From here we look at the push of the rail itself. It has nothing to do with the speed, but it is the force the rider actually feels.
Moving in a circle takes a force pointing at the centre. Its size is , and it is called the centripetal force. The faster the motion and the tighter the turn, the larger it is.
The centripetal force is not a new force. It is a way of saying that the forces actually acting, once added together, point at the centre and come to that size.
Only two forces act on the cart, gravity and the push of the rail. So the sum of those two has to be . Everything that follows is just doing that addition.
Take the bottom of the dip at the end of the descent. The rail curves upward here, so the centre of the curve is straight overhead.
The forces are gravity downward and the normal force upward. The centre is above, so taking upward as positive gives .
Solved, , which is larger than the weight. The faster the cart and the tighter the dip, the larger it grows.
This is the force with which the seat pushes on the rider. The body feels heavy through the bottom of a valley because has been added on.
Now the crest of a hill. The rail curves downward here, so the centre of the curve is straight below.
The forces point the same way as in the valley, gravity down and the normal force up. Only the direction of the centre has changed. Taking downward as positive gives .
Solved, . This time it is a subtraction, and the result is smaller than the weight.
The figure raises the speed and the arrow shrinks. The faster the cart crosses the hill, the more weakly the seat pushes on the rider. That is the feeling of floating out of the seat.
Being a subtraction, it must reach eventually. when , that is, when .
At that moment the seat is not pushing on the rider at all. Gravity alone is bending the body along the curve. The rider is falling, and falling exactly along the shape of the hill.
Go faster than this and the push of the rail would have to be negative, which is to say it would have to hold the cart down rather than push it up. A cart with no wheels gripping from above leaves the rail there and flies.
That same equation becomes the condition for the loop in the next lecture. Whether the rider floats at the top of a hill and whether the cart makes it round a loop are one and the same equation.