A car speeds over the top of a hill. If the radius of curvature of the hill at the top is 14.27 m, how fast can the car be traveling and maintain constant contact with the ground?
To find the maximum speed at which the car can be traveling and maintain constant contact with the ground, we need to consider the gravitational force and the centrifugal force acting on the car.
The maximum speed will occur when the gravitational force and the centrifugal force are equal. At this point, the car will not be able to maintain contact with the ground if it goes any faster.
The gravitational force is given by the equation:
F_gravity = m * g
where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The centrifugal force is given by the equation:
F_centrifugal = m * v^2 / R
where v is the velocity of the car and R is the radius of curvature of the hill.
Setting the gravitational force equal to the centrifugal force, we can solve for the maximum velocity:
m * g = m * v^2 / R
Canceling out the mass (m) from both sides of the equation, we get:
g = v^2 / R
Rearranging the equation to solve for v, we have:
v = sqrt(g * R)
Substituting the values for g (9.8 m/s^2) and R (14.27 m), we can calculate the maximum velocity:
v = sqrt(9.8 * 14.27) ≈ 10.68 m/s
Therefore, the car can travel at a maximum speed of approximately 10.68 m/s (or about 38.45 km/h) and maintain constant contact with the ground.