Do smaller aircraft have lower take-off speeds? Assuming there are two aircraft, each of the same density and each the same shape, am I correct in understanding that the smaller aircraft would have a lower take-off speed? I have explained how I have come to this conclusion below;
Given both aircraft are the same density, the weight of each aircraft would be proportional to the length cubed. Therefore;
$W = A l^3$
Likewise, given both aircraft have the same shape, the surface area would be proportional to length squared. Therefore;
$S = Bl^2$
The lift equation is of course;
$L = \frac{1}2C\rho v^2 S$, where;


*

*$C$ = Co-efficient of lift

*$\rho$ = Air density

*$v$ = Velocity of aircraft

*$S$ = Surface area of wings


Given the shape of both aircraft are the same, $C$ should be the same for both aircraft. The same will be true for $\rho$, as they are travelling through the same air density, so these (and $\frac{1}2$) can be replaced by a constant, $D$. Therefore;
$L = Dv^2S$
As $S=Bl^2$, $L = DBv^2l^2$, where $D$ and $B$ can be combined to form a new constant, $E$. Therefore;
$L = Ev^2l^2$
At take-off speed, lift will equal weight. Therefore;
$Al^3=Ev^2l^2$, so $v^2=\frac{Al^3}{El^2}$. With $\frac{A}{E}$ forming a new constant, $F$, this simplifies to $v^2=Fl$. Therefore;
$v \propto \sqrt{l}$.
Does this therefore mean that a 1:100 scale model of an aircraft would have a takeoff speed 10 times lower than that of the original aircraft, or is there something that I have not taken into account?
 A: It is true in general that smaller aircraft have lower takeoff speeds than larger ones, but the relationship is complicated by a variety of factors, as follows.
A small aircraft (say, an Ercoupe) has a relatively small speed range over which it operates: takeoff speed ~70 MPH, maximum speed ~110 MPH and its wing profile is a compromise between low and high speed performance. That compromise allows it to get away without the flaps, slats and/or leading-edge droop that large planes use to configure their wings for landings and takeoff (150-175 MPH) and then reconfigure them for high speed cruise (550-575 MPH). This means that the large plane's wing area and coefficient of lift are not constant: they are radically different at takeoff than they are for cruise in a big plane. 
A: Twice as big airplane has 8 times bigger mass, but just 4 times bigger wing area, giving at the same speed just 4 times bigger air lift. So it needs $\sqrt{2}$ higher speed, as the lift is proportional to $v^2$.
Edit: Similarly, 1:100 scale model would need  $\sqrt{(10^6/10^4)}=10$ times lower speed.
Note that all is considered as a rough estimation, based on pure geometrical similarity. If material strength and total flight mass are considered, the exponent $a$ in $m=L^a$ will be less then 3,but more then 2.
Another factor is that bigger airplanes can afford variable wing geometry due flaps, slots and similar.
A: The formulas are correct, the conclusion is also definitely correct. A paper airplane flies at a lower minimum speed than a commercial airplane, even though its density is higher! This would seem to be an advantage in favor of small flying objects but actually the opposite is true:


*

*First of all you DO WANT to fly faster - you will get to the destination earlier ;-)

*Small, light objects flying much faster than their stall speed are very unstable because some unbalanced or asymmetric force from the air flow ($L$ is your example) can make them tumble or flip, since it is much bigger than their gravity and inertia (which is $W$ in your notation), so keep in mind that $W<<L$ is highly unstable.


There is a just a small inadequacy in your argument. There are limits because of the fixed strength of materials, so in practice the assumption of fixed density may be wrong. I am not saying it is wrong, but that requires further examination at engineering level.
A: Your conclusion is generally correct, but if you have ever been inside a big airliner, you will have noticed that it is not a solid piece of metal, but hollow on the inside. So the assumption that mass scales with the cube of length is not entirely correct.
If you look at existing designs, mass scales with an exponent of length that is between 2.2 and 2.4 given that the remaining parameters stay comparable (please do not compare jets with piston aircraft, for example!). So there is indeed an increase in wing loading as scale goes up, but a very moderate one. The cube law does help the larger airplane, however, to achieve a much higher range and limits its cargo capacity by mass rather than volume.
As @NielsNielsen points out correctly, larger aircraft change their wing shape a lot in order to combine high wing loading with low landing speeds. They can afford to do this because the thrust of jets (and even turboprops) does not drop with speed as much as that of a piston aircraft does. That allows them to cover a higher speed range, and for high speed a high wing loading is a big advantage.
