A transportation engineer would like to design a banked highway curve so that the tires on a vehicle going 45 mph will not experience any transverse force. What is the best angle?

Forces on the Vehicle

An object moving in a curved path will experience acceleration in the normal direction. This translates to a force that must be resisted or the object will slide off the curved path. For a car, the resistance to sliding off a road is the friction between the tires and road.

First, find the velocity of the car in standard units,
     v = (45 mph)(5,280 ft/mile)(1 hr/3,600 s)
       = 66.00 ft/s

Normal acceleration,
     a_n = v²/ρ = v²/r = 66.00² / 200
         = 21.76 ft/s²

Construct a free-body diagram with acceleration, and then sum forces,
     ΣF_x = m a_x
     m g sinθ = m [cosθ a_n]
     g tanθ = a_n
     (32.2 ft/s²) tanθ = 21.76 ft/s²

θ = 34.07^o