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Math Help - Diff EQ Falling body

  1. #1
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    Diff EQ Falling body

    Consider the falling body problem where:
    m(dv/dt) = W - B - CV. W is weight, B is boyancy and CV is drag force. W,B,C are constants, V is dependant on t of course V(t).

    Consider an additional force acting upon the object that depends only on time symbolized by n(t). Assume the body is dropped from rest.

    I must set up an IVP for V(t), t > or = 0.
    Im a little confused how to approuch this with the extra n(t).
    It also doesnt explicity say if n(t) is + or -, but i assume its -, do you agree?

    m(dv/dt) = W - B - CV - n(t)

    How shall i separate these variables and attempt to solve?
    Thanks to all in help.

    Chris
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by chrsr345 View Post
    Consider the falling body problem where:
    m(dv/dt) = W - B - CV. W is weight, B is boyancy and CV is drag force. W,B,C are constants, V is dependant on t of course V(t).

    Consider an additional force acting upon the object that depends only on time symbolized by n(t). Assume the body is dropped from rest.

    I must set up an IVP for V(t), t > or = 0.
    Im a little confused how to approuch this with the extra n(t).
    It also doesnt explicity say if n(t) is + or -, but i assume its -, do you agree? I would assume it to be positive. That's what I usually tend to do in situations like these...

    m(dv/dt) = W - B - CV - n(t)

    How shall i separate these variables and attempt to solve?
    Thanks to all in help.

    Chris
    First divide the equation through by m to get:

    \frac{\,dV}{\,dt}=\frac{W}{m}-\frac{B}{m}-\frac{C}{m}V+\frac{n(t)}{m}

    This DE is linear:

    \frac{\,dV}{\,dt}+\frac{C}{m}V=\frac{W}{m}-\frac{B}{m}+\frac{n(t)}{m}

    Apply the method of the integrating factor, where P(t)=\frac{C}{m}

    Can you try to take it from here?

    --Chris
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