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Classical mechanics - finding distance D in terms of velocity

  1. Sep 14, 2011 #1
    1. The problem statement, all variables and given/known data

    "A passenger (mass m) initially at rest steps out of an airplane. Assume down is the positive x-axis and put the origin at the airplane. Assume the air resistance force is linear in the velocity so F(air)= -mbv. Find the distance D he has fallen when his velocity is v."


    2. Relevant equations

    Equations of motion

    The "vdv/dx trick": d2z/dt2 = (z-dot)*(d(z-dot)/dz)

    F(tot) = ma

    F(air) = -mbv

    Weight = mg


    3. The attempt at a solution

    Here's how far I've gotten:

    Since the skydiver is only falling in the x direction, there's only one equation of motion, which I found to be ma = -mbv + mg [or, alternatively, m(x-double dot) = -mb(x-dot) + mg]. Now, I know I want the relation of distance and velocity, without time, so I use the "vdv/dx" trick (so that there's no longer time in the equation).

    That makes this mv*(dv/dx) = -mbv + mg, or m(x-dot)*(d(x-dot)/dx) = -mb(x-dot) + mg. I rearranged this to get dx = (-m/b)*(vdx/v-(mg/b)), where -(mg/b) is the terminal velocity.
    I'm sorry for all the writing, but am I correct so far? And how do I continue to solve the problem from here? Any help would be appreciated.
     
    Last edited: Sep 14, 2011
  2. jcsd
  3. Sep 14, 2011 #2

    kuruman

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    First off, divide through by m to simplify the expression a bit. Secondly, can you separate variables, i.e have things with "v" on one side and "x" on the other?
     
  4. Sep 14, 2011 #3
    First, since m is common to every term, I would eliminate that to make things easier. Next, you have vdv/dx = g-bv. Try to isolate v and dv on one side, and dx on the other side.
     
  5. Sep 15, 2011 #4
    Alright, so...

    I went back and simplified what I had first and ended up with v(dv/dx) = -bv - g. So, after isolating dz, I end up with dz = -(vdv/(bv+g)).

    Now, I know my next step is to integrate this, but I'm not sure what the limits would be.
     
  6. Sep 16, 2011 #5

    kuruman

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    The problem tells you what the limits should be.

    "A passenger (mass m) initially at rest steps out of an airplane. Assume down is the positive x-axis and put the origin at the airplane. Assume the air resistance force is linear in the velocity so F(air)= -mbv. Find the distance D he has fallen when his velocity is v."
     
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