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Deriving acceleration and velocity for a model car

  1. Oct 25, 2011 #1
    1. The problem statement, all variables and given/known data
    I am not sure that my work is correct, and some guidance would be much appreciated.

    An electric toy car has mass m and has a power source of constant power P.

    The power source converts electrical energy directly into kinetic energy. If the car is initially stationary and on a level surface, calculate the velocity and acceleration as functions of time. Find the limiting value of the velocity and of the acceleration at very large times.

    2. Relevant equations

    [itex]P =\frac W t[/itex]

    [itex]W=\Delta K[/itex]

    3. The attempt at a solution

    [itex]P = \frac W t[/itex]

    [itex]W=P t[/itex]

    [itex]\Delta K = P t[/itex]

    Because the initial velocity is zero,

    [itex]\frac 1 2 m v^2 \equiv P t[/itex]

    [itex]v(t) = \sqrt{\frac{2Pt}{m}}[/itex]

    And acceleration is the derivative:

    [itex]a(t) = \frac{d}{dt} \left(\sqrt{\frac{2Pt}{m}}\right) = \frac{d}{dt} \left(\sqrt{\frac{2P}{m}} \sqrt{t}\right) = \frac{\sqrt{\frac{2P}{m}}}{2 \sqrt{t}}[/itex]

    There are probably some mistakes thus far. I am asked to provide the limiting value of acceleration and velocity at very large times; based on my work, velocity will continually increase and acceleration will approach zero as t increases. Are there no "limiting values", then, other than 0 and infinity?
    Any help is much appreciated.
  2. jcsd
  3. Oct 26, 2011 #2
    That is all correct what you did. There is no limit of the velocity with a power source like that which stays constant always. The thing is its useless to talk about the situation when time is large as the model is not good for that....
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