Question

What is the minimum speed, in meters per second, required to take a 108 m radius curve banked at 19° so that you don't slide inwards, assuming there is no friction?

Answer 1

The minimum speed to take a 108 m radius curve banked at 19° so that a car doesn't slide inward, without friction, is calculated using the equation for centripetal force provided by banking of the road. The calculated speed is approximately 13.4 m/s.

Step-by-step explanation:

To find the minimum speed needed for a car to take a curve banked at a certain angle without sliding inwards, we can use the principles of circular motion and the forces involved in a banked turn. When a car drives around a banked curve without friction, the only forces acting on it are the normal force and the weight of the vehicle. The condition for the car not to slide inward is that the horizontal component of the normal force must provide the centripetal force necessary for circular motion.

For a curve of radius 108 m banked at 19°, the minimum speed can be calculated as follows:

1. Denote v as the minimum speed, r as the radius of the curve and θ as the banking angle.
2. For ideal conditions where friction is absent and no sliding occurs, the following relationship holds: v2/r = g ∗ tan(θ) where g is the acceleration due to gravity (approx. 9.8 m/s2).
3. Plug in the values: θ = 19°, r = 108 m, and g = 9.8 m/s2.
4. Calculate the minimum speed v using the equation: ∗ v = ∗ √ (g ∗ r ∗ tan(θ)) = ∗ √ (9.8 ∗ 108 ∗ tan(19°)) ≈ 13.4 m/s

Therefore, the minimum speed required to take a 108 m radius curve banked at 19° without sliding inward, assuming there is no friction, is approximately 13.4 m/s.