Final answer:
The range of speeds can a car safely make the curve, the ideal speed to take the banked curve is approximately 25.41 m/s and the coefficient of static friction needs to be greater than or equal to approximately 0.27 for the car to safely make the curve at any speed.
Step-by-step explanation:
When a car takes a banked curve, the force of gravity and the normal force combine to provide the centripetal force required to keep the car moving in a circular path. The ideal speed to take a banked curve can be calculated using the formula:
ideal speed = √(g * r * tanθ)
where
g is the acceleration due to gravity (9.8 m/s^2)
r is the radius of the curve (72 m)
θ is the angle of banking (given as 15.0°).
Plugging in these values, we get:
ideal speed = √(9.8 * 72 * tan(15.0°)) ≈ 25.41 m/s.
The minimum coefficient of static friction needed for a car to safely make the curve at a particular speed can be calculated using the formula:
μ = tanθ
where
μ is the coefficient of static friction
θ is the angle of banking.
Plugging in the given angle of banking (15.0°), we get:
μ = tan(15.0°) ≈ 0.27.
Therefore, for a car to safely make the curve, it should be driven at a range of speeds less than or equal to approximately 25.41 m/s and the coefficient of static friction should be greater than or equal to approximately 0.27.