Final answer:
The speed required for a 500 kg car to navigate a banked turn at a radius of 65 m and angle of 30 degrees without frictional assistance can be calculated using the balance of centripetal and gravitational forces.
Step-by-step explanation:
The speed at which a 500 kg car should travel around a corner with a radius of 65 m, banked at 30 degrees, to need no assistance from friction can be found using the concept of centripetal force and the force due to gravity. In the absence of friction, the only forces acting on the car are the normal force and the gravitational force, which must provide the necessary centripetal force to keep the car on a circular path. By balancing the forces using Newton's second law and the condition for circular motion, we can derive an expression for the required speed.
Using the centripetal force equation Fc = m*v2/r, where Fc is the centripetal force, m is the mass, v is the velocity, and r is the radius of the curve, and equating this to the component of the gravitational force acting down the slope of the banked curve, we can solve for v. The banking angle (θ = 30 degrees) allows the car to maintain a speed where all of the centripetal force needed is a component of the gravitational force, meaning no additional frictional force is required.
Therefore, the formula relating the required speed (v), the radius of the curve (r), the mass of the car (m), and the angle of the bank (θ) is given by v = √(rg tan θ). This equation can be used to calculate the ideal speed for this scenario.