Newton's Second Law
Newton's Second Law states that the total force acting on an object is equal to the product of its mass and its acceleration:
$Fequals;ma$


where

$F$ is the total force acting on the object
$m$ is the (inertial) mass of the object
$a$ is its acceleration






Gravity
In a vacuum, an object on the surface of the earth accelerates downwards at g = 9.81$m$/${s}^{2}$ due to gravity. This is known as free fall. The force of gravity is given by
${F}_{g}equals;mg$


where

${F}_{g}$ is the gravitational force acting on the object
$m$ is the (gravitational) mass of the object
$g$ is the gravitational acceleration






If gravity is the only force acting ($F\={F}_{g}$), the object experiences a constant acceleration. Assuming the object has been dropped from rest, its speed and distance traveled can be determined through integration:

$aequals;g$
$vequals;gt$
$dequals;\frac{g{t}^{2}}{2}$

where

a is the acceleration
$v$ is the speed
$\phantom{\rule[0.0ex]{0.0em}{0.0ex}}d$ is the distance traveled
$t$ is the time elapsed






Notice that the object's motion is not affected by its mass, weight, density, or any other measurement of its size. In fact, all objects fall at the same rate in a vacuum as long as the only force acting on them is gravity.
Air resistance
An object that falls in real life is subject to air resistance. Air resistance is a type of drag, the frictional force slowing an object moving through a fluid medium. The formula for the drag is known as the drag equation:
${F}_{d}equals;\frac{1}{2}{\mathrm{rho;}}_{f}{v}^{2}{C}_{d}A$


where

${\mathrm{\ρ}}_{f}$ is the density of the fluid
$v$ is the speed of the object through the fluid
${C}_{d}$ is the drag coefficient
$A$ is the crosssectional area of the object in a plane perpendicular to the motion






Note that for a spherical object made from a given material with a given shape moving at a given speed, the drag force is proportional to its area, while the inertial mass is proportional to the volume. By Newton's Second Law, the acceleration the object experiences due to the drag force is inversely proportional to its radius.
Drag coefficient
The drag coefficient is a dimensionless number defined by the drag equation. It depends on the shape of the object and the nature of the fluid flow around the object, in other words. whether the fluid flows smoothly or turbulently. At many common speeds, the drag coefficient is dependent only on the shape of an object:
Shape

Drag Coefficient (${\mathit{C}}_{\mathit{d}}$)

Sphere

0.47

HalfSphere

0.42

Cube

1.05

Cylinder

0.82



At lower speeds with smooth flow, the viscosity of the fluid is more important and tends to create more drag than at higher speeds. The nature of the fluid and speed of the flow are captured by another dimensionless number, the Reynolds number $R$:
$Requals;\frac{{\mathrm{rho;}}_{f}vL}{\mathrm{mu;}}$


where

${\mathrm{\ρ}}_{f}$ is the density of the fluid
$v$ is the speed of the object through the fluid
$L$ is a characteristic linear dimension (diameter of object)
$\mathrm{\μ}$ is the (dynamic) viscosity of the fluid.






For a given shape of object, the drag coefficient can be written as a function of the Reynolds number. In particular for spheres, for small values of R, $\left(R<10\right)$, the flow is approximately smooth (laminar); ${C}_{d}$ is roughly $\frac{24}{R}$. For values of R between ${10}^{3}$ and ${10}^{5}$, the flow is turbulent, and ${C}_{d}$ is roughly constant at 0.47. For values of $R$ between 10 and ${10}^{3}\,$${C}_{d}$ is between $\frac{24}{R}$ and 0.47. For the purpose of this demonstration assume the following function for ${C}_{d}\left(R\right)$:
${C}_{d}\=\left\{\begin{array}{cc}\frac{24}{R}\,& R<\frac{24}{.47}\\ .47comma;& \mathrm{R}\ge \frac{24}{.47}\end{array}\right.$
Stokes' Law
Stokes' Law, applicable for laminar flow, expresses the drag force on a sphere in terms of the speed. Substituting ${C}_{d}\=\frac{24}{R}$ in the drag equation and using $L\=2r$ and $A\=\mathrm{\π}\cdot {r}^{2}$, you get:
${F}_{d}equals;6\mathrm{pi;}\mathrm{mu;}rv$
Terminal velocity
As the object falls, its speed increases, and so does the amount of drag it experiences. In the limit, at a certain speed called terminal velocity, the net downward force of gravity is balanced exactly by the drag. The terminal velocity can be determined by equating the force of drag to the force of gravity and solving for the speed. If the flow is laminar, the terminal velocity is:
${v}_{\infty}\=\frac{mg}{6\mathrm{pi;}\mathrm{mu;}r}$
If the flow is turbulent, the terminal velocity is:
${v}_{\infty}\=\sqrt{\frac{2mg}{{\mathrm{rho;}}_{f}{C}_{d}A}}$
Equations of motion
The equations of motion are found by setting $F\={F}_{g}\+{F}_{d}$ and integrating:
Laminar flow:
$\frac{a\left(t\right)}{g}\=\frac{v\left(t\right)}{{v}_{\mathrm{\∞}}}1$

