The speed of cherry blossom petals My high school students competed with other students on a test called webtrotter. There is a question in this competition:

"5 cm per second" is the falling speed of cherry petals, in Makoto Shinkai's animated film.

I know that for a uniformly accelerated motion the speed of fall is given by the relation
$$v=\sqrt{2g \ \Delta h} \tag 1$$
and the fall time is
$$t=\sqrt{\frac{2\Delta h}{g}} \tag 2$$
Evidently in both formulas mass $m$ is not taken into consideration for the speed of cherry blossom petals and if I remember correctly we don't consider air resistance.
My question is this:

is it possible that the previous relationships can be changed in the presence of air resistance and for the mass $m$? Or do they always have general validity?


Is it possible that a cherry petal, which has a different mass than a fig leaf, falls at a velocity of $5 \text{ cm/s}$?

 A: It's true that in the absence of air resistance and in a uniform gravitational field, objects of different masses will accelerate at the same rate, as Galileo showed, i.e. independent from $m$. This is why the period of a simple pendulum is independent from the mass of the bob.
For a cherry petal and fig leaf, however, air resistance plays quite a dominant role in the motion. Much of the motion is not uniformly accelerated motion. A cherry petal can fall at $5\,\text{cm}/\text{s}$ in calm air if that is its terminal velocity. You can read about the Stokes drag and quadratic drag, two models of air resistance in the low and high speed regimes, respectively. (I am guessing that at the speed that you quoted, the drag would be linear, i.e. $F_D\propto v$).
Note that when drag is applicable, mass indeed plays a role in the net acceleration of the projectile, since drag force is independent from mass. A high mass object will experience less of an effect from air resistance in its motion than a low mass object of the same shape.
From personal experience, however, one often sees cherry petals drifting on the wind. This situation, albeit pleasing, is probably not easily modeled at all.
A: 
is it possible that the previous relationships can be changed in the presence of air resistance and for the mass m? Or do they always have general validity?

The idea that freefall is uniformly accelerated is of course an idealization. This is primarily due to 2 things:

*

*Neglecting that the gravitational force changes with height, and,

*Neglecting drag.

One thing that is worth nothing is that whether or not you neglect the gravitational dependence on height, the acceleration will not depend on the mass.
For short distances like 5 cm, neglecting gravitational force dependence on height is completely reasonable.
However, drag, especially for light objects, should not be neglected. A simple model for the drag force $f$ is $$f=-kv$$ where $k$ is some proportionality constant and $v$ is the speed. In vector form, this becomes $\vec f = - k \vec v$ but the 1-D analysis will suffice.
The constant $k$ depends on a variety of factors. One amongst many is the shape of the object in question. The more cross-sectional area an object has, the more air particles it will hit, so the greater the drag force. This is why you can drop a sheet of paper of the same mass in a variety of ways and record very different flight times.
Second, recall that $F=ma$, so the net force on your object is $ma=mg-kv\implies a=g-\dfrac km v$. The larger the mass, the smaller the k/m term, so the smaller the drag.
The above situation yields a very different equation of motion, which will yield a different answer for the flight time than your initial model. Hopefully that answers your first question.

Is it possible that a cherry petal, which has a different mass than a fig leaf, falls at a velocity of 5 cm/s?

Check out this page. The drag there is quadratic (such that $f=kv^2$, but the concept is the same). And yes, it is possible to get your cherry leaf to impact with such a speed, although you will have to adjust the drop height to account for drag.
