Dave said:
If the helicopter only needs to hover, then the induced velocity can be 500 ft per second (through a minuscule rotor) or 0.5 ft per second (through a monstrous rotor), since the craft is not going anywhere.
If the craft is required to climb at, say, 50 ft per second, then the induced velocity must be at least 50 ft per second.
The same thing applies for forward flight and this is probably one reason why faster helicopters have a higher disk loading and a smaller rotors.
Not quite right.
The induced velocity is the vector difference between the inflow velocity and the outflow velocity. It can be small compared to these velocities. For example, a helicopter that has a 10 ft/sec downwash hovering will have something like (but not exactly) a 50 ft/sec inflow and a 60 ft/sec outflow while climbing at 50 ft/sec. It's the change in momentum that matters. I believe you have a copy of Stepniewski's "Rotary-Wing Aerodynamics", it is explained in the early chapters of that book.
I'm not quite sure why higher disk loadings are favoured for faster helicopters. I'm sure Nick Lappos has posted about it in the past. Maybe I'll check Stepniewski tonight.