How much speed does a down ramp impart to minecarts?

Solution 1:

While I have not done the !!Science!! myself, according to the wiki speed is reduced by a certain amount, based on the tile type, per tick the cart stays on the tile. Thus the calculations for exactly balancing this equation are somewhat complicated and depend strongly on the initial speed of the minecart going into the system. If an actual minecart enters the system going too fast it will continue to speed up and if it enters too slowly it will crawl to a stop.

However, later on the same page of the wiki there are suggestions for controlling speed using controlled derailment. They show a single Z-level 5x5 construction which will slow any entering minecart to less than 1 tile every 2 ticks (derailment speed). I will reproduce (read: copy and paste) it here:

     OOOO
in  ═╔═╗O
out ═╬═╝O <- if going to fast, derails onto track stop path from here.
    O╚S╝O <- start of track stop path.
    OOOOO

Where O are the walls and S is a track stop at high friction or less.

Putting one of these every few levels (probably at least 3 levels between control points), will keep your maximum speed from getting too high, and also produces an output speed that is stable under variation of the input speed, unlike trying to exactly balance friction and acceleration.

In regards to your secondary question. Since the same page in the wiki states that rollers can be built on ramps to drag the cart up them, you should not need more rollers than you have z-levels. Beyond that it is rather vague on the physics of rollers. That is a rather large upper bound to work with though, since you have stated that this will cover fifty z-levels.

However, if you are not above using something that could be classified as a bug, take a look at the impulse ramps, which would allow you to move cart upwards without using rollers. Specifically if a cart derails onto an upward ramp it gains speed as though it had come down the ramp. This allows you to build up speed without dropping z-levels or using power. This technique, however, seems rather sketchy to me, and while I might use it I wouldn't feel comfortable showing the resulting fortress to other people.