Position as a function of speed

[Johnny Blade]

Homework Statement

There's an object with mass m in movement in the horizontal axes. There's a force \textbf{P} of constant power acting on the object. Another force is the air drag which has the magnitude of \beta m v^{2}. I need to find the position x as a function of the speed v.

Homework Equations

\textbf{P} = \vec{F} \cdot \vec{v} = Fv because the vectors are parallel

\Rightarrow F = \frac{\textbf{P}}{v}

\left|\vec{f}\right| = \beta m v^{2}

F = ma

The Attempt at a Solution

With these equation I plug them in F = ma and I get \frac{\textbf{P}}{mv}-\beta v^{2}=\frac{dv}{dt} then by multiplying by \frac{dx}{dx} I got \frac{\textbf{P}}{mv}-\beta v^{2}=v\frac{dv}{dx}.

Then I don't know how to solve this. Or perhaps there's an easier way to this problem?

 

[rl.bhat]

P/mv - βv^2 = v*dv/dx.
Multiply by v on both the side. You get
P/m -β*v^3 = v^2*dv/dx.
So
dx = v^2*dv/( P/m -β*v^3 )
Now find the integration to find x.