Calculating distance from acceleration as function of speed

[Andrew97]

I'm trying to calculate the displacement d of a plane as it accelerates towards lift-off velocity.
The acceleration is given by the function a = a₀ - kv² where a₀ is the constant acceleration from the plane's engine and the term -kv² is caused by air resistance.

I can't seem to find a way to integrate the equation above so that I get the plane's velocity nor it's displacement.

The answer is meant to be:
d = - ln (1 - kv² / a₀ ) / 2k

 

[BvU]

Hello Andrew,

This homework (*) ? For the first step click 'show' here

(*) PF rules are rather strict on this. Need to use the template and show an attempt at solution and such.

 

[Andrew97]

Hello Andrew,

This homework (*) ? For the first step click 'show' here

(*) PF rules are rather strict on this. Need to use the template and show an attempt at solution and such.

Ah, yes sorry I did not know.

I kept getting stuck because I tried moving over the velocity to the left side of the equation, and since a=dv/dt I could move over dt and integrate both sides, however I did not find a way to isolate v on the left side of the equation.

The link you sent me made me realize that the mistake I was making was that I should have divided both sides of the equation by the entire right side so that the velocity exists only on the left side of the equation allowing me to integrate both sides and proceed to solve the problem. Thank you.