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Speed as a function of time

  1. Aug 7, 2011 #1
    How could I express the speed of an object as a function of time if the object is accelerating with a constant power output and is being affected by quadratic (Rayleigh) drag?
     
  2. jcsd
  3. Aug 7, 2011 #2

    SteamKing

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    What type of object?
     
  4. Aug 7, 2011 #3
    Well, it's supposed to be a submarine.
    As far as the physics go, it's a solid object with density equal to the density of the liquid
    Anything else?
     
  5. Aug 7, 2011 #4

    SteamKing

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    As a start, I would draw a free-body diagram and add the relevant forces. Since it is a submarine, the power produced by the propulsion machinery is converted into thrust, either by turning a propeller or by means more exotic. The drag is proportional to the velocity squared and acts to oppose the thrust. Since the sub is accelerating, then T - D is positive, and T - D = m a, where m is the mass of the submarine, and a = dv/dt.
     
  6. Aug 7, 2011 #5

    A.T.

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    And in general the thurst will not be constant for constant power input, but depend on the speed.
     
  7. Aug 8, 2011 #6
    And that's the point where I got stuck. I found an equation that works if the thrust is constant but that is not the case here.

    Perhaps if I expressed drag as a loss in energy (speed3) it might be simpler.

    Not quite sure though, we haven't done anything like this at school so far and it might be a few years till we do.
     
  8. Aug 8, 2011 #7

    A.T.

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    You won't get an analytic solution for this. The thrust itself is not some simple function but usually derived empirically or numerically. You have a chart like this:

    9043d1157980313-prop-efficiency-opc-diagram.jpg

    And get the thrust by:

    thrust = (efficiency * shaft_power) / velocity

    To get the speed as function of time you have to integrate the acceleration from the net force (thrust - drag) numerically. For low velocities there is static thrust data.
     
    Last edited: Aug 8, 2011
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