Dimensional analysis check

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.

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Dick
Homework Helper

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