Rocket science

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Launch of Space Shuttle Columbia on STS-1 (Image credit: NASA)
People often use the term “rocket science” to refer to a task that is really
difficult or even impossible. Well, that’s because launching a rocket is really
difficult. Fortunately, many of the key concepts in rocket science can be
explained using basic math and physics.
Rocket science lies at the heart of aerospace engineering, and deals with the
design, development, testing, and operation of rockets that launch humans
and spacecraft into orbits around the Earth and beyond. In 16.00x, we will
introduce you to the basics of rocket science, including the fundamentals of
propulsion, the famous Rocket Equation, the different types of rocket
engines, and the effects of gravity and atmospheric drag. Prof. Hoffman will

also give his personal account of what it is like to ride on a rocket and launch
into space.
Upon completing Unit 2, students should be able to:

MO 2.1: Describe the basic principles of rocket propulsion and explain the different forces acting
on a rocket

MO 2.2: Identify the different types of rocket propulsion systems and describe the operating
principles, advantages, and disadvantages of each system

MO 2.3: Apply the Ideal Rocket Equation to calculate the performance of single- and multi-stage

MO 2.4: Explain why rocket staging is important for achieving Earth orbit

MO 2.5: Explain the unique structural requirements for rockets

MO 2.6: Describe the basic concept of a control feedback loop that keeps a rocket flying along
the desired trajectory

We begin 16.00x by tackling one of the most crucial and challenging
topics in aerospace engineering — Rocket Science. Because of the
complexity of this subject, we will devote 1.5 weeks to Unit 2 instead of the
1 week that will be typical of later topics.
Please be aware that Unit 2 has a particularly large number of lecture
videos, so we suggest that you allow adequate time to finish the
material. The embedded questions in Unit 2, which will count for 25% of the
final grade, will be due on Friday, September 23 at 14:00 UTC. The
subsections that contain embedded questions will have “Review of Concepts”
in the title.
After the completion of Unit 2, Unit 3: Environmental Control and Life
Support Systems will be released on Friday, September 23 at 14:00 UTC. As
always, be sure to check the Syllabus & Calendar tab to view the full course schedule.


Three, two, one, zero, lift off.
We have a lift off.
32 minutes past the hour, lift off on Apollo 11.
If we want to fly out of the Earth’s atmosphere
and go up into space, we need to use rockets
and it takes a lot of energy.
And so in order to build rockets we really
push our technology, in many cases,
to the limits, which is why when we want
to talk about something being really hard,
we say oh, that’s rocket science.
So we’re going to learn a little bit about rocket science, how
rocket engines work.
But first, let’s talk about really what
we mean when we talk about rocket propulsion.
The pro is forward, pulsare is to push, Latin roots.
Any system which uses a thrust (push suddenly or violently in a specified direction)of
some sort to make itself go forward is a propulsion system.
I can actually be my own propulsion system.
I’ve got a space qualified skateboard here.
It’s got its own solid booster motors and the MIT AeroAstro
sticker on it.

So let’s go outside with this and we’ll turn this
into a propulsion machine.

So you saw how I could push on a lamp post
and make myself move forward.
Cars, the tires push on a road and the cars go forward.
And you can sort of have the feeling when
you think about it in that way that you’re
pushing on something external to make yourself go forward.
It doesn’t have to be that way.
I could actually carry something.
I could carry a big, heavy weight, stand on my skateboard,
and I could push against that weight,
throw it in one direction, and I’ll move the other direction.
Let’s try that out.

