V1 is the decision speed with respect to a single engine failure in a multi-engine aircraft. It's the speed below rotation speed at which the decision to abort safely can no longer be made.

Captains can make the decision to abort the takeoff in the case of absolute power loss or for 'failure to fly' (where the aircraft is clearly not going to fly, e.g. the elevator/pitch controls aren't responding). But the training is adamant: if you're uncertain what has happened after V1 you try to fly the plane away from the runway.