Final answer:
The ideal banking angle for a curve depends on the intended speed and curve radius and can be found using the formula θ = tan⁻¹(v²/rg). An ideally banked curve allows for navigation without relying on tire friction. This angle varies for highways and racetracks with steep banked curves for higher speeds.
Step-by-step explanation:
The ideal banking angle for a curve depends on the speed at which vehicles are expected to travel and the radius of the turn. We can use the formula θ = tan⁻¹(v²/rg), where θ is the banking angle, v is the velocity, r is the radius of the curve, and g is the acceleration due to gravity (approximately 9.8 m/s²). An ideally banked curve is designed such that a vehicle can navigate the curve at a certain speed without relying on friction.
For example, if a curve on a highway has a radius of 1.20 km and vehicles travel at 105 km/h, the ideal banking angle can be calculated using the formula provided. Similarly, for a race track designed with a steep banking angle, vehicles can negotiate curves at higher speeds.
To calculate the ideal banking angle for the provided scenarios, one would perform the following steps:
- Convert all units to SI units (meters for distance, seconds for time).
- Substitute the values of v and r into the formula θ = tan⁻¹(v²/rg).
- Use a calculator to find the inverse tangent of the calculated value to find the angle in degrees.