Question

You are upgrading an amusement park ride by adding a circular loop to an existing roller coaster ride. The first hill for the existing roller coaster is 55 m tall, and you are to build, right at the bottom of this hill, the tallest loop possible without having the cars fall out of the track or the passengers fall out of the cars. The roller coaster starts from rest at the top of the hill. What is the maximum radius for the loop?

Answer 1

Here, H = 55 m = height from which the roller coaster will be released from rest

and lets say that R is the radius of tallest loop possible

Also, we know that for completing a circle, velocity required at the bottom of circle: v

So, by conservation of energy, we can say that Kinetic energy at the start of vertical loop = Potential energy of the roller coaster at height H:

Now,

On solving,

R = 22 m

EDIT : The expression of velocity of roller coaster at the bottom of the loop (i.e., just before entering the loop)

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