A car travels along a horizontal circular road of radius R , with...

The car experiences two types of acceleration while moving on the circular path:

Tangential acceleration, $a_t = \dfrac{dv}{dt} = a$

Radial (centripetal) acceleration, $a_r = \dfrac{v^2}{R}$

The net acceleration of the car is given by:

$a_{net} = \sqrt{a_t^2 + a_r^2} = \sqrt{a^2 + \left(\dfrac{v^2}{R}\right)^2}$

The net force required to keep the car moving without skidding is provided by static friction. The car will just begin to skid when this required force equals the maximum static friction force:

$m a_{net} = \mu N$

Since the road is horizontal, the normal reaction is $N = mg$. Substituting the values:

$m \sqrt{a^2 + \dfrac{v^4}{R^2}} = \mu mg$

Dividing by $m$ and squaring both sides:

$a^2 + \dfrac{v^4}{R^2} = \mu^2 g^2$

$\dfrac{v^4}{R^2} = \mu^2 g^2 - a^2$

$v^4 = \left(\mu^2 g^2 - a^2\right)R^2$

$v = \left[\left(\mu^2 g^2 - a^2\right)R^2\right]^{1/4}$