So what we’ve seen is you don’t necessarily
have to push on something that’s external to you.
When I was pushing against a lamp post
I was essentially pushing against the earth.
When a car’s wheel spins on the road,

it’s pushing against the earth.
The earth is very big, and so it feels
like it’s not moving at all.
In fact, it’s moving a very tiny, little bit.
Probably can’t even measure it.
On the other hand, when I take a heavy weight
and I throw it in one direction, I’m
actually pushing against that weight
even though it was something I was originally carrying.
That weight has inertia.
It doesn’t want to move.
That’s Newton’s First Law. (Newton’s First Law states that an object will remain at rest or
in uniform motion in a straight line unless acted upon by an external force. It may be seen as a
statement about inertia, that objects will remain in their state of motion unless a force acts to
change the motion.)( Newton’s First Law of Motion says that an object will remain in its state of
motion unless acted upon by force.)(
Première loi du mouvement de Newton dit qu’un objet restera dans son état de mouvement à
moins sollicité par la force)

In order to make it move in that direction,
I have to exert a force and push on it.
That’s Newton’s Second Law. {Acceleration is produced when a force acts on a mass. The
greater the mass (of the object being accelerated) the greater the amount of force needed (to
accelerate the object).

And when I do that, this object exerts
an equal and opposite force back on me. {These two forces are called action and
reaction forces and are the subject of Newton’s third law of motion. Formally stated, Newton’s

third law is: For every action, there is an equal and opposite reaction. The statement means
that in every interaction, there is a pair of forces acting on the two interacting objects.}

That’s Newton’s Third Law.
And so the essence of engines that we use in aerospace,
whether flying through the atmosphere or a rocket flying
through space, is we carry a certain amount of material
inside and we push that material out at a certain speed
and that makes us go forward.
The fluid that we push is called the working fluid
of the engine.(KEROSEN OR …)
Now in the case of a propeller plane,
the working fluid is the air.
And of course, we don’t have to carry that.
It’s basically all around us.
So the propellers make the air move,
they push against the air, and that creates a thrust,
which makes us move.
Now a jet plane is a little bit different.
Sometimes we call them gas turbine engines.
They carry their fuel, which is burned together with oxygen,
which is brought in.
You can see here the air intake.

And then you take the hot combustion gases
and you push them out the end of the jet engine
and that gives you thrust to go forward.
What is the critical difference between a jet
engine and a rocket engine?
Why don’t you pause a little bit if you don’t know the answer
and think about it.

OK, the difference between a jet engine
and a rocket engine, the jet engine
only needs to carry its own fuel.
A rocket, on the other hand, has to carry not only its own fuel,
but also the oxidizer, which in most cases
is oxygen either compressed gas or liquid oxygen.
But the point is you need something
to burn the fuel with.
Here the terminology is a little bit critical.
Instead of talking just about putting rocket fuel in,
we talk about loading it up with propellants.
And there’s two types of propellants for a rocket.
You need the fuel, whether it’s kerosene or liquid hydrogen
or alcohol methane.

That’s the fuel, and then you need the oxidizer.
Those two come together, they burn,
and voila, you’re ready to go into space.
What we’ll learn in the next section
is that you can have several types of rocket engines.
You can have the fuel and the oxidizer in different forms.
They can be liquids, they can be solids,
or you can have hybrids.
And that’s what we’ll deal with in the next segment
of the lecture.

MO 2.1

Newton’s Laws of Motion were first published in 1687 in his seminal
work Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of

Natural Philosophy), often referred to simply as the Principia. These three
laws are used to describe the motion of an object resulting from the forces
acting upon that object. Together, they form the foundation of classical
1. An object in a state of uniform motion (i.e., at rest or moving at a constant
velocity) will remain in that state of motion unless acted on by an external force.
2. The vector sum of the forces F→ acting on an object is equal to the mass m of the
object multiplied by its acceleration vector a→.


3. Every action has an equal and opposite reaction.
Explore the simulations below to see Newton’s Laws of Motion in action. Try
changing the mass of the object being moved, the applied force, the amount of
friction, etc. How do these parameters affect the speed and acceleration of the
resulting motion? Are there any results that surprise you? Use the discussion forum
below the simulation to discuss your findings.


