How the heights of the hills are designed to allow an empty cart to reach the end of the ride

As it is important to not stop, the friction must be estimated bearing in mind the curves, and the alterations to forces at different points.
This is far from simple.
And the friction may alter as the equipment ages.

So to help, at Luna park in Melbourne there was a brakeman riding on the back of the carriages.
The cars were designed to go a bit too fast and the brakeman would slow them down a bit where necessary.
This gave the margin of error necessary to cope with wind, rain, ice or heat. Which couldn't be factored in at design time.

At the end there is always some surplus speed and once more brakes are applied to ensure the cars stop at the correct point.

It would be impossible to get the cars to travel at the correct speed in a range of different weather conditions. So some process for adjustment at some point, is vital.

If you know the rolling resistance of the cars you can work out the MINIMUM height of the hill required but you must always make it higher than this and that is done by experience