Forces are vector quantities having both a magnitude and a direction

LECTURE 1 (background reading)

Isaac Newton Biography Portrait of Isaac Newton and listing of this Three Laws of Motion

Computer drawing of the forces on a rocket.

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)

Photo of the Space Shuttle and an astronaut in orbit. All objects
fall at the same rate in a vacuum.

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.

Computer drawing of a model rocket turning into the wind during
ascent. Also a picture of a weather vane indicating wind direction.

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.

Computer drawing of three model rockets showing the restoring
force present when cp is below cg.

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)

Computer drawing of a rocket showing simple translation
and the definitions of average velocity and acceleration.

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: