# Circular motion with tangential acceleration limited by grip

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?

• You can invert both sides and integrate wrt v on one side and wrt t on the other side. – Dr jh Oct 15 '20 at 8:21
• 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? – Mattia Oct 21 '20 at 9:45