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Separable first order diff eq

  1. Sep 18, 2008 #1
    Ok heres my problem:

    The acceleration of a car is proportional to the difference between 250 km/h and the velocity of the car. If this machine can accelerate from rest to 100 km/h in 10s, how long will it take for the car to accelerate from rest to 200 km/h?

    Here is what ive done so far:

    dv/dt = k(250-v)
    integrating that...
    [tex]\int[/tex]dv/(250-v) = [tex]\int[/tex]kdt which equates to
    -ln|250-v| = kt + c solving for v gives me
    v = e^c * e^(-kt) - 250

    Now, i proceeded to solve for k, making e^c = 1 since i assumed c = vnaught and vnaught = 0

    When i solved for t i ended up getting like 7.2 seconds which isnt right if it takes 10 seconds to accelerate to 100 km/h. My main question is: how do i equate vnaught into the equation? Because obviously i was mistaken to think vnaught = c.

    I have also tried to make e^c = B, but once again am stuck as to how to equate B to vnaught. Any suggestions would be helpful, my book doesnt really explain much.
  2. jcsd
  3. Sep 19, 2008 #2


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    Homework Helper

    v0 is the initial velocity = 0. That's where t=0. That doesn't make e^c = 1 in any way.
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