LECTURE 1 (background reading)
Forces are
vector quantities having both a magnitude and a direction. When
describing the action of forces, one must account for both the magnitude and
the direction. In flight, a rocket is subjected
to four forces; weight, thrust, and the aerodynamic forces, lift and drag. The
magnitude of the weight depends on the mass of all of the parts of the rocket. The weight
force is always directed towards the center of the earth and acts through the
center of gravity, the yellow dot on the figure. The magnitude of the thrust
depends on the mass flow rate through the engine and the velocity and pressure
at the exit of the nozzle. The thrust force normally acts along the
longitudinal axis of the rocket and
therefore acts through the center of gravity. Some full scale rockets can move,
or gimbal, their nozzles to produce a force which is not aligned with the center
of gravity. The resulting torque about the center of gravity can be used to
maneuver the rocket. The
magnitude of the aerodynamic forces depends on the shape, size, and velocity of
the rocket and on properties of the atmosphere.
The aerodynamic forces act through the center of pressure, the black and yellow
dot on the figure. Aerodynamic forces are very important for model rockets, but may
not be as important for full scale rockets, depending
on the mission of the rocket. Full scale
boosters usually spend only a short amount of time in the atmosphere.
In flight the magnitude, and sometimes the
direction, of the four forces is constantly changing. The response of the rocket depends on
the relative magnitude and direction of the forces, much like the motion of the
rope in a "tug-of-war" contest. If we add up the forces, being
careful to account for the direction, we obtain a net external
force on the rocket. The
resulting motion of the rocket is described
by Newton's laws of motion.
Although the
same four forces act on a rocket as on an airplane, there are some important
differences in the application of the forces:
On an airplane, the lift force (the aerodynamic force
perpendicular to the flight direction) is used to overcome the weight. On a
rocket, thrust is used in opposition to
weight. On many rockets, lift is
used to stabilize and control the direction of flight.
On an airplane, most of the aerodynamic forces are generated by
the wings and the tail surfaces. For a rocket, the
aerodynamic forces are generated by the fins, nose cone, and body tube. For
both airplane and rocket, the
aerodynamic forces act through the center of pressure (the yellow dot with the
black center on the figure) while the weight acts through the center of gravity
(the yellow dot on the figure).
While most airplanes have a high lift to drag ratio, the drag of
a rocket is usually much greater than the
lift. While the magnitude and direction
of the forces remain fairly constant for an airplane, the magnitude and
direction of the forces acting on a rocket change
dramatically during a typical flight.
Thrust is the force
which moves the rocket through the
air, and through space. Thrust is generated by the propulsion system of the
rocket through the application of Newton's third law
of motion; For every action there is an equal and
opposite re-action. In the propulsion system, an engine does work on a gas or
liquid, called a working fluid, and accelerates the working fluid through the
propulsion system. The re-action to the acceleration of the working fluid
produces the thrust force on the engine. The working fluid is expelled from the
engine in one direction and the thrust force is applied to the engine in the
opposite direction.
The direction of the thrust is normally along the longitudinal
axis of the rocket through the
rocket center of gravity. But on some rockets, the
exhaust nozzle and the thrust direction can be rotated, or gimbaled. The rocket can then be
maneuvered by using the torque about the center of gravity. The magnitude of
the thrust can be determined by
the general thrust equation. The magnitude of the thrust depends on the mass
flow rate of the working fluid through the engine and the exit velocity and
pressure of the working fluid. The efficiency of the propulsion system is
characterized by the specific impulse; the ratio of the amount of thrust
produced to the weight flow of the propellants.
All rocket engines
produce thrust by accelerating a working fluid. But there are many different
ways to produce the acceleration, and many different available working fluids.
Let's look at some of the various types
of rocket engines and how they produce thrust.
The simplest rocket engine uses
air as the working fluid, and pressure produced by a pump to accelerate the air.
This is the type of "engine" used in a toy balloon or a stomp rocket. Because the
weight flow of air is so small, this type of rocket engine does
not produce much thrust. A bottle rocket uses water
as the working fluid and pressurized air to accelerate the working fluid.
Because water is much heavier than air, bottle rockets generate
more thrust than stomp rockets.
Model rockets, and most
full scale rockets use
chemical rocket engines.
Chemical rocket engines use
the combustion of propellants to produce exhaust gases as the working fluid.
The high pressures and temperatures of combustion are used to accelerate the
exhaust gases through a rocket nozzle to
produce thrust. There are two important parts of a chemical rocket engine; the
nozzle, and the propellants. The nozzle design determines the
mass flow rate, exhaust velocity, and exit pressure for a given initial
pressure and temperature. The initial pressure and temperature are determined by
the chemical properties of the propellants. Propellants are composed of a fuel
to be burned and an oxidizer, or source of oxygen, for combustion. Under normal
temperature conditions, propellants do not burn, but require some source of
heat, or igniter, to initiate combustion. Chemical rocket engines do
not typically rely on the surrounding atmosphere as a source of oxygen.
