If a flight crew begins their takeoff roll and don't have slats and flaps configured properly, when does the system know when to sound the alarm? If it actually sounds during the roll itself, that seems a bit too late, as there then isn't enough time to deploy those surfaces. And it doesn't seem likely that the crew would have the time to look up a new V2 to figure out at what speed they can take off without flaps and slats.
For the Boeing 727-100 and 747-100/200 the answers are:
when does the system know when to sound the alarm?
When the thrust levers are advanced to begin the takeoff, if flaps and slats and a few other things are not in takeoff position, a loud horn sounds.
figure out at what speed they can take off without flaps and slats.
Takeoff is not permitted without flaps and slats. You have to have them. Depending on how many were not in the takeoff position, the takeoff speed would be inordinately high, require runway lengths not available, and might be above the maximum tire speed. In short, as I said, you have to have them.
On the aircraft I mentioned, if you're starting the takeoff roll from a stationary position and advance the thrust levers at the usual rate, you'll get the horn before your ground speed is little more than walking speed, if that. The thrust lever position, while above anything you would use for taxi, is not anywhere near what you could take off with. It's not far along the travel of the thrust levers from idle to full throw.
The procedure taught for a takeoff warning horn at the carriers I flew for was to bring the thrust levers all the way back to idle, brake to a stop, and request clearance from the tower to vacate the runway. Once off the runway, brake to a stop and troubleshoot the problem. In no case were you to troubleshoot the problem while moving on the runway.
In practice, if there was no traffic, once stopped some would stay on the runway to troubleshoot after getting tower clearance to do so. The assumption was that the problem would be quickly corrected and takeoff resumed.
Troubleshooting was a simple matter of checking that everything that was supposed to be done had been done. For the 747, the items that would trigger the takeoff warning horn were:
- leading edge flaps not extended, there were lights on the f.e. panel that showed their individual positions,
- stabilizer not in the green band, there was an indicator by the trim wheel on the center console.
- flaps not in 10 or 20 degrees position
- body gear not centered, lights on the f.e. panel showed its position.
- speed brake lever not in the down detent.
Recalling these items from memory was a standard question on oral exams.
If everything had been done that was supposed to be and you still got the horn, it was usual to recycle everything. If that didn't fix the problem, which if the problem was the warning system itself would be the case, it was back to the gate for maintenance.
On most aircraft I have flown, the Takeoff Config warning sounds when the thrust levers are advanced for takeoff power.
The configuration horn goes off as soon as the aircraft decides the crew is trying to take off instead of just taxi - it's based on the throttle setting. But the rest of your question has an unspoken inference that continuing a takeoff beyond the point where a configuration warning activates is even conceivable in a professional environment. The correct crew response would be an immediate abort and identification of the problem once clear of the runway. Anything else would be unacceptably risky and not justifiable under any circumstances.