# Deriving acceleration and velocity for a model car

## 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

$P =\frac W t$

$W=\Delta K$

## The Attempt at a Solution

$P = \frac W t$

$W=P t$

$\Delta K = P t$

Because the initial velocity is zero,

$\frac 1 2 m v^2 \equiv P t$

$v(t) = \sqrt{\frac{2Pt}{m}}$

And acceleration is the derivative:

$a(t) = \frac{d}{dt} \left(\sqrt{\frac{2Pt}{m}}\right) = \frac{d}{dt} \left(\sqrt{\frac{2P}{m}} \sqrt{t}\right) = \frac{\sqrt{\frac{2P}{m}}}{2 \sqrt{t}}$

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.