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Solve DE for approaching terminal velocity

  1. Oct 25, 2012 #1
    1. The problem statement, all variables and given/known data
    I'm trying to find the function, that describes the velocity approaching to a terminal velocity.


    2. Relevant equations
    [itex]F_{net}=mg-\frac{1}{2}\rho v^2 AC_d[/itex]


    3. The attempt at a solution
    [itex]F=ma[/itex]
    [itex]a=F/m[/itex]
    [itex]\dot{v}=F/m=g-\frac{1}{2m}\rho v^2 AC_d[/itex]
    [itex]\dot{v}=g-kv^2[/itex]
    [itex]\dot{v}+kv^2=g[/itex]

    (k and g are constants)
    I have very few knowledge of DEs and it seems hard to guess a solution.
    Can somebody help me?
     
    Last edited: Oct 25, 2012
  2. jcsd
  3. Oct 25, 2012 #2
    Rewrite the equation as
    [tex]\frac{dv}{dt}=g-kv^2[/tex]
    [tex]\frac{dv}{g-kv^2}=dt[/tex]

    Now it should be easy to solve.
     
  4. Oct 25, 2012 #3
    thank you!
    now it seems obvious. :D
     
  5. Oct 29, 2012 #4
    The expression can be rewritten as:
    [tex]\frac{dv}{k((\sqrt{\frac{g}{k}})^2-v^2)}=dt[/tex]

    Integrating LHS is same as integrating [itex]\frac{dx}{a^2-x^2}[/itex] where a is some constant. Integrate [itex]\frac{dx}{a^2-x^2}[/itex] using partial fractions.
     
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