Ah, the coveted V2 speed....
So, many teach, incorrectly, that V2 is your best angle of climb speed with one engine inoperative. This is incorrect. The truth is that V2 is the MINIMUM speed at which the aircraft will achieve the certification-required climb gradient with one engine inoperative.
In reality, a speed slightly higher than V2 will provide a slightly better climb gradient with one engine inoperative. "Slightly" depends on a number of factors...I do not have the expertise to precisely delineate those factors....suffice it to say, it depends on wing design, etc.
The aircraft, when certified, must meet certain performance criteria with one engine inoperative. Ultimately, the aircraft, in order to meet these requirements, must adjust takeoff weight. Thus, for a given runway length, existing obstacles, etc., the aircarft has a maximum weight (temperature, pressure, etc.) to achieve these requirements.
You are to be at this minimum speed (V2) by 35 feet, gear retracted, after losing an engine at V1 and continuing the takeoff. You need "X" amount of runway to achieve this....given weight, temperature, pressure. (15 feet, instead of 35 feet, if using a 'wet' runway.)
But, let's say that you have a really long runway, but you have obstacles in the departure path. In this hypothetical example, runway limits are not a problem (accelerate stop, accelerate go).....since you have much more runway than needed....but, need a really good second-segment climb gradient to miss the obstacles. Or, to put it another way, your take off is second-segment-climb limited. (In this example....)
So, why not allow the aircraft accelerate to a slightly faster speed for V1, for Vr, and for V2? Why not! You have plenty of runway in this example. No need to worry about stopping in the case of a rejected takeoff, since you have much more runway than needed. Use a higher V2 to get a slightly better second-segment climb gradient, taking advantage of the excess runway you have.
So, this is what you do....you increase the speeds....so that, after takeoff, your 'new' V2 speed gives you a slightly better climb gradient.
So, typically, as a general rule, you use this 'improved climb' when you are second-segment climb limited, but have excess runway for your takeoff run.
Fly safe,
PantLoad