It is certainly true that when looking up the various V speeds (V1, Vr V2 etc) you do not need to know the field length.
Vmu is the minimum speed at which the aircraft can generate enough lift to rise up off the runway surface. But if we permit it to lift off at Vmu then any minor reduction in airspeed, due for example to a drop in headwind, will cause the aircraft to sink back onto the runway. To avoid this we plan to make the aircraft lift off a Vlof (Velocity lift off) which is slightly higher than Vmu. I cannot recall the exact figure, but it is about 5% higher than Vmu.
To ensure that the aircraft lifts off at Vlof we must begin to rotate it into the take-off attitude at the correct speed. The speed at which we must begin this rotation is Vr. The difference between Vr and Vlof is determined by aerodynamics.
Before beginning the nose up rotation we must make the decision to continue or to abort the take-off. The time between rotation and lift-off is often very short. So if we rotate then change our minds and push it back down again, it is quite possible that we will have-lifted off and landed again. That may be OK when you have lots of spare runway available, but that will not always be the case.
V1 is the take-off decision speed. There has been much debate over the years about how exactly it would be better described as the take-off action speed. Essentially V1 is the speed at which, the take-off must be continued unless the first actions have been taken to reject the take-off. But there may of course be some overriding reason to prevent the take-off from being continued. If for example a wing drops off at V1 + 1 knot, your only choice is to attempt to stop.
Logically we must made the decision to take-off at or before the point at which we begin the rotation to the take-off attitude. So V1 must not be greater than Vr.
We now have a logical sequence of speeds V1, Vr, Vmu, Vlof. The magnitudes of these speeds are determined by aerodynamics and have nothing to do with the field lengths available (TORA, TODA, ASDA).
But that does not mean that V1 has nothing to do with field lengths available!
Imagine that we have an extremely long field in which the distances available are far in excess of those that will be required by the aircraft at its current take-off wait.
We start the take-off roll and accelerate down the runway. Eventually we will reach the speed at which we can suffer a single engine failure and still have enough TODA to reach screen height at V2 within the remaining TODA. The speed at this point is called Vgo and it is the minimum value for V1 for this combination of aircraft weight, atmospheric conditions and field length.
Now let’s suppose that instead of rotating, we keep the nose on the runway and keep on accelerating. We will eventually reach a speed at which we have just enough ASDA remaining to bring the aircraft to a stop if we decide to reject the take-off. This speed is called Vstop and it is the maximum value of V1 for this combination of aircraft weight, atmospheric conditions and field length.
Because the field is much longer than the aircraft requires, there is a gap between Vgo and Vstop. But at any speed within this gap the aircraft is capable of continuing the take-off following a single engine failure, or aborting the take-off, within the distances available. This means that we have a speed range within which we can select our decision speed V1.
If we gradually increase the aircraft weight, the value of Vgo and the amount of runway used to accelerate to Vgo will increase. And the value of Vstop and the distance required to accelerate to Vstop will both decrease. This means that the allowable range of speeds between Vgo and Vstop, within which we must select V1, will gradually reduce.
If we continue to increase the aircraft weight (or reduce the field lengths available) we will eventually get to a point where Vgo and Vstop are equal. We then have only one choice for the value of V1. In this condition when we reach V1 we have just enough TODA available to complete the take-off, with one engine failed, and get to screen height at V2 within the remaining TODA. We also have just enough distance to abort the take-off and stop the aircraft within the remaining ASDA. In this condition the aircraft is said to be field limited.
We now have a relationship between V1 and the field lengths in that when carrying out a field limited take-off V1 is the speed at which we have just enough TODA available to complete the take-off, with one engine failed, and get to screen height at V2 within the remaining TODA. We also have just enough distance to abort the take-off and stop the aircraft within the remaining ASDA.