Therefore, chemical rocket engines can
be used in space, where there is no atmosphere present.
There are two main types of chemical rocket engines;
liquid rockets and solid
rockets. In a liquid rocket, the fuel
and the oxidizer are stored separately and pumped into the combustion chamber
of the nozzle where the burning occurs. In a solid rocket, the fuel
and oxidizer are mixed together into a
solid propellant which is packed into a cylinder. The propellant only burns on
the surface. So, as the propellant burns, a "flame front" is produced
which moves into the propellant. Once the burning starts, it will proceed until
all the propellant is consumed. With a liquid rocket, you can
stop the thrust by turning off the flow of fuel or oxidizer; but with a solid
rocket, you must destroy the casing to stop
the engine. Liquid rockets tend to be
heavier and more complex because of the pumps used to move the fuel and
oxidizer, and you usually load the fuel and oxidizer into the rocket just before
launch. A solid rocket is much
easier to handle and can sit for years before firing.
Weight is the force
generated by the gravitational attraction on the rocket. We are more
familiar with weight than with the other forces acting on a rocket, because
each of us have our own weight which we can measure
every morning on the bathroom scale. We know when one thing is heavy and when
another thing is light. But weight, the gravitational force, is fundamentally
different from the other forces acting on a rocket in flight.
The aerodynamic forces, lift and drag, and the thrust
force are mechanical forces. The rocket must be in
physical contact with the the gases which generates these forces. The
gravitational force is a field force and the rocket does not
have to be in contact with the source of this force.
The nature of the gravitational force has been studied by
scientists for many years and is still being investigated by theoretical
physicists. For an object the size of a rocket, the
explanation given three hundred years ago by Sir Isaac Newton is sufficient to
describe the motion of the object. Newton developed
his theory of gravitation when he was only 23 years old and published the
theories with his laws of motion some years later. As Newton observed,
the gravitational force between two
objects depends on the mass of the objects and the inverse of the the square of
the distance between the
objects. More massive objects create greater forces and the farther apart the
objects are the weaker the attraction. Newton was able to
express the relationship in a single weight equation. The gravitational force,
F, between two particles equals a universal
constant, G, times the product of the mass of the particles, m1 and m2, divided
by the square of the distance, d, between the
particles.
F = G * m1 * m2 / d^2
If you have a lot of particles acting on a single particle, you
have to add up the contribution of all the individual particles. For objects
near the Earth, the sum of the mass of all the particles is simply the mass of
the Earth and the distance is then measured from the center of the Earth. On
the surface of the Earth the distance is about 4000 miles. Scientists have
combined the universal gravitational constant, the mass of the Earth, and the
square of the radius of the Earth to form the Earth's gravitational
acceleration, ge .
ge = G * m Earth / (d Earth)^2
ge = 9.8 m/sec^2 = 32.2 ft/sec^2
The weight W, or gravitational force, is then just the mass of
an object times the gravitational acceleration.
W = m * ge
An object's mass does not change from place to place, but an
object's weight does change because the gravitational acceleration ge depends on the square of the distance from the center of
the Earth. Let's do a
calculation and determine the
weight of the Space Shuttle in low Earth orbit. On the ground, the orbiter
weighs about 250,000 pounds. In orbit, the Shuttle is about 200 miles above the
surface of the Earth; the distance from the center of the Earth is 4200 miles.
Then:
m = Ws / ge = Wo / go
Wo = Ws * go / ge
where Ws = surface weight (250,000
pounds), Wo is the orbital weight, and go is the orbital value of the
gravitational acceleration. We can calculate the ratio of the orbital
gravitational acceleration to the value at the surface of the Earth as the square of Earth radius
divided by the square of the orbital radius.
go / ge = (d Earth)^2 / (d orbit)^2
go / ge = (4000/4200)^2 = .907
On orbit, the shuttle weighs 250,000 * .907 = 226,757 pounds.
Notice: the weight is not zero. There is a large
gravitational force acting on the Shuttle at a distance of 200 miles. The
"weightlessness" experienced by astronauts on board the Shuttle is
caused by the free-fall of all objects in orbit. The Shuttle is pulled towards
the Earth because of gravity.
But the high orbital speed, tangent to the surface of the Earth,
causes the fall towards the surface to be exactly matched by the curvature of
the Earth away from the shuttle. In essence, the shuttle is constantly falling
all around the Earth.
Because the weight of an object depends on the mass of the
object, the mass of the attracting object, and the square of the distance between them,
the surface weight of an object varies from planet to planet. We have derived
a gravitational acceleration for the surface of the Earth, ge
= 9.8 m/sec^2, based on the mass of the Earth and the radius of the Earth.
