How to Find the Velocity of a Gliding Object After Time t?

[suchara]

Homework Statement

Finding the velocity of a gliding object with an initial thrust-velocity of V after time t.
Constants are:-
g: gravity
a: constant for angle of attack, lift coefficient, air density and wing-area

Homework Equations

L \propto a*V^2
acceleration = v (dv/dt)
V is the airspeed of the object

The Attempt at a Solution

I know lift is a force and has an acceleration. So thinking of Lift in terms of velocity-dependant acceleration and only focusing on Vy

acceleration = -g + a*v^2
v (dv/dt) = -g + a*v^2
dv/dt = -g/v + a*v
dv/((-g/v) + a*v) = dt

\int_{v_0}^v dv/(-g/v + a*v) = \int_{0}^t dt

I haven't done integration in a long time.. also I am confused about the notation
Heres the equation I get
\Big|_{v_0}^v (g*ln((a*v^2)-g) + a*v^2)/(a*v^2) = t
After this.. I am unsure on how to proceed further

 

[Tomsk]

Try writing your integral as
\int_{v_0}^v \frac{v}{-g+av^2}dv
Then substitute in u=-g+av^2

 

[suchara]

Try writing your integral as
\int_{v_0}^v \frac{v}{-g+av^2}dv
Then substitute in u=-g+av^2

thanks for the reply.. after integrating that i get \Big|_{v_0}^v \frac{ln(a*v^2+g)} {(2*a)} = t ..which looks a lot more workable, but is the integration correct?
and how do i proceed from here for coming up with a v = v0(t) equation? (i.e. a "Vf= Vi+at" -type equation where given an initial velocity I can come up with a velocity after time t)
Im not asking about the actual math involved.. I am just unsure as to what this:- " \Big|_{v_0}^v " notation means, since it will determine the initial equation i need to simplify
I tried the following way but..

1) \Big|_{v_0}^v \frac{ln(a*v^2+g)} {(2*a)} = t

2) (v-v0) ( \frac{ln(a*v^2+g)} {(2*a)}) = t

3) (v-v0) = \frac{t*2*a} {ln(a*v^2+g)}

4) v = v0 + \frac{t*2*a} {ln(a*v^2+g)}

I get stuck again since I have an unknown v^2 in the fraction on the right side.. should i try another method from step 2 onwards or is the step wrong to begin with?