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v = √gr

Explanation:

In order for the roller coaster to make it all the way around the loop, the centripetal force at the top of the rider must be equal to the weight the student. So, that the student does not fall during the motion. Therefore,

Centripetal Force = Weight

mv²/r = mg

v²/r = g

v² = gr

Taking square root on both sides, we get:

v =  √gr

where,

v = speed of roller coaster

r = radius of curvature

g = acceleration due to gravity

Hence, the minimum speed the roller coaster can maintain and still make it all the way around the loop, is found to be:

v = √gr