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Let's suppose I'm running in a circle. The maximum acceleration is limited by a value $\mu$:

$v^2/R = \mu*g$

So, if I'm running in that circle at a speed lower than the max speed (the lateral acceleration is lower than $\mu*g$), I can accelerate. The longitudinal acceleration is:

$a = ((\mu*g)^2 - v^4/R^2)^{0.5}$

If I accelerate, the speed increases, so there will be less longitudinal acceleration. I can write:

$dv/dt = ((\mu*g)^2 - v^4/R^2)^{0.5}$

How can I solve this? I need to remove $v$ from the square root, but how?

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    $\begingroup$ You can invert both sides and integrate wrt v on one side and wrt t on the other side. $\endgroup$ – Dr jh Oct 15 '20 at 8:21
  • $\begingroup$ You're right, I didn't thought about that. But doing this I don't know how to integrate it. It exist an analytical for of this problem, or must I solve it numerically? $\endgroup$ – Mattia Oct 21 '20 at 9:45

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