There are similar gravitational accelerations for every object in the solar
system which depend on the mass of the object and the radius of the object. Of
particular interest for the Vision for Space Exploration, the gravitational
acceleration of the Moon gm is given by:
gm = G * m Moon / (d Moon)^2
gm = 1.61 m/sec^2 = 5.3 ft/sec^2
and the gravitational acceleration of
Mars gmar is given by:
gmar = G * m Mars / (d Mars)^2
gmar = 3.68 m/sec^2 = 12.1 ft/sec^2
The mass of a rocket is the same
on the surface of the Earth, the Moon and Mars. But on the surface of the Moon,
the weight force is approximately 1/6 the weight on Earth, and on Mars, the
weight is approximately 1/3 the weight on Earth. You don't need as much thrust
to launch the same rocket from the
Moon or Mars, because the weight is less on these planets.
For a rocket, weight is a
force which is always directed towards the center of the Earth. The magnitude
of this force depends on the mass of all of the parts of the rocket itself, plus
the amount of fuel, plus any payload on board. The weight is distributed
throughout the rocket, but we can
often think of it as collected and acting through a single point called the
center of gravity. In flight, the rocket rotates
about the center of gravity, but the direction of the weight force always
remains toward the center of the Earth.
During launch the rocket burns up and
exhausts its fuel, so the weight of the rocket constantly
changes. For a model rocket, the change
is a small percentage of the total weight and we can determine the
rocket weight as the sum of the component
weights. For a full scale rocket the change
is large and must be included in the equations of motion. Engineers have
established several mass ratios which help to characterize the performance of a
rocket with changing mass. Full scale rockets are often
staged or broken into smaller rockets which are
discarded during flight to increase the rocket's
performance.
Aerodynamic forces are
generated and act on a rocket as it flies
through the air. The magnitude of the aerodynamic forces depends on the shape,
size and velocity of the rocket and some
properties of the air through which it flies. By convention, the single
aerodynamic force is broken into two components: the drag force which is opposed to the direction of motion, and the lift force which acts perpendicular to
the direction of motion. The lift and drag act through the center of pressure
which is the average location of the aerodynamic forces on an object.
Aerodynamic forces are mechanical forces. They are generated by
the interaction and contact of a solid body with a fluid, a liquid or a gas.
Aerodynamic forces are not generated by a force field, in the sense of the
gravitational field,or an electromagnetic field.
For lift and drag to be generated, the rocket must be in
contact with the air. So outside the atmosphere there is no lift and no drag.
Aerodynamic forces are generated by the difference in velocity between the rocket and the air.
There must be motion between the rocket and the air.
If there is no relative motion, there is no lift and no drag. Aerodynamic
forces are more important for a model rocket than for a
full scale rocket because the
entire flight path of the model rocket takes place
in the atmosphere. A full scale rocket climbs above
the atmosphere very quickly.
Aerodynamic
forces are used differently on a rocket than on an airplane. On an
airplane, lift is used to overcome the weight of the aircraft, but on a rocket, thrust is
used in opposition to weight. Because the center of pressure is not normally
located at the center of gravity of the rocket, aerodynamic
forces can cause the rocket to rotate in
flight. The lift of a rocket is a side
force used to stabilize and control the direction of flight. While most
aircraft have a high lift to drag ratio, the drag of a rocket is usually
much greater than the lift.
We can think
of drag as aerodynamic friction, and one of the sources of drag is
the skin friction between the
molecules of the air and the solid surface of the moving rocket. Because the
skin friction is an interaction between a solid
and a gas, the magnitude of the skin friction depends on properties of both
solid and gas. For the solid, a smooth, waxed surface produces less skin
friction than a roughened surface. For the gas, the magnitude depends on the
viscosity of the air and the relative magnitude of the viscous forces to the
motion of the flow, expressed as the Reynolds number. Along the surface, a
boundary layer of low energy flow is generated and the magnitude of the skin
friction depends on the state of this flow. We can also think of drag as aerodynamic
resistance to the motion of the object through the fluid. This source of drag
depends on the shape of the rocket and is
called form drag. As air flows
around a body, the local velocity and pressure are changed. Since pressure is a
measure of the momentum of the gas molecules and a change in momentum produces
a force, a varying pressure distribution will produce a force on the body. We
can determine the magnitude of the force by integrating, or adding up the local pressure times the
surface area around the entire body. The base area of a model rocket produces
form drag.
Lift occurs
when a flow of gas is turned by a solid object. The flow is
turned in one direction, and the lift is generated in the opposite direction,
according to Newton's third law and
Bernoulli’s refinements. For a model rocket, the nose
cone, body tube, and fins can turn the flow and become a source of lift if the
rocket is inclined to the flight direction.
LECTURE 2 (background reading)
Ballistic
Equations (neglecting thrust and air – atmospheric Drag) above
An object that falls through a vacuum is subjected to only one
external force, the gravitational force, expressed as the weight of the object.
