Final answer:
The maximum speed at which the car can navigate the 65 m radius curve without slipping, given a coefficient of static friction of 0.70, is calculated to be approximately 21.4 m/s.
Step-by-step explanation:
To determine the maximum speed at which a car can round an unbanked curve without slipping, we use the concept of centripetal force and friction. The frictional force provides the centripetal force required to keep the car moving in a circle. This force is determined by the coefficient of static friction (μ) between the tires and the road and the normal force, which for an unbanked curve equals the weight of the car. The formula used is:
Fc = m · v2 / r = μ · m · g
Where Fc is the centripetal force, m is the mass of the car, v is the velocity, r is the radius of the curve, and g is the acceleration due to gravity (9.8 m/s2). In this scenario, we are given the radius of the curve (65 m) and the coefficient of static friction (0.70). Solving for v gives us the maximum speed:
v = √(μ · r · g)
Plugging in the values:
v = √(0.70 · 65 m · 9.8 m/s2)
v ≈ 21.4 m/s
Therefore, the car can take the curve at a maximum speed of 21.4 m/s without slipping.