A toy roller coaster contains a loop of radiusR. A toy train, which consistsof a lot of small waggons is moving towards the loop with constant velocity. The lengthof the train isL >2Rπ, and the radius of the loop is much larger than the height of thewaggons. The distance between the waggons and the friction can be neglected. What isthe velocity of the train before the loop if all of the waggonspush the track during themovement of the train?

Question

I tried to solve this question many times but ti does not show any improvement by circular motion , newtons laws , and even equations of motion

Answer 1

at the top of the loop, the force between the track and the wheels becomes zero when the centripetal acceleration is the same as g

g = v^2/R = 9.81 m/s^2
so at the top
v = sqrt(R g)

now we need the speed at the bottom of the loop that will give us that speed at the top
potential energy at top + kinetic energy at top = kinetic energy at bottom if we define potential zero at bottom.
so
(1/2) m V^2 = (1/2) m (R g) + m g (2R)
so
V^2 = R g + 4 R g = 5 R g
or
V = sqrt (5 R g)