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Integrate equation

  1. Dec 4, 2016 #1
    1. The problem statement, all variables and given/known data
    I'm trying to integrate this equation so that I can find my position using my initial drag force and initial velocity.

    V/(-g-kv) dv
    2. Relevant equations

    3. The attempt at a solution
    U substitution won't work
    Integration by parts won't work
    Partial fractions won't work.
    It doesn't match any trigonometric equations

    I'm not sure how to integrate it.
    Basically a potato is being shot up in the air with air resistance and gravity affecting it.
    This is what I started with
    M(dv/dt)= -mg-kmv
    well v=dx/dt so dt=dx/v

    Separation of variable from D.E
    Now I want to integrate the left side which I don't know how to do.

    Thanks for any help
    Last edited: Dec 4, 2016
  2. jcsd
  3. Dec 4, 2016 #2
    Do you know have to compute the indefinite integral
    $$\int \frac{dx}{x}\quad ? $$
  4. Dec 4, 2016 #3
  5. Dec 4, 2016 #4


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    Gold Member

    This is the equation you should integrate.
    Are you asked to find the displacement as a function of time?
  6. Dec 4, 2016 #5
    If the
    V dv

    If the v wasn't at the stop I could integrate it but I don't know what to do with the V at the top
  7. Dec 4, 2016 #6
    I'm not given time which is why I had to use the substitution dt=dx/v. Because I am given initial velocity and k
  8. Dec 4, 2016 #7
    I did that integral first then realized I didn't have time so I had to take a different route
  9. Dec 4, 2016 #8


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    Hint: Isolate the v in the denominator (get rid of the k attached to it).
  10. Dec 4, 2016 #9
    Take first out the minus sign from the denominator. Do then the substitution ##x=g+kv##.
  11. Dec 4, 2016 #10
    just a suggestion :- try binomial expansion on the denominator.
  12. Dec 4, 2016 #11
    So I did some rearrangements and got

    --------- =- dx
    g/v +k

    Then integrating both sides I get

    Assuming c is the initial velocity
  13. Dec 4, 2016 #12


    Staff: Mentor

    First off, this isn't an equation -- an equation always has '=' somewhere in it.
    Second, are V and v different quantities?
    Substitution will work if V and v are different. If they are different, I'm assuming that V is a constant, and v is the variable velocity. If that's the case, substitution will work.
    Now it appears that V and v both represent velocity, a variable. Writing both V and v in the same expression is very confusing.
    Write the equation as ##\int \frac {v~dv}{g + kv} = -\int dx##. On the left side, use polynomial long division.
  14. Dec 4, 2016 #13
    Sorry when I type in the first letter it always wants to capitalize it. Well I shall search for understanding on polynomial long division. Thank you
  15. Dec 4, 2016 #14


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    That's not correct.
    Try the methods in #9 and #12. If it's not too much, you can try this too.. Take the k out from the denominator and you'll have the equation in the form y*dy/(a+y). Just a simple rearrangement in the numerator and it will be integrable.
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