Driving in your car with a constant speed of 12 m s you encounter a bump in the road that has a circular cross-section. The radius of curvature of the bump is 35.2 m. At what speed must you go over the bump if people in your car are to feel "weightless"?
To calculate the speed at which you need to go over the bump for people in your car to feel "weightless," we can use the concept of centripetal force.
When a car goes over a bump in the road, the normal force acting on the car changes due to the curvature of the bump. The normal force is the force perpendicular to the surface, which in this case, is the force exerted by the bump on the car.
When the car moves over the bump with enough speed, the normal force becomes zero at the top of the bump, and at that moment, people in the car can feel weightless.
To determine the required speed, we can set up an equation using the centripetal force acting on the car:
Centripetal force = Weight of the car
At the top of the bump, the centripetal force is provided by the normal force:
Centripetal force = Normal force
The weight of the car is given by:
Weight of the car = Mass of the car * Acceleration due to gravity
Since the car is moving with a constant speed, there is no vertical acceleration, and the weight of the car can be expressed as:
Weight of the car = Mass of the car * g
Now, the normal force can be expressed as:
Normal force = Weight of the car - Centripetal force
Since we want the normal force to be zero at the top of the bump for people to feel weightless, we have:
0 = Weight of the car - Centripetal force
Therefore:
Centripetal force = Weight of the car
Now, we need to calculate the centripetal force:
Centripetal force = Mass of the car * (Velocity^2 / Radius of curvature)
Setting the centripetal force equal to the weight of the car, we can solve for the required velocity (v) to go over the bump:
Mass of the car * (v^2 / Radius of curvature) = Mass of the car * g
Canceling out the mass of the car on both sides, we get:
v^2 = Radius of curvature * g
Now, we can solve for the velocity (v) by taking the square root of both sides:
v = √(Radius of curvature * g)
Substituting the given values:
v = √(35.2 m * 9.8 m/s^2)
v ≈ 18.8 m/s
So, you must go over the bump with a speed of approximately 18.8 m/s for people in your car to feel "weightless".