Here is how I see this, theoretically speaking. For some reason an actual Boeing take off speed table tells a different story...
According do Oxford aviation academy performance book, chapter 14 pages 375 and 376
Regarding
Vr:
May not be less than:
V1
1,05 VMC
Speed such that V2 may be attained before 35ft
A speed that, if the aeroplane is rotated at its maximum practical rate, will result in VLOF of not less than 1,1VMU (1,05VMU one eng. inop case)
At a low density altitude (high density) the 1,05Vmc requirement will be limiting for Vr, i.e. the rotation speed is governed by the requirement that we can not start the rotation at a speed lower than 1,05Vmc .
As density altitude increases (lower density) the Vmc is reduced as well, causing the Vr to reduce with it.
Eventually the Vr becomes limited by the maximum practical rotation rate due to the 1,1 Vmu requirement. That is, if we would rotate at a speed equal to 1,05Vmc we would reach Vlof at a speed lower than 1,1Vmu. So from a certain density altitude, going up, the Vr will increase.
As for the
V2 speed:
V2min in terms of CAS may not be less than
1,13 VSR
1,1 Vmc
At a low density altitude the V2 will initially be limited by the 1,1Vmc requirement. As the density altitude gets higher, the Vmc is reduced and so is the V2 with it until reaching the 1,13Vsr limit where the V2 becomes a fixed number.
