A highway is being constructed to accommodate traffic for speeds of 22.8 m/s. If the angle of the bank is 6.28° and there is no friction force, what is the highway's curve radius?
To find the curve radius of the highway, we can use the equation that relates the speed of the vehicle, the angle of the bank, and the radius of the curve. The equation is:
(radius) = (speed)^2 / (acceleration)
Since there is no friction force perpendicular to the banked road, the only force acting on the vehicle in the horizontal direction is the centripetal force, which is provided by the gravitational force:
centripetal force = gravitational force
(mass of the vehicle) * (acceleration) = (mass of the vehicle) * (gravitational acceleration)
Since the mass of the vehicle cancels out, we can write:
acceleration = gravitational acceleration
The gravitational acceleration is approximately 9.8 m/s^2.
Now, let's substitute these values into the equation:
(radius) = (speed)^2 / (acceleration)
(radius) = (22.8 m/s)^2 / (9.8 m/s^2)
(radius) = 530.74 m^2/s^2 / 9.8 m/s^2
(radius) = 54.19 m
Therefore, the curve radius of the highway is approximately 54.19 meters.