After a quick and dirty launch experiment i dedicated an entire embark to the advancement of the supersonic minecart. Calculations predicted that the height to which transterminal carts can be launched will escalate very quickly: a two-tier launch should be able to hit ~130 z, a three-tier one ~400z, a height that you're very unlikely to actually have available without editing in extra sky levels via DFHack, and even then you'd need a 13-tile long embark. Even on a maximum-length embark, no more than 500 zlevels of height are possible due to the fixed launch angle.
So first order of business - find a nice embark bordering on mountains, make it long (i settled for 8x2), build a collision array as deep underground as reasonably possible, carve out a flight path so the cart can hit the ceiling. Carving out the path was expectedly the hardest part, mainly because i originally underestimated the steepness of the cart's ascent. It made the full flight from 90z above map bottom to 247. 157 z-levels up in 21 steps, 243 tiles forward. The perfect 45° ascent would have hit the ceiling after "only" 235 tiles, the path didn't curve very much.
The cart sequence used was: bloodthorn cart loaded with one iron goblet (57kg) -> applewood minecart (29kg) -> willow minecart (15kg) -> adamantine minecart (8kg).
A bit more precisely, it was
bloodthorn-->apple=>apple(2) apple-->willow=>willow(2) apple-->willow=>willow(2) willow-->adamantine=>adamantine(2)
apple(2)-->willow(2)=>willow(4) willow(2)-->adamantine(2)=>adamantine(4)
willow(4)-->adamantine(4)=>adamantine(eight)
I.e. in the first tier we collide eight carts in pairs to get four carts with double starting velocity; in the second tier, those four accelerated carts are bashed together to give two carts with four times starting speed and the third tier combines the last pair to reach a single cart going at 7-8 times starting speed.
I had speculated on adding one or two tiers to the minecart chain, which would require aluminium for the upper end (108 kg, can be made to pass impulse to a preloaded bloodthorn minecart, e.g. loaded with one bronze pick (4kg) to reach 54 and thus 1/2 the weight of the aluminium cart) and featherwood for the lower end (4kg, can take full impulse from adamantine carts).
The embark had featherwood trees and the dwarven civ had native aluminium, so i was all set. I carefully worked out the build order of the minecarts and the floor plan and got to the four-tier shot without errors (added featherwood cart, no aluminium yet), reaching a blistering 36 tiles per step (13,3 times terminal velocity). During the first shot, a crippled cat was crawling around the test track and thus had the dubious distinction to become
the first known supersonic creature. The cart stopped moving entirely and the cat shot away at 20 tiles per step, by the canonical conversion 530 m/s, i.e Mach 1,55. After seven steps (and ~130 tiles) it unfortunately touched the wall and blew apart.
After an undisturbed test shot i doubled the size of my collision assembly yet again (to 32 carts) and got the aluminium cart into the mix. After two false starts (vitally important tracks upside down and thus the wrong collision order; wrong distance to the final collision) the third attempt worked out perfectly and i got my five-tier transterminal shot. The featherwood cart reached a top speed of 62,5 tiles per step, 23 times terminal velocity. By the standard conversion of 1 tile/step = 96 km/h (60mph, 26,667 m/s), that's 1667 m/s - a bit over one mile per second or just under Mach 5. Even the exceptional transterminal friction took almost half a dwarf day to drag the cart down to terminal velocity; the cart could have travelled through 23 full world tiles in that time. A ballistic shot could have reached a height of 4300 zlevels.
Since the fully elastic collisions have rather strict conditions - pushing cart no more than twice as heavy than the pushed cart, no excessive speed difference - each tier of collisions takes twice the number of carts to achieve (you have to "pre-heat" the carts so they can actually take the heightened impulses). A five-tier shot takes thirty-two carts (one featherwood, five adamantine, ten willow, ten apple, five bloodthorn, one aluminium). I used impulse-ramp cyclotrons to reach terminal velocity, with carts dropped in via hatch and released into the collision array via doors. Everything was operated by two levers, one linked to 32 hatches, the other to 32 doors. Much fun was had linking it all up.