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MO 2.1 MO 2.2
To begin our study of chemical rocket propulsion, we first introduce solid
rockets, sometimes known as solid-fuel rockets. The use of solid rockets
dates back many centuries to the introduction of fireworks by the
Chinese. In modern-day aerospace engineering, solid rockets are still used
in a number of applications, from sounding rockets that carry scientific
experiements into sub-orbital flight to the enormous solid rocket boosters
that helped propel the Space Shuttle out of the Earth’s atmosphere.

We’re first going to talk about solid rocket motors.
They were really started many, many centuries

ago by the Chinese.
Sounding rockets typically use solid propulsion.
This is where you take the fuel and the oxidizer
and you sort of mix them together into a slurry,
pour it into the rocket, you leave a hole in the middle,
and then you ignite it.
The largest solid rockets ever made
were the Boosters for the Space Shuttle,
and this is a picture of the inside of one
of those Boosters.
You get some sense of the scale by looking at the people
The black material on the inside is the solid propellant,
and it contains both a fuel and an oxidizer.
You have to start it off with essentially
a little miniature rocket motor up at the very top,
and that shoots a flame all the way down through, and that
starts the segment burning from the inside to the outside.
So you can see one of the fundamental advantages
of a solid rocket motor is it’s simple-almost no moving parts.
It’s just the flames moving through the combustion chamber
No motion of the propellant– you don’t need any pumps.
Of course, one of the disadvantages
is once you’ve lit it, it’s going to go.

You can’t turn off a solid rocket motor
any more than you can turn off a fireworks
display once you’ve lit it and once it starts its way up
into the sky.
It is possible in a very limited way to control the thrust.
Here’s a diagram of the solid rocket booster,
and actually the segments we were looking at
were these kind of segments in the middle where
you have a round cross section.
Towards the end, the cross section gets a little bigger,
but up at the front you can actually
see there’s little stars in the middle– it’s a star pattern.
And this is actually kind of an amazing picture.
It was taken by putting a video camera
right at the tail end of the solid booster
when they were doing a ground test,
and this is looking right up to the very front of the booster
when it ignites, and you can see those little fin patterns.
And here as you’re burning and then as you burn these out,
the thrust level will change.
And this is important for being able to control the thrust just
a little bit when we’re flying through the area
in the atmosphere where we get the most
vibrations and dynamic pressure.
All of the fire and flame that’s going on
inside this central section is very turbulent,

so there’s a tremendous amount of shaking and vibration, which
you can see in this next scene which actually shows
the very final flight of the Space Shuttle
with a view inside the cabin.
So you’ll see at T-0, the crew will start to shake
and it gets pretty violent for the first two
minutes of flight.
Between a solid rocket motor and a liquid rocket motor,
there is another type of engine which is still
more or less in the research stage.
It’s being developed and we don’t have any large engines,
but it’s called a hybrid.
And again, if you go back to this engine
and you just imagine that all of this material on the inside-instead of being fuel and oxidizer,
let’s just imagine that it’s fuel,
and now we pour some liquid oxidizer down the center
section such that they ignite spontaneously.
You can use nitrous oxide– laughing gas-with a kind of a rubber material.
That’s what’s been used in small hybrid rockets.
The best known example of hybrid engines in real use
have been with SpaceShipOne.
There’s an example.
These are hybrid engines, and there’s
the SpaceShipOne on its record-breaking flight.

So hybrids are a possibility.
We don’t at the moment have the ability
to make really large hybrid engines that could replace
things like the shuttle solid boosters
or some of the big boosters that are being planned for the Space
Launch System, but it’s another type of engine.
So now, we finished talking about solids,
we’ve looked a little bit of hybrids,
and it’s time to look at the basics of how liquid rocket
engines work.


Robert Goddard is often regarded as the father of modern rocketry. His
groundbreaking work, A Method of Reaching Extreme Altitudes, published in 1919,

laid the theoretical foundation for spaceflight. In this photo, Prof. Goddard is
standing next to the launch frame of his most famous invention, the liquidpropellant rocket.
In the following video, Prof. Hoffman describes the development and
operation of modern-day liquid rockets.

