A car approaches the top of a hill that is shaped like a vertical circle with a radius of 58.2 m. What is the fastest speed that the car can go over the hill without losing contact with the ground?
If the car just barely leaves the ground, the centripetal force and gravity are in balance.
V^2/R = g
Solve for V
To find the fastest speed that the car can go over the hill without losing contact with the ground, we need to consider the forces acting on the car at the top of the hill. At the top of the hill, there are two main forces to consider: the gravitational force (mg) and the normal force (N) exerted by the ground on the car.
For the car to stay in contact with the ground, the normal force N must be greater than or equal to zero. If N becomes zero, it means that the car has lost contact with the ground and is in freefall.
At the top of the hill, the gravitational force acting downward is mg, where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s²).
The centripetal force required to keep the car moving in a circular path at the top of the hill is given by Fc = mv²/r, where m is the mass of the car, v is its velocity, and r is the radius of the circular path (the radius of the hill).
At the top of the hill, the net force acting on the car is the sum of the gravitational force (mg) and the centripetal force (mv²/r):
Net force = mg + mv²/r
The normal force (N) that counteracts the gravitational force is given by N = mg.
So, we can rewrite the net force equation as:
mg = mg + mv²/r
Simplifying the equation, we get:
0 = mv²/r
To find the fastest speed without losing contact, we need to find the velocity (v) when the net force is zero. Here, the net force becomes zero when mv²/r = 0.
mv²/r = 0
Since the radius (r) cannot be zero, the only way for the equation to hold true is if the velocity (v) is also zero. However, a car moving at zero velocity would not be able to reach the top of the hill.
Thus, the conclusion is that there is no velocity at which the car can go over the hill without losing contact with the ground. The car would lose contact at the top of the hill regardless of its speed.