This might look like a homework question, but really isn't. I am teaching myself some basic physics and am shamefully stuck in one place. I have designed the problem below to help myself understand the situation better.
Suppose a motorbike weighing 100 kg is at rest and begins producing a constant force of $F=$ 800 N. The resistance to motion is $R=$ 500 N, so the the net force of 300 N causes acceleration in the positive direction, $a = (F-R)/m = 300/100 = 3$ m/s. The acceleration is constant, so the speed is given by v = at. The power produced is given by $P = Fv = Fat$, so it is increasing linearly over time and in direct proportion to speed. Suppose the bike accelerates for the first 5 seconds and then reaches a certain maximum power output that remains constant. Using the numbers above, the speed reached after 5 s is 15 m/s, and so the maximum power is 12 kW.
If my understanding is correct, at an instant when $P$ becomes constant, $v$ will still be increasing, so $F$ will have to be decreasing in inverse proportion, $F = P/v$. Since $F$ is decreasing, the acceleration will also be decreasing, $a = (F - R)/m$. So we have a three-way circular relationship where to find the function of $F$ we need the function of $v$, to find the function of $v$ we need the function of $a$, and to find the function of $a$, we need the function of $F$. How we find the curves for $F$, $a$ and $v$?
Also, is my understanding correct in that $v$ will drop down until $a = 0$, at which point $F = R$, so the maximum velocity allowed by the maximum power output $= P/F = P/R = $ 12000/500 = 24 m/s ?