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Dynamics problem

  1. Oct 29, 2011 #1
    1. The problem statement, all variables and given/known data

    A boat has mass m=1000 kg and is moving at velocity v0=324 ms^-1. Friction force btw the boat and water is proportional to velocity v, Fd=70*v. How long it takes to slow down to 162ms^-1 ?


    2. Relevant equations
    I'm not sure which function should I integrate, because acceleration is not constant.


    3. The attempt at a solution
    I understand Friction force and acceleration as functions of v, but I have no idea how to express these as functions of time, since acc is not constant. Then I would integrate a(t) with respect to time and substitute final velocity for v(t) and from that I'd get the answer.
     
  2. jcsd
  3. Oct 29, 2011 #2
    Draw a force diagram, and apply Newton's 2nd law. You can get the acceleration that way. Then use [itex]a = \frac{dv}{dt}[/itex] to get a differential equation. Solve it and find [itex]v[/itex] in terms of [itex]t[/itex]. Solve for [itex]t[/itex] and substitute correct value of [itex]v[/itex] to find the time.
     
  4. Oct 29, 2011 #3
    a=-70v/m
    ∫dv=∫(-70v/m)dt
    v=-70/m ∫vdt - the problem is I do not know v in terms of t :(
     
  5. Oct 29, 2011 #4
    Instead of your second line, do this,
    [tex]\int{\frac{dv}{v}}=\int{\frac{-70 dt}{m}}[/tex]
     
  6. Oct 29, 2011 #5
    I have to correct given informtion v0=25 ms^-1 and v=12.5 ms^-1

    Ok, I tried your suggestion and from that I get:

    t = (-m*(ln(v) + v0))/70 -what doesn't seem right

    After substituting v = 12.5 I get t = 393 s and that is wrong (correct answer should be 9.9s)
     
  7. Oct 29, 2011 #6
    This does not agree with my final answer.

    Check whether you applied the initial boundary condition (v = v0 when t=0 ) correctly.
     
  8. Oct 29, 2011 #7
    At t = 0, I'm pretty sure, the integration constant is equal vo (=25).
     
  9. Oct 29, 2011 #8
    Not ln(vo) ?
     
  10. Oct 29, 2011 #9
    Yes, you're right, thanks. I finally got it correct. My mistake was I put the constant directly from initial conditions, not solving from integrated function.
     
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