A car that is initially at rest at the top of a roller-coaster track moves down the track without friction. What must be the maximm initial height from which the car descends so that it will not fly off the 10-m-high circular hill at the bottom?

To determine the maximum initial height from which the car can descend without flying off the circular hill at the bottom, we need to consider the forces acting on the car at different points along the track.

At the top of the roller coaster track, the car is at rest, so it has potential energy (mgH) but no kinetic energy. As the car descends, this potential energy is converted into kinetic energy.

When the car reaches the circular hill at the bottom, the car is moving with a certain velocity (v) which depends on its initial height (H). At the top of the hill, the car experiences two forces: the gravitational force (mg) acting vertically downwards and the centripetal force (mv²/r) acting towards the center of the circular path.

To prevent the car from flying off the circular hill, the centripetal force must be greater than or equal to the gravitational force. Therefore, we can set up the following inequality:

mv²/r ≥ mg

where m is the mass of the car, v is its velocity at the top of the hill, r is the radius of the circular hill, and g is the acceleration due to gravity.

Rearranging the equation, we can solve for the minimum velocity required:

v ≥ √(rg)

Now, we can relate the velocity to the initial height using conservation of energy. At the bottom of the hill, the car has lost the height H and gained kinetic energy:

mgH = 1/2 mv²

Rearranging the equation, we find:

v² = 2gH

Substituting this expression for v² into the inequality, we have:

2gH ≥ rg

Cancelling out 'g' on both sides, we get:

2H ≥ r

Since we know the radius of the circular hill (r = 10 m), we can solve for the maximum initial height (H):

2H ≥ 10 m

H ≥ 5 m

Therefore, the maximum initial height from which the car must descend so that it will not fly off the 10-m-high circular hill at the bottom is 5 meters.