The weight equation defines the weight W to be equal to the mass of the object
m times the gravitational acceleration g:
W = m * g
the value of g is 9.8 meters per
square second (32.2 feet per square
second) on the surface of the Earth, and has different values on the surface of
the Moon and Mars. The gravitational acceleration g decreases with the square
of the distance from the center of the planet. But for
many practical problems, we can assume this factor to be a constant. The mass
of an object does not depend on the location, the weight does.
An object that moves because of the action of gravity alone is
said to be free falling. If the object falls through an atmosphere, there is an
additional drag force acting on the object and the physics involved with the
motion of the object is more complex than in free fall. For an object in free
fall, we can easily predict the motion of the object.
Assuming the mass of the object remains constant, and the size
and speed of the object is not so small or so fast that we must consider
relativistic effects, the motion of the object is described by Newton's second
law of motion, force F equals mass m times acceleration a:
F = m * a
We can do a little algebra and solve for the acceleration of the
object in terms of the net external
force and the mass of the object:
a = F / m
For a free falling object, the net external
force is just the weight of the object:
F = W
Substituting into the second law equation gives:
a = W / m = (m * g) / m = g
The acceleration of the object equals the gravitational
acceleration. The mass, size, and shape of the object are not a factor in
describing the motion of the object. So all objects,
regardless of size or shape or weight, free fall with the same acceleration.
In the figure, we show an orbiting Space Shuttle and a space walking astronaut.
The astronaut and the Shuttle have very different weight, size and shape. But
objects in orbit are in a free fall and the only force acting on the objects is
the gravitational attraction of the Earth. So both the astronaut and the
Shuttle are accelerated towards the Earth with the same acceleration. Because
the objects orbit at some altitude above the Earth's surface, the acceleration
is slightly less than the surface value. At a 200 mile orbit the acceleration
is about 90% of the surface value. Since both Space Shuttle and astronaut are
falling with the same acceleration, the astronaut appears to be
"weightless" and "floats" relative to the Shuttle.
If you know the local value of the gravitational acceleration,
you can use the equations for translation of an object to obtain the
instantaneous velocity and location as a function of time. The mass must remain
constant for a constant acceleration to occur. If one launches an object from
the surface of a planet, and there
is only gravity acting on the object (no thrust and no drag), the resulting
trajectory is described by the ballistic flight equations.
The remarkable observation that all free falling objects fall
with the same acceleration was first proposed by Galileo, nearly 400 years ago.
Galileo conducted experiments using a ball on an inclined plane to determine the
relationship between the time
and distance traveled. He found that the distance depended on the square of the
time and that the velocity increased as the ball moved down the incline, and
the weight of the ball didn’t make a difference.
GETTING TO
‘ORBIT’… What does it Take?
Re is
the mean Earth radius (3963 miles), and h is the height of the orbit in
miles.
NOW… LET’S ADD THE ATMOSPHERIC (DRAG – LIFT)
COMPONENTS!
What is the
DENSITY of AIR?
ρ = 14.4 psi/0 ft…
14 @ 1000 ft…
13.1 @ 3,000 ft… 10 @
10,000 ft… 5 @40,000 ft
Notice please, the higher we go, the less ‘dense’ the air, and
therefore, the more efficient our flight becomes! Jet aircraft use
this to their advantage, by flying at 25,000 feet or higher.
Following the liftoff of a model rocket, it often
turns into the wind. This maneuver is called weather cocking and it is caused
by aerodynamic forces on the rocket. The term
weather cocking is derived from the action of a weather vane which is shown in
black at the top of the figure. A weather vane is often found on the roof of a
barn. It pivots about the vertical bar and always points into the wind. Older,
more artistic weather vanes used the figure of a rooster with large flaring
tail feathers instead of the wing shown on the figure. This type of weather
vane was called a weather cock.
Why does weather cocking occur? As the rocket accelerates
away from the launch pad, the velocity increases and the aerodynamic forces on
the rocket increase. Aerodynamic forces depend
on the square of the velocity of the air passing the vehicle. If no wind were present, the flight path would be vertical as shown
at the left of the figure, and the relative air velocity would also be vertical
and in a direction opposite to the flight path. If you were on the rocket, the air
would appear to move past you toward the rear of the rocket.
The velocity of an object is a vector quantity having both a
magnitude and a direction and when discussing velocities we must account for
both magnitude and direction. The wind introduces an additional velocity
component perpendicular to the flight path, as shown in the middle of the
figure. The addition of this component produces an effective flow direction
shown in red on the figure. The effective flow direction is inclined to the
horizontal at an angle which we shall call angle b. The size of angle b depends
on the relative magnitude of the wind and the rocket velocity.
Since the effective flow is inclined to the rocket axis, an
aerodynamic lift force is generated by the rocket body and
fins. The lift force acts through the center of pressure cp of the rocket. For stability reasons, the cp is located
below the center of gravity cg. The lift force generates a torque about the cg which causes the rocket to rotate.
The rotation of the rocket produces a
new flight path into the wind, as shown on the right of the figure. When the
new flight path is aligned with the effective flow direction, there is no
longer any lift force generated and the rocket continues to
fly in the new flight direction. The flight path is inclined to the horizontal
at angle b.
