In the figure below, a stuntman drives a car (without negative lift) over the top of a hill, the cross section of which can be approximated by a circle of radius R = 240 m. What is the greatest speed at which he can drive without the car leaving the road at the top of the hill?
To determine the greatest speed at which the car can drive without leaving the road at the top of the hill, we can use the concept of centripetal force.
The centripetal force required to keep the car moving in a curved path is provided by the normal force acting on the car. At the top of the hill, the normal force must be equal to the weight of the car, otherwise the car would lose contact with the road.
Let's break down the steps to find the greatest speed:
Step 1: Find the weight of the car.
The weight of an object is given by the formula:
Weight = mass x gravity
Step 2: Calculate the normal force.
At the top of the hill, the normal force acting on the car must be equal in magnitude to the weight. Therefore, the normal force will also be equal to the weight of the car.
Step 3: Determine the maximum velocity.
The maximum velocity occurs when the normal force is equal to the weight. At this point, the car is just about to lose contact with the road. This means the centripetal force is equal to the maximum frictional force, which is the product of the normal force and the static coefficient of friction.
The centripetal force is given by the formula:
Centripetal force = mass x (velocity^2) / radius
The maximum frictional force is given by:
Maximum frictional force = static coefficient of friction x normal force
Setting the centripetal force equal to the maximum frictional force, we can solve for the maximum velocity.
Step 4: Plug in the values and solve the equation.
By substituting the known values into the equation, you can calculate the greatest speed at which the car can drive without leaving the road at the top of the hill.