So we’ve looked at solid rockets and hybrid rockets.
Let’s take a look now at liquid rockets, which really
are a 20th century invention.
It was early in the 20th century that Professor Robert Goddard,
who was working at Clark University in Worcester,
Massachusetts at the time, developed
the concept of liquid rockets.
The liquid rocket would carry its own fuel and oxidizer,
and you could control how much of that
went into the rocket engine, ignited,
and the rocket would take off.
Professor Goddard actually appreciated in a way
that few people did at the time the capabilities of rocket
engines, because he realized that, since they carry
their own oxidizer, they don’t need to operate in the Earth’s
And in 1919, he published what he

thought was just a simple academic paper explaining
how his engines work.
And he postulated in that paper that, with a sufficiently
powerful rocket, you could actually leave the earth
and fly all the way to the moon.
In 1919.
The New York Times somehow or other
came across this scholarly article,
and they really pounced on the poor physics professor.
This was in January 13th of 1920.
The New York Times editorial said,
space travel is impossible because, without an atmosphere
to push against, a rocket simply cannot move so much as an inch.
Professor Goddard, it is clear, lacks
the knowledge ladled out daily in high schools
about basic physics.
Of course, July 20th, 1969, about 50 years
after Goddard published his article,
the Apollo 11 astronauts landed on the moon.
And I’m happy to say The New York
Times did publish a retraction.
The Times said, and I quote, further investigation

and experimentation have confirmed the findings
of Sir Isaac Newton in the 17th century,
and it has now been definitively established
that a rocket can function in a vacuum
as well as in an atmosphere.
The Times sincerely regrets its error.
So it goes.
So here’s a very simple conceptual diagram
of how a liquid rocket engine works.
As I pointed out before, you need a fuel and an oxidizer.
They’re carried in their own tanks.
And then pipes take the fuel and the oxidizer
to a combustion chamber, where they’re ignited and they burn.
And then hot gas comes out the back,
and that makes us go forward.
And that’s basically how a rocket engine works.
This is simple.
We call it a pressure-fed rocket engine.
Obviously, you need something to push the oxidizer and the fuel
into the combustion chamber.
And inside the combustion chamber, of course,
things get pretty hot.

This is the inside of a combustion chamber.
You can see what we call a shower head.
And the fuel and the oxidizer are
pushed through those little holes,
where they sort of get pre-mixed,
which helps the combustion.
And actually, how you design those shower heads,
which are formally called injector plates,
because you use them to inject the fuel and the oxidizer
into the combustion chamber.
The combustion chamber is inside here, and all of this area
is the rocket nozzle, which we’ll talk about shortly.
You can imagine that the pressure when
you start burning the fuel and oxidizer inside this combustion
chamber gets pretty high.
And if all you have is to push the fuel and oxidizer
into the combustion chamber using pressure in those tanks,
that pressure is going to get pretty high, which
means the walls of those tanks are
going to have to get thicker and thicker to hold the pressure.
And it runs away.
So you can use pressure feeding for rocket engines that

are relatively small, but if you want
to get a lot more power out of it,
then you need a better way of getting the fuel and oxidizer
into the combustion chamber.
And to do that, we use pumps.
So the critical part of almost any liquid rocket engine
is a turbo pump to get both the fuel and the oxidizer
into the combustion chamber.
So now we don’t have to have high pressure
inside the oxidizer or the fuel tank,
because the pumps will create that pressure
to get into the combustion chamber, where it gets burned.
Just to give you an example of how powerful these pumps have
to be, the high-pressure oxygen turbo pump in the space shuttle
main engine is about the size of a large automobile engine.
Maybe a V8 engine in a Cadillac or something.
And yet, it puts out 50,000 horsepower.
Man, I’d love to see what my car would do with a 50,000
horsepower engine in it.
Well, maybe I wouldn’t.
But in any case, that’s what you need
to pump enough propellant into the combustion