We can determine this
angle by considering the middle of the figure. If the wind velocity is w and
the flight velocity is V, then: tan b = V / w
where "tan" is the trigonometric tangent
function. Weather cocking reduces the maximum altitude which a model rocket can achieve,
since the flight path is inclined from the vertical.
HOW TO BUILD
A “STABLE” ROCKET !
During the flight of a model rocket small gusts
of wind, or thrust instabilities can cause the rocket to
"wobble", or change its attitude in flight. Like any object in
flight, a model rocket rotates
about its center of gravity cg, shown as a yellow dot on the figure. The
rotation causes the axis of the rocket to be
inclined at some angle a to the flight path. Whenever the rocket is inclined
to the flight path, a lift force is generated by the rocket body and
fins, while the aerodynamic drag remains fairly constant for small
inclinations.
Lift and drag
both act through the center of pressure cp of the rocket, which is
shown as the black and yellow dot in the figure.
In this figure we show three cases for which the flight direction is exactly vertical.
In the center of the figure, the rocket is
undisturbed and the axis is aligned with the flight direction. The drag of the
rocket is along the axis and there is no
lift generated. On the left of the figure, a powered rocket has had the
nose of the rocket perturbed to
the right. On the right of the figure, a coasting rocket has had the
nose of the rocket perturbed to
the left.
We denote the angle in both cases by the symbol ‘a’. Considering
the powered rocket case, we see
that a lift force is generated and directed towards the right or downwind side
of the rocket. On the
coasting rocket case, the
lift is directed towards the left, also the downwind side of the rocket. For the
powered case, both the lift and the drag produce counter-clockwise torques, or
twists, about the center of gravity; the tail of the rocket will swing
to the right under the action of both forces and the nose will move to left.
For the coasting case, both lift and drag produce clockwise torques about the
center of gravity; the tail of the rocket will swing
to the left under the action of both forces and the nose will move to the
right. In both cases, the lift and the drag forces move the nose back towards
the flight direction. Engineers call this a restoring force because the forces
"restore" the vehicle to its initial condition and the rocket is determined to be
stable.
A restoring
force exists for this model rocket because the center of pressure is below the
center of gravity. If the center of pressure is above the center of gravity, the
lift and drag forces maintain their directions but the direction of the torque
generated by the forces is reversed. This is called a de-stabilizing force. Any
small displacement of the nose generates forces that cause the displacement to
increase. The conditions for a stable rocket are that the
center of pressure must be located below the center of gravity.
There is a relatively simple test that you can use on a model
rocket to determine the
stability. Tie a string around the body tube at the location of the center of
gravity. Be sure to have the parachute and the engine installed. Then swing the
rocket in a circle around you while holding
the other end of the string. After a few revolutions, if the nose points in the
direction of the rotation, the rocket is stable
and the center of pressure is below the center of gravity. If the rocket wobbles, or
the tail points in the direction of rotation, the rocket is unstable.
You can increase the stability by lowering the center of pressure, increasing
the fin area, for example, or by raising the center of gravity, adding weight
to the nose.
NOTE: Modern
full scale rockets do not usually rely on aerodynamics
for stability. Full scale rockets pivot their
exhaust nozzles to provide stability and control. That's why you don't see fins
on a Delta, Titan, or Atlas booster.
LECTURE 3 (background reading)
We live in world that is defined by three spatial dimensions and one time dimension. Objects can move
within this domain in two ways. An object can translate, or change location,
from one point to another. And an object can rotate, or change its attitude. In
general, the motion of an object involves both translation and rotation. The
motion of a rocket is
particularly complex because the rotations and translations are coupled together; a
rotation affects the magnitude and direction of the forces which affect
translations.
On this page we will consider only the translation of a rocket within our
domain. We can specify the location of our rocket at any time
t by specifying three coordinates x, y, and z on an orthogonal coordinate
system. An orthogonal coordinate system has each of its coordinate directions
perpendicular to all other coordinate directions. Initially, our rocket is at point
"0", with coordinates x0, y0, and z0 at time t0. In general, the rocket moves
through the domain until at some later time t1 the rocket is at point
"1" with coordinates x1, y1, and z1. We can specify the displacement
- d in each coordinate direction by the difference in coordinate from point
"0" to point "1". The x-displacement equals (x1 - x0), the
y-displacement equals (y1 - y0), and the z-displacement equals (z1 - z0). On
this page we only present displacement in the y-coordinate to help the student
better understand the fundamentals of
motion.
d = y1 - y0
The total displacement is a vector quantity with the x-, y-, and
z-displacements being the components of the displacement vector in the
coordinate directions. All of the quantities derived from the displacement are
also vector quantities.
