How Is Velocity Expressed as a Function of Position with Given Power Input?

[e2e8]

Power to a vehicle moving along a track is alpha*x^1/2. No dissipation. alpha is constant x is distance from beginning of the track. zero initial velocity.
What is Velocity as function of position?

P=d KE/dt
P=d(1/2 m v^2)/dt
P=1/2 m d((dx/dt)^2)/dt
P=1/2 m 2 v dv/dt
alpha x^1/2 = 1/2 m 2 v dv/dt

And that is as far as I get. After some random manipulations I was not able to yield anything useful. Some assistance or guidance would be greatly appreciated. Thank you.

e2

 

[ehild]

dv/dt=dv/dx*dx/dt = dv/dx *v.

use this in your last formula then solve the differential equation for v(x).

ehild

 

[kerimek]

=alpha*x^1/2
d(e2)/dt=d(1/2 m v^2)/dt
2*alpha*(dx/dt)*(d2x/dt2)=m*2v*(dv/dt)
alpha*dx/dt*d2x/dt2=m*v*dv/dt
alpha*dx/dt*d2x/dt2=alpha*x^1/2*dv/dt
dx/dt*d2x/dt2=x^1/2*dv/dt
d2x/dt2=x^-1/2*dv/dt
Integrating both sides:
dx/dt=x^-1/2*v + C1
Integrating again:
x=-2*v*x^1/2 + C1*t + C2
Since initial velocity is zero, C1=0.
Therefore, velocity as a function of position is:
v=-1/2*x^-1/2*x + C3
where C3 is the constant of integration.