Flying model rockets is a relatively
safe
and inexpensive way for students
to learn the basics of forces and the
response
of a vehicle to external forces.
A model rocket is subjected to
four forces in flight;
weight,thrust, and the
aerodynamic forces,
lift and drag.
There are many different types of model rockets.
The simplest type of rocket that
a student encounters is the
compressed air rocket.
The air rocket system consists of two main parts,
the launcher and the rocket.
On this page we show the equations which model the
launch of a compressed air rocket.
On the figure we show a generic launcher, although launchers come in a
wide variety of shapes and sizes. The launcher has a base to support
the rocket during launch and a hollow launch tube mounted perpendicular
to the base. The launch tube is inserted into the base of the rocket
before launch and forms a closed pressure vessel with the sides and
nose cone of the rocket.
We have cut the rocket along its axis so that you can see
the launch tube inside the body of the rocket; the inside of the rocket
is colored yellow to the right, the outside surface of the rocket is in
silver to the left,and
the launch tube is colored green at the bottom.
We will denote the length of the launch tube inside the body
of the rocket by the symbol L.
The launch tube is connected to an air pump
which is used to increase the pressure inside the body tube.
We will denote the launch
pressure
by the symbol ps.
The pressure exerts an equal force on all sides of the pressure vessel.
Because the rocket is free to move along the launch tube, the pressure
exerts a vertical force given by:
Fp = (ps - po) * A
where Fp is the pressure force, po is the atmospheric pressure,
and A is the cross-sectional area of the rocket tube.
The net force acting on the rocket is the difference of the vertical pressure
force and the weight W of the rocket:
Fnet = (ps - po) * A - W
To simplify the analysis,
we are going to assume that the length of the launch tube is small
relative to the length of the rocket. As the rocket moves, the
pressure remains fairly constant inside the tube. In reality, there is a
slight decrease in pressure as the volume
increase, but we are going to neglect that effect to simplify this analysis.
With a constant pressure, and constant pressure force, we can use a simple
algebraic form of
Newton's second law
to determine the acceleration a, velocity v
and the distance y which the rocket moves along the
launch tube:
Fnet = m * a
m * a = (ps - po) * A - W
where m is the mass of the rocket
This equation can also be written as:
W * a = g * ((ps - po) * A - W)
where g is the gravitational acceleration, which is equal to 32.2 ft/sec^2 or
9.8 m/sec^2 on the surface of the
Earth.
The value of g is different on the
Moon, and on
Mars.
The
weightW is equal to the mass times the gravitational
acceleration. Now divide both sides by the weight to
determine the rocket acceleration along the launch tube:
a = g * ((ps - po) * A / W - 1)
For a constant acceleration,
the velocity v and distance traveled y
are given by:
v = a * t
y = .5 * a * t^2
where t is the time. Using the
length of the launch tube L,
we can now solve for the lift off time - TLO
by using the second equation:
TLO = sqrt (2 * L / a)
and then using this time, we can use the first equation to solve for the
velocity at the end of the tube:
v = TLO * a
v = sqrt ( 2 * L * a)
As the rocket clears the end of the launch tube (t = TLO+), the air inside the rocket,
at pressure ps, will rush out the exit and the pressure will equalize
with free stream inside the tube. In reality, this will provide an additional
amount of thrust T given by:
T = m dot * Ve
where m dot is the air mass flow rate out of the tube, Ve is the
exit velocity of the air. For an air rocket, with no nozzle and small volume,
this additional thrust is small relative to the weight of the rocket, and
we will ignore this effect.
The velocity v at the end of the launch tube is used as the
initial velocity in the
flight equations
to determine the flight trajectory of a compressed air rocket.
You can study the flight of a stomp rocket by using the
RocketModeler II
simulation program.
Here is a computer animation of the launch of a compressed air rocket:
And here is an interactive, Java simulation of the launch process:
You can launch the air rocket by using the buttons at the
bottom of the simulator. "Start" brings the rocket back to its
original configuration.
You can download your own copy of this simulator for use off line. The program
is provided as Stomp.zip. You must save this file on your hard drive
and "Extract" the necessary files from Stomp.zip. Click on "Stomp.html"
to launch your browser and load the program.
