# Kinematics of a gliding object

## 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 havent done integration in a long time.. also Im 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.. Im unsure on how to proceed further

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$$\int_{v_0}^v \frac{v}{-g+av^2}dv$$
Then substitute in u=-g+av^2

$$\int_{v_0}^v \frac{v}{-g+av^2}dv$$
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.. im 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?

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