car is moving with acceleration a starting from rest along a circular track of radius R, if friction coefficient is , then at any time t car will be moving without skidding if :

The condition for the car to move without skidding is that the frictional force should be equal to or greater than the centripetal force required for circular motion.

The frictional force (F_friction) is given by the equation:

F_friction = μN

where μ is the coefficient of friction and N is the normal force. For a car moving in a horizontal circle, the normal force is equal to the weight of the car, mg, where m is the mass of the car and g is the acceleration due to gravity.

So, F_friction = μmg

The centripetal force (F_centripetal) required for circular motion is given by the equation:

F_centripetal = mv²/R

where m is the mass of the car, v is the velocity of the car, and R is the radius of the circular track.

Since the car is moving with acceleration a starting from rest, its velocity at any time t is given by the equation:

v = at

Substituting this into the equation for F_centripetal gives:

F_centripetal = ma²t²/R

For the car to move without skidding, F_friction must be equal to or greater than F_centripetal. Therefore, the condition for the car to move without skidding is:

μmg ≥ ma²t²/R

Solving this inequality for t gives:

t ≤ sqrt((μmgR)/(ma²))

So, at any time t less than or equal to sqrt((μmgR)/(ma²)), the car will be moving without skidding.