chamber in a really powerful rocket engine.
This is a picture of the F1 engine.
There were five of them in the first stage
of the huge Saturn 5 rocket.
You can get some sense here of how big the turbo
pumps are to pump all that oxidizer
and fuel into the engine.
Now, the other characteristic of every rocket engine
is it has a nozzle on the end of it.
And you may be asking, well, why do you need this big nozzle?
It’s heavy, and it creates drag.
Well, here’s the thing.
The trick with a rocket engine is what
we call controlled combustion.
So first of all, we don’t allow everything to burn at once,
like in a stick of dynamite.
In the combustion chamber here, we
start out with a very high temperature and pressure.
What we really need to do is run the combustion products
through what we call a converging-diverging nozzle
with a thin throat in the middle.
This is the throat.

And when the randomly moving molecules
go through that throat, they actually
get up to a supersonic velocity.
And this random thermal motion is
converted into a directed kinetic energy
where the random motion is pretty much removed.
And there may be a little bit of motion side to side,
but most of the kinetic energy is now directed backwards.
And that is what gives you your forward thrust.
So we’ll just finish up with a look
again at one of the examples of the biggest liquid engines that
were ever built. And those were the engines
for the space shuttle, which were also
the first large, fully reusable engine.
And here is the fuel tank.
We call it an external tank because it
was so big we couldn’t put all the fuel we
needed inside the shuttle orbiter itself.
And of course, it’s huge.
You can look at the people down here at the bottom,
and you get some sense of how big it is.
And this is the tank when it was being

manufactured in Louisiana.
And you can see here the hydrogen
tank for the liquid hydrogen and the liquid oxygen. Liquid
oxygen is much denser than liquid hydrogen,
so it doesn’t take up nearly as much volume.
So most of the external fuel tank for the shuttle
was for the hydrogen.
And now that we’ve looked at how liquid engines work,
let’s put it all together and look at a space shuttle launch
where of course, in addition to the liquid engines, which
ignite about six seconds before the solids,
they come up to speed and they don’t have quite enough thrust
to get the shuttle off the ground.
That’s why we have the solid boosters, which
have just ignited.
And now the whole stack lifts off
on one of the shuttle’s many successful missions.
But we need to find out how much we can get out
of a limited amount of propellant
that we can carry on board.
And that is what we’ll learn about in the famous rocket
equation, which we’ll cover in the next segment

MO 2.2

Rocket Propellant
(2 points possible)

What are the two components that make up the propellant of a rocket? Be
sure to check your spelling, and do not leave any extra spaces before or
after your answer.
First component:


MO 2.1 MO 2.2

Schematic of a water bottle rocket (Image credit: NASA)
Building a rocket that can launch into orbit is a highly complex feat of
engineering. However, the principles of propulsion can be demonstrated
using a much simpler water bottle rocket. In a water bottle rocket, the frame
is constructed from plastic soda bottles, the fins are typically made from
materials such as cardboard or balsa wood, and the propellant consists of
water, which is expelled from the rocket by pressurized air supplied from a
bicycle pump.
In the following video, watch as Prof. Hoffman explains how a water bottle
rocket works and launches one from the MIT athletic fields.

Video Segment: Water Bottle Rockets
Well, here we are out on the MIT athletic fields,
and I’d like to demonstrate the principles of rocketry using

not a chemical rocket but a water rocket, which
maybe some of you have seen things like this.
This is a pretty fancy water rocket, I have to say.
It was built by one of our MIT freshman.
What’s the difference between a chemical rocket and a water
rocket and what are the principles which are the same?
Well, in both cases, of course, we
generate the thrust by pushing something out the end.
For the water rocket, of course, instead of hot gases
we’ll be pushing out water.
The other big difference is the energy
that we use to push it out.
For a chemical rocket, we actually
burn the propellants, the fuel, and the oxidizer.
And that provides the energy to push the hot gases out.
In this case, after we put the water in– I’m
going to put the rocket on here.
And this is connected to a bicycle pump,
so the energy for this water rocket
comes not from burning chemicals but from the compressed air
which I’m going to pump in.
And then we have little pin here, which I’ll put in.

