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I've found two simple relationships that both make sense, but both (seem to) contradict each other.

Thrust is equivelant to change in momentum:

T = Mdot*v

Power is the integral of thrust with respect to velocity:

P = int(Mdot*v)dv = 0.5*Mdot*v^2

Mass flow rate is ρAv, hence:

T = ρAv^2 and

P = 0.5*ρAv^3

Or, rearranging gives:

P^2 = (T^3)/(4ρA)

I can follow that fine and it makes sense to me. But there's also an alternative solution:

Work = Force * distance, so:

Power = Work/Time = Force *Velocity = Tv = ρAv^3

Rearranging gives:

P^2 = (T^3)/(ρA)

This is the same as before but without the factor of 1/4. Both make sense to me but obviously can't both be correct. Could anyone provide some guidance on the matter?

(If it helps, both formulas are contained on wikipedia: http://en.wikipedia.org/wiki/Thrust . My scenario concerns a motor spinning a propeller to provide thrust for an aeroplane, which seems to be more in line with the first equation. But given that the first equation gives a lower power requirement I'm really more inclined to go with the second just to be safe.)