This question is similar, but I am a bit out of practice on mathematics and struggle to apply the steps to my problem.
How can the $\triangle$$t$ be calculated for a given $\triangle$$v$ when acceleration changes as a function of $v$?
Given:
$a(v) =- \frac{8.69+0.0252v^{2}}{200}$
My current goal is to find the time required to change from $v=70$ to $v=60$. (velocity: $\frac{km}{hr}$, acceleration: $\frac{km}{hr/s}$)
My approach:
I have tried to use the linked question to rewrite $a(v)$ as $v(t)$. Then I would set the equation equal to 70 and then 60 respectively and solve for $\triangle$$t$.
I do not understand where $U$ comes from in the linked problem.
Semi-related question, are the steps above necessary for expressing $a(v)$ in terms $a(t)$?
$$ \frac{dv}{dt}=a+b v^2\implies \frac{dv}{a+bv^2}=dt\implies \Delta t=\frac1{\sqrt{ab}}\left[\arctan\sqrt{\frac ab}v\right]_{v_1}^{v_2}. $$