And that will hold the rocket in place so that it can’t come up.
And then at the right time, I will do a countdown,
pull the pin.
And if everything works right, the rocket will come up.
The water will be pushed out by the compressed air,
and we’ll have a nice launch.
Take one.
Take two.
Take three.
Four, five, six, whatever.
All right, so here’s the water rocket.
And now we have to put the propellant
in it, which is the water.
So I’m going to fill it about a third of the way.
We need to leave enough room for the air.

All right, that should do it.
So now we have the water, which is our propellant,
and I’m going to put it down on the launch apparatus.

Now we’re going to put in the restraining pin.

OK, we are ready to go.
Now I’m going to put in the energy
that we need for the rocket.
And since everything we do in space flight,
we’re very concerned with safety,
I’m going to put on my safety goggles.
And now we’ll pump it up to about three atmospheres.

And I think we’re ready to go.
So five, four, three, two, one, launch.


That’s a pretty tough rocket.



Timeline – Brief History of Rocketry

Roll your mouse over a picture to learn more.

Humanity has always looked to the stars. Throughout time, people have
dreamed of traveling to the far distant points of light in our night sky. It is
only in the latter half of the twentieth century, however, that humans have
actually left the Earth and set foot on the Moon or sent robotic spacecraft
throughout the solar system.
The vehicle that has made such travel possible is the rocket. Today’s rockets
are remarkable collections of human ingenuity that have their roots in the
science and technology of the past. They are natural outgrowths of literally
thousands of years of experimentation and research on rockets and rocket

300 B.C.
Steam Powered Rockets
One of the first devices to successfully employ the principles
essential to rocket flight was a model pigeon made of wood and
suspended from the end of a pivot bar on wires. The writings
of Aulus Gellius, a Roman, tell the story of a Greek
named Archytas who lived in the city of Tarentum, now a part of
southern Italy. Somewhere around the year 300 B.C., Archytas
mystified and amused the citizens of Tarentum by flying a model
pigeon. Escaping steam propelled the bird, which was suspended
on wires. The pigeon used the same action-reaction principle as
the rocket, which was not stated as a scientific law until the 17th
About three hundred years later, another Greek, Hero of
Alexandria, invented a similar rocket-like device called
an aeolipile. It, too, used steam as a propulsive gas.
Hero mounted a sphere on top of a water basin. A fire below the
basin turned the water to steam, which traveled through pipes and

into the sphere. Two L-shaped tubes on opposite sides of the sphere allowed the steam to escape
and provided a thrust that caused the sphere to rotate.

100 A.D.
First True Rockets
Just when the first true rockets appeared is unclear. Stories of early rocket-like devices appear
sporadically throughout the historical records of many cultures. It is likely that the first true rocket
flights were the result of accidents. In the third century B.C., the Chinese reportedly developed a
simple form of gunpowder made from saltpeter, sulfur, and charcoal dust. The powder was used to
create explosions during religious festivals in order to frighten away evil spirits. Bamboo tubes were
filled with the powder and tossed into fires. It may be that some of the tubes failed to explode and,
instead, skittered out of the flames and along the ground,
propelled by hot, leaking gases.
Early observations of such phenomena almost certainly
led to more coordinated activity. The Chinese are known
to have experimented with gunpowder-filled tubes of
different designs. Among other things, they attached
bamboo tubes to arrows and launched them with bows,
creating a device called the fire arrow. Fire arrows, having better range than ordinary arrows,
eventually found their applications in battle. The Chinese also discovered that gunpowder tubes
could be launched by simply igniting the powder and releasing the tube. The bow was not essential
to getting the fire arrow aloft! Thus, the first true rockets were born.
The first recorded use of fire arrows occurred in 1045 A.D. An official named Tseng Kung-Liang
wrote a complete account of the Chinese use of gunpowder called The Wu-ching Tsungyao (Complete Compendium of Military Classics).

