Finding the Minimum Velocity for Sustaining Flight Using Dimensional Analysis

[Beer-monster]

Homework Statement

An flying object of mass M experiences a lift force and a drag force dependent on the it's velocity:

$$F_{l} \propto Wv^{2}$$
$$F_{d} \propto Av^{2}$$

Where W and A are wing surface area and "cross-sectional" area.

To sustain flight the object must fly at a minimum velocity which scales as $M^{k}$ where k is a constant.

What is k?

Homework Equations

For the object to remain in flight it must produce a lift force equal or greater than it's weight i.e.

$$Mg = F_{l(min)} \propto Wv^{2}$$

The Attempt at a Solution

I have found an answer but I'm not too happy with how I got there. Basically, I subbed the scaling factor for minimum velocity into the above equation:

$$Mg \propto W(M^{k})^{2} \propto WM^{2k}$$

As the left side only has mass to the power 1 and the right has mass to the power 2k we can get:

$$2k=1$$

So k = 1/2 or $v_{l(min)} \propto \sqrt{M}$.

This results makes some sense to me physically, but I'm not happy with my approach as it seems a bit simplistic.

 

[Dick]

Homework Statement

An flying object of mass M experiences a lift force and a drag force dependent on the it's velocity:

$$F_{l} \propto Wv^{2}$$
$$F_{d} \propto Av^{2}$$

Where W and A are wing surface area and "cross-sectional" area.

To sustain flight the object must fly at a minimum velocity which scales as $M^{k}$ where k is a constant.

What is k?

Homework Equations

For the object to remain in flight it must produce a lift force equal or greater than it's weight i.e.

$$Mg = F_{l(min)} \propto Wv^{2}$$

The Attempt at a Solution

I have found an answer but I'm not too happy with how I got there. Basically, I subbed the scaling factor for minimum velocity into the above equation:

$$Mg \propto W(M^{k})^{2} \propto WM^{2k}$$

As the left side only has mass to the power 1 and the right has mass to the power 2k we can get:

$$2k=1$$

So k = 1/2 or $v_{l(min)} \propto \sqrt{M}$.

This results makes some sense to me physically, but I'm not happy with my approach as it seems a bit simplistic.

Seems fine to me. If you increase the mass by a factor of $2$ you need to increase the velocity by a factor of $\sqrt{2}$, all else being fixed. What seems simplistic about it to you?

 

[haruspex]

Maybe you are supposed to assume that the object is to be scaled linearly at constant density. Thus W and A will also change.