Deriving acceleration and velocity for a model car

  • #1

Homework Statement

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.

Homework Equations

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

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

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.

Answers and Replies

  • #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....