# Derive formula for air resistance $F = \frac{1}{2}CAdv^2$ through dimensional analysis

I have an assignment, where I’m required to derive the formula for air resistance for a falling object.

$$F = \frac{1}{2}CAdv^2$$

where $$C$$ is drag coefficient, $$A$$ is the cross-sectional area, $$d$$ is the density of the air and $$v$$ is the speed, through dimensional analysis.

In the process, I’m required to explain why the air resistance on the object is independent of the mass, $$m$$, and the position, $$x$$, of the object. However, I can’t really explain why that is? How do I clearly explain and prove, that those things do not have an effect on the air resistance?

Edit
Now I have a pretty good understanding of why air resistance is independent of mass, since it's actually quite straight forward. However, something that is not so straight forward, is explaining why air resistance is also independent of acceleration, $$a$$. How would you convince someone, that this is the case, because I don't see a clear way of doing that?

• Essentially a dupe of physics.stackexchange.com/q/197106/25301 – Kyle Kanos Dec 7 '18 at 23:50
• Carl, you should use $\rho$ for density, not $d$. – David White Dec 8 '18 at 1:01
• It's weird something depends on acceleration. Anyways, think this way: speeding from $1m/s$ to $2m/s$ in $1s$ is the same acceleration as in $80m/s$ to $81m/s$ in the same interval. Our experience tells us that the second case has more resistance. That's because it depends on $v$, not $a$. – FGSUZ Dec 10 '18 at 11:51
• Drag (say as measured in wind tunnels) is a time-average for steady states. There could be transient effects related to acceleration. – Bert Barrois Dec 10 '18 at 14:51

Air resistance can be thought of in the same way. A bunch of particles are in the way. They have to be moved away. The "heavier" they are, the tougher (density). The more of them you need to move, the tougher (size or frontal/perpendicular area). And the faster they must be moved, the tougher (speed). The constant $$C$$ encompasses all other constant influences.