The velocity -V of the rocket through the
domain is the derivative of the displacement with respect to time. In the Y -
direction, the average velocity is the displacement divided by the time
interval:
V = (y1 - y0) / (t1 - t0)
This is only an average velocity; the rocket could speed
up and slow down inside the domain. At any instant, the rocket could have a
velocity that is different than the average. If we shrink the time difference
down to a very small (differential) size, we can define the instantaneous
velocity to be the differential change in position divided by the differential
change in time;
V = dy / dt
where the symbol d / dt is the
differential from calculus. So when we initially specified the location of our
aircraft with x0, y0, z0, and t0 coordinates, we could also specify an initial
instantaneous velocity V0. Likewise at the final position x1, y1, z1, and t1,
the velocity could change to some V1. We are here considering only the
y-component of the velocity. In reality, the rocket velocity
changes in all three directions. Velocity is a vector quantity; it has both a
magnitude and a direction.
The acceleration (a) of the rocket through the
domain is the derivative of the velocity with respect to time. In the Y -
direction, the average acceleration is the change in velocity divided by the
time interval:
a = (V1 - V0) / (t1 - t0)
As with the velocity, this is only an average
, At any instant, the rocket could have
an acceleration that is different than the average. If we shrink the time
difference down to a very small (differential) size, we can define the
instantaneous acceleration to be the differential change in velocity divided by
the differential change in time:
a = dv / dt
From Newton's second law
of motion, we know that forces on an object produce accelerations. If we can determine the
forces on a rocket, and how
they change, we can use the equations presented on this slide to determine the
location and velocity of the rocket as a function
of the time.
From the diagram we show a simple way to determine the
altitude of a model rocket. The
procedure requires two observers and a tool, like the one shown in the upper
right portion of the figure, to measure angles. The observers are stationed
some distance L along a reference line. You can lay a string of known length
along the ground between the
observers to make this reference line. A long line produces more accurate
results. As the rocket passes its
maximum altitude, observer #1 calls out "Take Data", and measures the
angle a between the ground and the rocket. This
measurement is taken perpendicular to the ground. Observer #1 then measures the
angle b between the rocket and the
reference line. This measurement is taken parallel to the ground and can be
done by the observer facing the rocket, holding
position, and measuring from the direction the observer is facing to the
reference line on the ground. When the second observer hears the call,
"Take Data", the observer must face the rocket and measure
the angle d from the ground to the rocket. The second
observer must then measure the angle c, parallel to the ground, between the
direction the observer is facing and the reference line in the same manner as
the first observer. Angles a and d are measured in a
plane that is perpendicular to the ground while angles b and c are measured in
a plane parallel to the ground.
With the four measured angles and the measured distance between the
observers, we can use graph paper to build a scale model of the rocket in flight
and we can determine the
altitude h of the actual rocket. Scale
models depend on the mathematical ideas of ratios and proportions which you
learn in grade school. To determine the
altitude, we first draw the reference line L on the graph paper. Make the
length of the line on the graph paper some known ratio of the measured length.
The length of the line on the graph paper sets the scale
of the model. For instance, if the measured length was 100 feet, we might
make the line on the graph paper 10 inches long. Then one inch on the graph
paper equals 10 feet in the real
world. Now draw two lines beginning at the ends of the reference line and
inclined at the measured angles b and c
On the graph paper use a ruler to measure the distance w from
the beginning of the reference line, near observer #1, to the intersection of
the two drawn lines. The intersection point marks the location on the ground
that is directly beneath the flying rocket. Also
measure the distance x from the end of reference line, near observer #2.
As discussed on the web page with the derivation of the
equations, we really only need three measured angles and the reference length
to accurately determine the
altitude. So you can use either angle a measured by observer #1, or angle d
measured by observer #2, to determine the
altitude. If you measure all four angles, you can make two estimates of the
altitude h; they should be the same answer, but if they aren't, you can average
them.
For the measurments of observer #1, on another piece of graph
paper, draw a line of length w which you measured on the previous piece of
graph paper. At one end of this line, draw another line inclined at the angle
a. On the other end of the w line, draw a vertical line until it intersects the
blue line inclined at angle a.
Now count the blocks, or measure the length, of the vertical
line h. Convert this distance by the scale of the reference line, and you have
determined the altitude h of the actual
rocket. If you use the measurments of
observer #2, substitute the length x for length w and angle d for angle a. For
instance, in our example, 1 inch equals 10 feet. If your
measured height h is 10 inches, the actual rocket was flying
100 feet in the air. If you understand the
mathematical ideas of trigonometry you can
also calculate the altitude of model rocket and check
your graphical solution.
SUPPLEMENTAL NOTES
“POWER” in
Model Rocketry: The
relationship of Thrust to “Impulse”
In model rocketry, the power
of a motor is called “total impulse.” An impulse is the product of “force” and
the “time” over which the force is applied—which is, after all, the definition
of “power.” The total impulse is then
the product of the force and the duration over which it was applied. In rocketry, the force
is the “thrust” produced by the motor, and the time is the duration over which
the rocket motor is producing thrust. To
explain this concept, a number of examples will be used.
