(III) A curve of radius 78 m is banked for a design speed of 85km/h . If the coefficient of static friction is 0.30 (wet pavement), at what range of speeds can a car safely make the curve ? Hint:Consider the direction of the friction force when the car goes too slow or too fast.]

To determine the range of speeds at which a car can safely make the curve, we need to consider both the centripetal force and the frictional force acting on the car. The centripetal force is provided by the banking of the curve, and the frictional force acts to prevent the car from sliding.Given:- Radius of the curve, m- Design speed, km/h (which we will convert to m/s)- Coefficient of static friction, First, let's convert the design speed to meters per second: Next, we need to find the maximum and minimum speeds at which the car can safely navigate the curve. This involves finding the speeds where the net force providing the centripetal force is just enough to keep the car on the curve without relying on friction.### Case 1: No friction (ideal scenario)In the ideal scenario where there is no friction, the entire centripetal force is provided by the banking of the curve. The centripetal force is given by: Since the car is not sliding, the normal force equals the weight of the car : The banking provides the centripetal force: Where is the angle of the banking. Since we are not given , we assume the banking is designed such that at the design speed , the centripetal force is balanced by the normal force: ### Case 2: Frictional force is just enough to prevent slidingWhen the car is moving at a speed where it is on the verge of sliding, the frictional force equals the maximum static friction force: The net force providing the centripetal force is then: Since , we can write: Substituting : Simplifying: We need to solve for : ### Finding the range of speedsWe need to find the minimum and maximum speeds and for which the car can safely navigate the curve.1. **Minimum speed **: 2. **Maximum speed **: Since the car can safely navigate the curve between these two speeds, we have: ### ConclusionThe range of speeds at which a car can safely make the curve is: Where: These speeds ensure that the car remains on the curve without sliding, considering both the centripetal force provided by the banking and the frictional force.