An object which is falling through the
atmosphere is subjected to two external
forces. One force is the gravitational
force, expressed as the
weight
of the object. The other force is the
air resistance, or drag of the object.
If the mass of an object remains constant, the
motion of the object can be described by
Newton's second law of motion,
force F equals mass m times acceleration a:
F = m * a
which can be solved for the acceleration of
the object in terms of the net external force and the mass of the
object:
a = F / m
Weight and drag are forces
which are
vector quantities.
The net external force F is then equal to the
difference
of the weight W and the drag D
F = W - D
The acceleration of a falling object then becomes:
On the figure at the top, the density is expressed by the Greek symbol
"rho". The symbol looks like a script "p". This is the standard symbol used by
aeronautical engineers. We are using "r" in the text for ease of translation
by interpretive software.
Drag increases with the
square of the speed.
So as an object falls, we quickly reach conditions where the
drag becomes equal to the weight, if the weight is small.
When drag is equal to weight, there is no
net external force on the object
and the vertical acceleration goes to zero. With no acceleration,
the object falls at a
constant velocity as described by Newton's
first law
of motion. The constant vertical velocity is called the terminal
velocity .
Using algebra, we can determine the value of the terminal velocity.
At terminal velocity:
D = W
Cd * r * V ^2 * A / 2 = W
Solving for the vertical velocity V, we obtain the equation
V = sqrt ( (2 * W) / (Cd * r * A)
where sqrt denotes the square root
function.
Typical values of the drag coefficient are given
on a separate slide.
Here's a JavaScipt calculator which will solve the
equations presented on this page:
Terminal Velocity Calculator
Choose planet and units
Enter values for weight, area, and drag coefficient
Enter value for altitude or air density
Press Compute Button
Terminal Velocity
The chemistry of the atmosphere and the gravitational constant of a planet
affects the terminal velocity. You select the planet using the choice button
at the top left. You can perform the calculations in English (Imperial) or
metric units. You must specify the weight or mass of your object. You can
choose to input either the weight on Earth, the local weight on the planet,
or the mass of the object.
Then you must specify the cross sectional area and the drag
coefficient. Finally you must specify the atmospheric density. We have included
models of the atmospheric density variation with altitude for
Earth and
Mars
in the calculator.
When you have the proper test conditions, press the red "Compute" button to
calculate the terminal velocity.
You can download your own copy of this calculator for use off line. The program
is provided as TermVel.zip. You must save this file on your hard drive
and "Extract" the necessary files from TermVel.zip. Click on "Termvcalc.html"
to launch your browser and load the program.
When you have gained some experience with the terminal velocity calculator
and are familiar with the variables and operation, you can run a
simple version
of the program on-line. The simple version contains just the calculator and
no instructions and it loads faster than the version given above.
Notice In this calculator, you have to specify the
drag coefficient.
The value of the drag coefficient depends on the
shape.
of the object and on
compressibility effects
in the flow.
For airflow near and faster than the
speed of sound,
there is a large increase in the drag coefficient because of
the formation of
shock waves
on the object. So be very careful when interpreting results with large
terminal velocities. If your drag coefficient includes compressibility
effects, then your answer is correct. If your drag coefficient
was determined at low speeds, and the terminal velocity is very high,
you are getting the wrong answer because your drag coefficient does
not include compressibility effects.
The terminal velocity equation tells us that an object with a
large cross-sectional area or a high drag coefficient falls
slower than an object with a small area or low drag coefficient. A
large flat plate falls slower than a small ball with the same
weight. If we have two objects with the same area and drag
coefficient, like two identically sized spheres, the lighter object
falls slower. This seems to contradict the findings of Galileo that
all free falling objects fall at the
same rate with equal air resistance. But Galileo's principle only
applies in a vacuum, where there is NO air resistance and drag
is equal to zero.
We have also developed a simple simulation of a falling object to help
you study this interesting physics problem. The program is called
DropSim and is available for free at this
web site.