Suppose a rocket motor
produces 10 Newtons of thrust
force for one second.
So the total impulse of the rocket motor is:
IT=Thrust x time = (10N)x(1s) IT=10 N-s
This is the impulse of the rocket motor for
that one second. If we assume that all the propellant was consumed during that
one second, we would say that the total impulse (IT) that the motor could
produce is 10 Newton-seconds. A rocket motor that
has a thrust of 20 newtons for a duration of 0.5
seconds has the same amount of power
as a motor that produces 10 newtons of thrust for 1.0 seconds (10
Newton-seconds). Because the total impulse of the motor is not dependant on the
way the propellant burns, the level of thrust produced, nor even the type of
propellant burned, you can easily see that it is a useful indicator that can be
used to compare different rocket motors. If
one motor has a higher total impulse than another, it is said to have “more
power.”
[However,
THRUST IS IMPORTANT if you must consider the WEIGHT of the rocket you are trying to launch, as if there isn’t
enough thrust to overcome gravity, your rocket won’t lift off, but simply sputter on the
ground!]
The usefulness of the total impulse number has led to a simple
way of classifying model rocket motors
(professional rockets, like the
space shuttle are not classified this way).
Code Newton Seconds (there are 4.48N to a
lbf of thrust!)
|
1/4A
|
0-.625
|
|
1/2A
|
.626-1.25
|
|
A
|
1.26-2.50
|
|
B
|
2.51-5.00
|
|
C
|
5.01-10.00
|
|
D
|
10.01-20.00
|
|
E
|
20.01-40.00
|
|
F
|
40.01-80.00
|
|
G
|
80.01-160.00
|
|
H
|
161.01-320.00
|
|
I
|
320.01-640.00
|
The classification code for each motor can also tell us
approximately how long the motor burns. The burn time for each motor is equal
to the total impulse of the motor divided by its average thrust level: t b =
It / Tavg
So to determine the
burn time of a B2 motor, we simple divide the total impulse of the motor ( B motors have 5 N-s of total impulse) by the average
thrust of 2 newtons. The result is a burn time of 5 ÷ 2 = 2.5 seconds.
The average thrust is found by taking the total impulse that the
motor produces divided by the time which the motor produces thrust. Thus a
motor that has a total impulse of 10 Newton-seconds and burns for 2 seconds
would have an average thrust of fi ve newtons (10 ÷ 2 = 5). If the same motor
burned in 1.67 seconds, it would have an average thrust of 6 newtons. The last number after the “dash”) is the time
(in seconds) until the ejection charge fires.
Aero-Tech
Engine data
|
Single Use 18mm Motors
Time Motor Max
Total Average
Motor Delay Diam. Liftoff
Impulse Thrust
Product # Type
(sec) in/mm Wt. (oz) lbs/N-s lbs
ARO42104 D21 4T
2.75/70 16.0 4.5/20.0
4.7
ARO42107 D21 7T
2.75/70 11.2 4.5/20.0
4.7
ARO52504 E25 4T
2.75/70 16.0 4.5/22.0
5.6
ARO52507 E25 7T
2.75/70 12.0 4.5/22.0
5.6
|
|
Single
Use 24mm Motors
Time Motor Max
Total Average
Motor Delay
Diam. Liftoff Impulse Thrust
Product # Type (sec)
in/mm Wt.(oz) lbs/N-s lbs
ARO51504 E15 4W
2.75/70 16.0 9.0/40.0 3.4
ARO51507 E15 7W
2.75/70 9.5 9.0/40.0 3.4
ARO53004 E30 4T
2.75/70 16.0 9.0/40.0 6.7
ARO53007 E30 7T
2.75/70 10.6 9.0/40.0 6.7
|
Specifications:
Length: 24.5"
Diameter: 1.637"
Fin Span:
10.5"
Weight: 2.49 oz.
18 millimeters is equal
to 0.71 inches
29 millimeters is equal
to 1.14 inches.
70 millimeters is equal
to 2.76 inches
Engine:
C6-5
Max.Total
Impulse (Newton-Seconds): 10.00
Thrust Duration
(Seconds): 1.6
Propellant
Weight: 12.70g
HANDY-DANDY
CALCULATIONS
Equations for finding your rocket's peak altitude and motor delay.
Definition of Terms
- m = rocket mass in kg (see
below)
- g =
acceleration of gravity = 9.81 m/s2
- A = rocket cross-sectional
area in m2
- Cd
= drag coefficient = 0.75 for average rocket
- r (rho) =
air density = 1.22 kg/m3
- t = motor
burn time in seconds (NOTE: little t)
- T = motor
thrust in Newtons (NOTE: big T)
- I = motor
impulse in Newton-seconds
- v = burnout
velocity in m/s
- y1
= altitude at burnout
- yc
= coasting distance
- Note that
the peak altitude is y1 + yc
- ta
= coasting time => delay time for motor
- average mass during boost is mr
+ me - mp/2
use this value for all but the yc, qa, and qb
calculations.
