator

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    I think the best route to go is to let b be user inputted at this point.

    Thank you for the confirmation on a = g and understanding of terminal velocity.
    When t is infinite, terminal velocity is a/b which is 9.81/b.

    This is for a physics course entitled Mathematical and Computational methods for Physicists.

    Your help here has been extremely important in my understanding of what I have to do.
     
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