A car in an amusement park roller coaster ride rolls without friction at the top

Question

A car in an amusement park roller coaster ride rolls without friction at the top of a hill. The car begins at a height h from the top of a hill. A the bottom, the car then goes through a vertical loop where the car is upside down at the loop's top. If the radius of the loop is 20.0m, what is the minimum height h such that the car moves around the loop without falling off the track at the top of the loop while upside down?

If the height is instead 70m, what is the car's speed at the top of the loop?

From a 70m height, what is the magnitude of the car's radial acceleration?

From 70m, what is the magnitude of the car's tangential acceleration?

Explanation / Answer

Does the car start at height h and enter a loop of height 70 m? you can use the change in energy to determine the speed Initial energy: m*g*h Potential Energy at top of 70*g*h, so the rest of the energy m*g*(h-70)=1/2 *m *v^2 or if the speed is 0 at the top of the loop, then the speed at the bottom is given by the equation v^2 = 2*a*d, so v = sqrt(2*9.8*70)=37 m/s. Then the radial acceleration will be v^2/r = 37^2/35 meters. At the bottom of the loop, tangential acceleration would be 0.

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