- mass during coast is mr + me
- mp
use this value for the yc, qa, and qb
calculations.
ADVANCED TOPICS
On this page we derive the equations which are shown on the
figure to determine the
maximum altitude the rocket reaches
during the flight. The procedure requires two observers to measure several
angles. The observers are placed some distance L apart along a reference line
which is shown in white on the figure. As the rocket passes its
maximum altitude, observer #1 measures the angle a between the
ground and the rocket. This
measurement is taken perpendicular to the ground. Observer #1 then measures the
angle b between the rocket and the
reference line. This measurement is taken parallel to the ground. The second
observer measures the angle d from the ground to the rocket and the
angle c, parallel to the ground, between the
direction the observer is facing and the reference line.
With the four measured angles and the measured distance between the
observers, we can use some relations from trigonometry to derive
the equation for the altitude h. We will need four "construction"
triangles to derive the equation. The first two triangles are formed by the
altitude h, the line of sight from the observers to the rocket, and the
ground track of the line of sight. From observer #1, we have our first trigonometry relations:
A detailed
analysis of this trigonometry problem
indicates that we really only need three angle measurements along with the
reference length measurement to completely determine the
answer. The angles a, b, c, and d are related to each other and we can
eliminate one of the angle measurements and still determine the
altitude. We can equate the values of h from Eq. 1a and Eq. 2a:
You can use any of these equations to determine the
height of any object from a tall tree to a flying rocket. If you have
your observers take all four angle measurements, you can actually make three
calculations of the height, which can help to eliminate errors in the
measurments.
MASS RATIOS
and the IDEAL ROCKET EQUATION
The forces on a rocket change
dramatically during a typical flight. This figure shows a derivation of the
change in velocity during powered flight while accounting for the changing mass of the rocket. During powered flight the propellants
of the propulsion system are constantly being exhausted from the nozzle. As a
result, the weight of the rocket is
constantly changing. In this derivation, we are going to neglect the effects of
aerodynamic lift and drag. We can add these effects to the final answer.
Let us begin
with Newton's second law
of motion, shown in blue on the figure:
d (M u) / dt = F net
where M is the mass of the rocket, u is the
velocity of the rocket, F net is the net external
force on the rocket and the symbol
d / dt denotes that this is a differential equation in time t. The only
external force which we will consider is the thrust from the propulsion system.
On the web page describing the specific impulse, the thrust
equation is given by:
F = mdot * Veq
where mdot is the mass flow rate, and Veq
is the equivalent exit velocity of the nozzle which is defined to be:
Veq = V exit + (p exit - p0) * Aexit / mdot
where V exit is the exit velocity, p exit
is the exit pressure, p0 is the free stream pressure, and A exit is the exit
area of the nozzle. Veq is also related to the specific impulse Isp:
Veq = Isp * g0
where g0 is the gravitational constant. m dot is mass flow rate and is equal to the change in the
mass of the propellants mp on board the rocket:
mdot = d mp / dt
Substituting the expression for the thrust into the motion
equation gives:
d (M u) / dt = V eq * d mp / dt
d (M u) = Veq d mp
Expanding the left side of the equation:
M du + u dM = Veq d mp
Assume we are moving with the rocket, then the value of u is zero:
M du = Veq d mp
Now, if we consider the instantaneous mass of the rocket M, the mass
is composed of two main parts, the empty mass me and the propellant mass mp.
The empty mass does not change with time, but the mass of propellants on board
the rocket does change with time:
M(t) = me + mp (t)
Initially, the full mass of the rocket mf contains
the empty mass and all of the propellant at lift off. At the end of the burn,
the mass of the rocket contains
only the empty mass:
M initial = mf = me + mp
M final = me
The change on the mass of the rocket is equal to
the change in mass of the propellant, which is negative, since propellant mass
is constantly being ejected out of the nozzle:
dM = - d mp
If we substitute this relation into the motion equation:
M du = - Veq dM
du = - Veq dM / M
We can now integrate this equation:
delta u = - Veq ln (M)
where delta represents the change in
velocity, and ln is the symbol for the natural logarithmic function. The limits
of integration are from the initial mass of the rocket to the final
mass of the rocket.
Substituting for these values we obtain:
delta u = Veq ln (mf / me)
This equation is called the ideal rocket equation.
There are several additional forms of this equation which we list here: Using
the definition of the propellant mass ratio MR
MR = mf / me
delta u = Veq * ln (MR)
or in terms of the specific impulse of
the engine:
delta u = Isp * g0 * ln (MR)
If we have a desired delta u for a maneuver, we can invert this
equation to determine the
amount of propellant required:
MR = exp (delta u / (Isp * g0) )
where exp is the exponential function.
If you include the effects of gravity, the rocket equation
becomes:
delta u = Veq ln (MR) - g0 * tb
where tb is the time for the burn.
