V1: That speed at which the pilot can with the critical engine inoperative, An average pilot using normal control inputs can keep control of the A/C,and can decide to safely continue the take off, or safely stop the take off.
Where all this gets jumbled is when we assign V1 a speed, and in some cases where you are limited due to weight and temperature, available runway etc etc, you may need to alter things accordingly.
Ref I think from Pan's Ops 1.
There are some abnomalies with this in the sense of cross wind, VS's critical engine, and the amount of usable rudder you have in a significant cross wind.
In some cases the critical engine will be assessed due to wind direction and strength.
Remember it does indeed help quite significantly if one keeps all there potential debris "on the centerline".
