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    differential equation word problem.

    suppose an object of weight 128 lbs is projected downwards with an initial velocity of 10ft/s in medium that offers resistance of magnitude 8|v|. Assuming gravitational acceleration is constant (g=32ft/s^2) find the velocity at time t.
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    Quote Originally Posted by myoplex11 View Post
    suppose an object of weight 128 lbs is projected downwards with an initial velocity of 10ft/s in medium that offers resistance of magnitude 8|v|. Assuming gravitational acceleration is constant (g=32ft/s^2) find the velocity at time t.
    Defining positive to be upward:
    F = ma = 8|v| - mg

    v is always directed downward. (A resistance can't propel the object upward, after all.) So...
    m*dv/dt = -8v - mg

    dv/dt + (8/m)v = g with v(0) = -10.

    The homogeneous equation is
    dv_h/dt + (8/m)v_h = 0

    So
    v_h(t) = Ae^{-(8/m)t}

    And the particular solution looks like it's
    v_p(t) = B

    Putting this into the differential equation:
    dv_p/dt + (8/m)v_p = g

    0 + (8/m)B = g

    B = mg/8

    Thus
    v(t) = v_h(t) + v_p(t) = Ae^{-(8/m)t} + (mg/8)

    Now, v(0) = -10, so
    -10 = A + (mg/8)

    A = -10 - (mg/8) = -(mg + 80)/8

    Thus
    v(t) = -(1/8)(mg + 80)e^{-(8/m)t} + (mg/8)

    -Dan
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