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Integration of dynamics problem

  1. May 30, 2010 #1
    1. The problem statement, all variables and given/known data
    A package of mass m is released from an aircraft flying horizontally with a velocity v0. In addition to gravity, a drag force acts on the package given by F=-[tex]\lambda[/tex]v. Find the i and j components of the velocity as a function of the time (t) from when the package was released.


    2. Relevant equations
    F=ma


    3. The attempt at a solution
    Acceleration in x-direction:
    F=ma(x)
    -[tex]\lambda[/tex]v(x)=ma(x)
    a(x)=-[tex]\lambda[/tex]v(x)/m
    dv(x)/dt = -[tex]\lambda[/tex]v(x)/m
    The solution method then integrates, and gets:
    v(x) = v0exp(-[tex]\lambda[/tex]t/m)
    I don't know how to do the integration.
    Thanks.
     
  2. jcsd
  3. May 30, 2010 #2

    Cyosis

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    Homework Helper

    They move all v-s to one side and t-s to the other side.
    [tex]
    \frac{dv}{v}=-\frac{\lambda}{m} dt
    [/tex]

    You can now easily integrate this equation and solve for v.
     
  4. May 30, 2010 #3
    So,
    dv/v = -[tex]\lambda[/tex]dt/m
    ln v = -[tex]\lambda[/tex]t/m
    v = exp(-[tex]\lambda[/tex]t/m)

    But in the answer, where does the extra v0 come from that is before the exponent?
     
  5. May 30, 2010 #4

    Cyosis

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    Homework Helper

    You forgot the integration constant in combination with [itex]v(t=0)=v_0[/itex].
     
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