$\frac{v\left(t\right){v}_{\mathit{\infty}}}{{v}_{0}{v}_{\mathit{\infty}}}equals;{ExponentialE;}^{\frac{g}{{v}_{\mathit{\infty}}}\cdot \left(t{t}_{0}\right)}$

$\frac{y\left(t\right){y}_{0}}{{v}_{\mathit{\infty}}}\=\frac{{v}_{0}{v}_{\mathit{\infty}}}{g}\left({ExponentialE;}^{\frac{g}{{v}_{\mathit{\infty}}}\left(t{t}_{0}\right)}1\right)plus;t{t}_{0}$



Turbulent flow:
$\frac{a\left(t\right)}{g}\={\left(\frac{v\left(t\right)}{{v}_{\mathrm{\∞}}}\right)}^{2}1$

$v\left(t\right)\={v}_{\mathit{\infty}}\mathrm{tanh}\left(\mathrm{arctanh}\left(\frac{{v}_{1}}{{v}_{\mathit{\infty}}}\right)\frac{g}{{v}_{\mathrm{infin;}}}\left(t{t}_{1}\right)\right)$

$y\left(t\right){y}_{1}\=\frac{{v}_{\mathrm{\∞}}^{2}}{g}\mathrm{ln}\left(\frac{\mathrm{cosh}\left(\mathrm{arctanh}\left(\frac{{v}_{1}}{{v}_{\mathit{\infty}}}\right)\frac{g}{{v}_{\infty}}\left(t{t}_{1}\right)\right)}{\mathrm{cosh}\left(\mathrm{arctanh}\left(\frac{{v}_{1}}{{v}_{\mathit{\infty}}}\right)\right)}\right)$
Buoyancy
Buoyancy is the force, equal to the weight of the fluid displaced, that the surrounding fluid exerts on an object by virtue of the pressure differences at various points on the surface of the object. The effect of buoyancy is equivalent to increasing the inertial mass of the object $\left(m\right)$ by the mass of the fluid displaced $\left({m}_{f}\right)$, and decreasing the gravitational mass of the object by the same amount. To account for buoyancy, in the formulae for terminal velocity, replace $m$ by $m\+{m}_{f}$, and in the equations of motion, replace $g$ by $\frac{\left(m{m}_{f}\right)}{\left(m\+{m}_{f}\right)}g$.
Note that, if $m\={m}_{f}$, the object does not move, while if ${m}_{f}\>m$, the object moves upwards (floats).
Conclusion
In a vacuum at the surface of the Earth, all objects fall at the same rate, under the constant acceleration of gravity, equal to 9.81$m$/${s}^{2}$. Galileo's claim was correct, and in particular, Aristotle's claim that the rate of fall of an object was proportional to the weight was incorrect.
However, in air or any other dense fluid medium, objects fall more slowly due to two effects: a drag force exerted by the medium on the object, and the effect of the buoyancy of the medium. Due to both of these effects, heavier objects do indeed fall somewhat faster in a dense medium.
Given two objects of the same size but of different materials, the heavier (denser) object will fall faster because the drag and buoyancy forces will be the same for both, but the gravitational force will be greater for the heavier object.
Moreover, given two objects of the same shape and material, the heavier (larger) one will fall faster because the ratio of drag force to gravitational force decreases as the size of the object increases.
In air, however, these differences will be very small for most objects, becoming noticeable only for objects of relatively low density.
Aristotle was correct in claiming that heavier objects fall faster (in air or any other dense medium anyway), but his claim that an object's rate of fall is proportional to its weight was incorrect. Furthermore he was right to suggest that the rate of fall was slower in more dense media, but his claim that the rate of fall was inversely proportional to the density of the medium was not correct.
Finally, if two objects have similar masses but different densities and sizes, it is possible that at the beginning the denser one will fall faster, but if it is small enough, its terminal velocity may be lower, allowing the less dense object to eventually overtake it.



