The safety speed of a vehicle on a curve horizontal road is

Explanation

Consider a car of mass m taking a turn of radius r along a horizontal (level) road. If `mu_(s)` is the coefficient of static friction between the car tyres and the road surface, the limiting force of friction is `mu_(s)` mg.

Then, the maximum safe speed `v_(max)` with which the car can take the turn without skidding off is determined by the conditon,
maximum centripetal force = limiting force of static friction.
`therefore(mv_(max)^(2))/(r)=mu_(s)mg` or `v_(max)^(2)=rmu_(s)g" "thereforev_(max)=sqrt(rmu_(s)g)`
This is the required expression.
Significance : The above expression shows that the maxium safe speed depends critically upon friction which change with circumstances (e.g., the nature of the surfaces and presence of oil or water on the road). IF the friction is not sufficient to provide the necessary centripetal force, the variable is likely to skid off the road.