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For a long time, competitive teams have been using methods of self-damage such as the Boston Basher or explosive damage to keep themselves below full overheal and therefore maximise the speed at which they can obtain 100% ubercharge, and it had been thought that 40 seconds was the fastest possible time, but since the Crusader’s Crossbow was changed to give uber on hit, this time has actually been much shorter.
Ever since the Tough Break update, the Crusader’s Crossbow has also given a small amount of uber per arrow, but only recently has the power of arrow building really come to light (for example in the recent Grand Finals of ESEA’s invite 6s league). I thought I’d do some rough maths to see how fast an uber really could be built.

The most efficient method of building during a stalemate would be to have both a Demoman and Scout deal damage to themselves, the Medic arrowing the Demoman at range and healing the Scout with the Medigun while the Crusader’s Crossbow reloaded.
The Ubercharge of the stock Medigun builds at 2.5% per second if the heal target is at less than 142.5% health (150% being a full overheal). The Crossbow, however, provides 1% per 16 health healed by each shot. As it can heal a maximum of 150 health per arrow, it can give up to 9% uber per arrow.

the fun part

The Crossbow has a 1.6 second reload speed, meaning that using arrows alone it would be possible to go from 0-100% (Well, technically 99%) in a mere 17.6 seconds (not including the travel time of the arrow from the Medic to the heal patient.
The method used by top teams, however, also includes healing a second patient while waiting for the Crossbow to reload. Assuming that the Medic, being a human and therefore liable to judgmental errors, was to fire the Crossbow every 2 seconds and heal in between, then they would be gaining 9% uber every 2 seconds, as well as an additional 2.5% between firing (weapons have a 0.5 second switch speed, meaning that switching to and from the Medigun would take 1 second, so only 1 second of healing would actually take place), meaning that they would be gaining 11.5% uber every 2 seconds, or 5.75% per second; meaning that going from 0-100% uber would take 17.3913043478 seconds.
Not a huge difference, I know, but a difference nonetheless.

Ok. Say the Medic was some kind of cyborg that could perfectly deal with time in tenths of a second. What is the fastest that they could POSSIBLY get a full ubercharge?

They would be gaining 9% every 1.6 seconds from the crossbow, and then (2.5×0.6 = 1.5)% in between that from the Medigun, meaning that they would be building uber at a rate of 10.5% every 1.6 seconds (or 6.5625% per second), and therefore would have gone from 0-100% ubercharge in 15.2380952381 seconds.

Wow.

40 seconds build time, huh?

https://www.youtube.com/watch?v=SY4Vv0sjjtM