Final answer:
To find the ideal speed for a frictionless steeply banked tight curve with a 100 m radius and a 65.0° banking angle, we use the formula v = √(rg tanθ). The ideal speed is calculated based on the curve's radius and the gravitational force being the only source of centripetal force in the absence of tire friction.
Step-by-step explanation:
Calculating the Ideal Speed for a Steeply Banked Curve Without Friction
To calculate the speed at which a 100 m radius curve banked at 65.0° should be driven if the road is frictionless, we need to apply principles of circular motion and dynamics. Without relying on tire friction, only the component of gravitational force parallel to the surface of the road provides the necessary centripetal force to keep a vehicle on its circular path. The ideal speed, therefore, can be found using the formula:
v = √(rg tanθ)
where:
- v is the ideal speed
- r is the radius of the curve
- θ is the banking angle of the curve
- g is the acceleration due to gravity (9.81 m/s²)
Plugging in the values, we have:
v = √(100 m * 9.81 m/s² * tan(65.0°))
This gives us the ideal speed to safely navigate the steeply banked tight curve.
It's important to note that in real-world scenarios, tire friction does play a significant role and allows vehicles to take curves at higher speeds than what would be calculated for a frictionless situation.