A roller coaster car (car A) of mass M free to slide on a frictionless track and

Question

A roller coaster car (car A) of mass M free to slide on a frictionless track and initially sits at rest at the bottom of a circular loop of radius R. At some time later, another roller coaster car (car B) of the same mass crashes into it, and they both begin to move up the loop such that they remain in contact with each other. What is the minimum speed v that car B needs to have before the collision so that both cars make it to the top of the loop without losing contact with the track? (Answer using Momentum and Energy methods).

Explanation / Answer

At the top of the track for minimum velocity the normal force can be zero, hence we can write the centripetal force equation as

Mg - N = Mv^2/R

V =sqrt (gR)

Using the conservation of energy.

Mg2R + 1/2MgR= 1/2MV'^2

V' = sqrt (5gR)

Now using the conservation of linear momentum

MVb = 2MV'

Vb = 2sqrt (5gR)

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