THE RAMP
Loop-the-loop by the energy ledger — clear it only if released from h ≥ 2.5r, mass be damned.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
The question
A cart is released from height h on a frictionless track that runs down into a vertical loop of radius r. Will it make it all the way around?
Energy
With no friction, the total energy E = mgh never changes. Kinetic and potential trade back and forth, but at every point KE + PE adds up to the same E.
The loop test
At the top of the loop gravity supplies the turning pull, so the cart needs v² ≥ gr to stay on the track. Feeding that back through energy gives a minimum release height h_min = 2.5r.
The ride
Down the ramp and into the loop: speed is set only by the height change, v = √(2g(h−y)). If h < r the energy reaches zero before the cart can climb to the side, so it stops and rolls back; higher failed runs lose contact instead.
The verdict
Three branches close the books: below r the cart stalls and rolls back; from r to 2.5r it loses contact; at or above 2.5r it clears. Every boundary is mass-independent.