Main Concept

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:

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

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:

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:

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:

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 $R$:

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\&comma;$$C_d$ is between $\frac{24}{R}$ and 0.47. For the purpose of this demonstration assume the following function for $C_d\left(R\right)$:

$C_d\&equals;\left\{\begin{array}{cc}\frac{24}{R}\&comma;& R< \frac{24}{.47}\\ .47\&comma;& \mathrm{R}\ge \frac{24}{.47}\end{array}\right.$

Stokes' Law

Stokes' Law, applicable for laminar flow, expresses the drag force on a sphere in terms of the speed. Substituting $C_d\&equals;\frac{24}{R}$ in the drag equation and using $L\&equals;2 r$ and $A\&equals;\mathrm{\&pi;}\cdot r^2$, you get:

$F_d \&equals; 6 \mathrm{\&pi;} \mathrm{\&mu;} r v$

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:

$v_{\infty }\&equals; \frac{m g}{6 \mathrm{\&pi;} \mathrm{\&mu;} r}$

If the flow is turbulent, the terminal velocity is:

$v_{\infty }\&equals; \sqrt{\frac{2 m g}{{\mathrm{\&rho;}}_f C_d A}}$

Equations of motion

The equations of motion are found by setting $F\&equals;F_g\&plus;F_d$ and integrating:

Laminar flow:

Turbulent flow: