1. ## Kinetics and integration

Hello,

The speed of a regional airliner during its take off run is a=A-Bv^2. Where v is its speed and A and B are positive constants.

1. Starting from rest, how long does it take the airliner to reach its take off speed vr?
2. Find the speed v(t) during the take off as a function of time.
3. Find the distance st that the airliner requires to take off.

I think I understand the kinematics theory in this question I just really struggled with the integration. Any help would be amazing.
thank you!

2. ## Re: Kinetics and integration

Originally Posted by Trianagt
The speed of a regional airliner during its take off run is a=A-Bv^2. Where v is its speed and A and B are positive constants.
Why do you denote speed with two different letters: a and v?

3. ## Re: Kinetics and integration

The a stands for the acceleration. I hope that helps!

4. ## Re: Kinetics and integration

Since a(t) = dv / dt, we have a differential equation dv / dt = A - Bv^2. It can be solved using separation of variables. This gives v(t) and answers 2. Answering 1 involves finding the inverse of v(t). Answering 3 requires finding $\displaystyle \int_0^t v(u)\,du$ where t is the answer to question 1.

5. ## Re: Kinetics and integration

$\displaystyle \frac{1}{A- Bv^2}= \frac{1}{(\sqrt{A}- \sqrt{B}v)(\sqrt{A}+ \sqrt{B}v)}$
Use "partial fractions to separate those and integrate each using the substitution $\displaystyle x= \sqrt{A}- \sqrt{B}v}$ for the first, $\displaystyle y= \sqrt{A}+\sqr{B}v$ for the second.