A roller coaster consists of a large hill, followed by a loop with a radius of 2

Question

A roller coaster consists of a large hill, followed by a loop with a radius of 20 meters. The roller coaster car starts at the top of the hill, gains speed as it rolls down, and then travels around the loop, starting at the bottom and turning the riders upside down at the top. You may neglect friction and air resistance. (For clarification: the bottom of the hill is at ground level, as is the bottom of the loop).
a. How high must the hill be to ensure that the car doesn't fall off the loop? [Hint: first, find symbolically the minimum velocity required to complete the loop. Substitute numbers at the very end.]
b. Why does the exact shape of the hill not matter? A roller coaster consists of a large hill, followed by a loop with a radius of 20 meters. The roller coaster car starts at the top of the hill, gains speed as it rolls down, and then travels around the loop, starting at the bottom and turning the riders upside down at the top. You may neglect friction and air resistance. (For clarification: the bottom of the hill is at ground level, as is the bottom of the loop).
a. How high must the hill be to ensure that the car doesn't fall off the loop? [Hint: first, find symbolically the minimum velocity required to complete the loop. Substitute numbers at the very end.]
b. Why does the exact shape of the hill not matter?
a. How high must the hill be to ensure that the car doesn't fall off the loop? [Hint: first, find symbolically the minimum velocity required to complete the loop. Substitute numbers at the very end.]
b. Why does the exact shape of the hill not matter?

Explanation / Answer

The initial gravitational potential energy should be managed by the kinetic and potential energy at the top inverted position in the loop.

m g H = (1/2) mv^2 + m g (2R)

R is the radius of the loop, H is the initial height, v is the speed at the top position of the loop.

To complete the loop the minimum speed should be V2 = R g , now the equation

m g H = (1/2) m (Rg) + m g (2R)

H = 5R / 2

The minimum height to needed for the coaster to complete the loop is 5R/2.

so H = 5 (20) / 2

H = 50m

b)

Shape does not matter becomes energy is converted from one form to the other such that total energy of the system remains unchanged. m g H is the total energy possed which is a constant through out though.

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