How fast can the car take this curve without skidding to the outside of the curve? The acceleration of gravity is 9.8 m/s 2 . Answer in units of m/s.

Question

A highway curves to the left with radius of curvature of 34 m and is banked at 29 ◦ so that cars can take this curve at higher speeds. Consider a car of mass 1671 kg whose tires have a static friction coefficient 0.89 against the pavement. How fast can the car take this curve without skidding to the outside of the curve? The acceleration of gravity is 9.8 m/s 2 . Answer in units of m/s.

Answer 1

Hi Christina,

The formula for the maximum speed of a car taking a banked curve without skidding is derived by considering the forces acting on the car in the radial (horizontal) direction. The key forces involved are the gravitational force and the frictional force.



Here are the main steps in the derivation:

1. Forces in the Radial Direction:
2. The forces in the radial direction include the gravitational component mgsin(θ) (directed toward the center of the curve) and the frictional force f (directed toward the center of the curve).
3. Centripetal Force Requirement:
4. To keep the car moving in a circular path, the net radial force must provide the necessary centripetal force.
5. The centripetal force required is m⋅ac​, where ac​ is the centripetal acceleration.
6. Centripetal Acceleration:
7. The centripetal acceleration ac​ is related to the radius of curvature R and the velocity v by ac​=Rv^2​.
8. Equating Forces:
9. Equate the forces in the radial direction to satisfy the centripetal force requirement: mgsin(θ)+f=m⋅Rv^2​
10. Frictional Force:
11. The frictional force can be expressed as f=μs​⋅N, where μs​ is the static friction coefficient and N is the normal force.
12. Normal Force and Components:
13. Break down the normal force N into its vertical Ncos(θ) and horizontal Nsin(θ) components.
14. Substitute and Simplify

Substituting for the vertical normal component into the above equation and isolating for v, we get this equation.

1. v= sqrt [ R⋅g⋅( sin(θ)+μs​⋅cos(θ) )​ ]