13th Through 16th Centuries
Rockets as Weapons

Rockets were first used as actual weapons in the battle of
Kai-fung-fu in 1232 A.D. The Chinese attempted to repel
Mongol invaders with barrages of fire arrows and, possibly,
gunpowder-launched grenades. The fire-arrows were a
simple form of a solid-propellant rocket. A tube, capped at
one end, contained gunpowder. The other end was left
open and the tube was attached to a long stick. When the
powder was ignited, the rapid burning of the powder
produced fire, smoke, and gas that escaped through the
open end and produced a thrust. The stick acted as a
simple guidance system that kept the rocket headed in one
general direction as it flew through the air. It is not clear
how effective these arrows of flying fire were. But one
source reported that one grenade could incinerate a 2,000
square foot area.
Following the battle of Kai-Keng, the Mongols produced
rockets of their own. During the 13th to the 15th centuries,
the Mongols used rockets in their attacks on Japan and
Baghdad and may have been responsible for the spread of
rockets to Europe. In England, a monk named Roger Bacon worked on improved forms of
gunpowder that greatlyincreased the range of rockets. In France, Jean Froissart found that more
accurate flights could be achieved by launching rockets through tubes. Froissart’s idea was the
forerunner of the modern bazooka. Joanes de Fontana of Italy designed a surface-running rocketpowered torpedo for setting enemy ships on fire.
By the 16th century rockets fell into a time of relative disuse as weapons of war, though they were
still used extensively in fireworks displays. A German fireworks maker,Johann Schmidlap, invented
the first “step rocket,” a multi-staged vehicle for lifting fireworks to higher altitudes. A large rocket
was ignited initially and carried one or more smaller rockets. When the large rocket burned out, the
smaller rockets ignited and continued to a higher altitude before showering the sky with glowing
cinders. Schmidlap’s idea, known today as staging, is basic to all modern rocketry.
Nearly all uses of rockets up to this time were for warfare or fireworks; but there is an interesting old
Chinese legend that reports the use of rockets as a means of transportation. With the help of many
assistants, a Chinese official named Wan-Hu assembled a rocket-powered flying chair. The chair
was mounted between two wooden stakes. Attached to the chair were two large kites, and fixed to
the kites were forty-seven fire-arrow rockets.
On the day of the flight, Wan-Hu sat in the chair and gave
the command to light the rockets. Forty-seven assistants,
each armed with torches, rushed forward to light the
rockets. In a moment, there was a tremendous roar
accompanied by billowing clouds of smoke. When the
smoke cleared, Wan-Hu and his flying chair were gone. No
one knows for sure what happened to Wan-Hu, but it is
probable that the event really did take place. Fire-arrows
are still as apt to explode as to fly!

7th through 19th Centuries
Rocketry as a Science
During the latter part of the 17th century, the scientific
foundations that apply to all modern rocketry were laid by the
English scientist Sir Isaac Newton (1642-1727). Newton
expressed his understanding of physical motion via three
scientific laws. The laws explain how rockets work and why
they are able to function in the vacuum of outer space.
Newton’s laws had a practical impact on the design of
rockets. About 1720, a Dutch professor, Willem
Gravesande, built model cars propelled by jets of steam. At the same time, rocket experimenters in
Germany and Russia began working with rockets of greater and greater mass. Some of these
rockets became so powerful that their escaping exhaust flames bored deep holes in the ground
even before liftoff.
Toward the end of the 18th century and early into the 19th, rockets experienced a brief revival as a
weapon of war. The success of Indian rocket barrages against the British in 1792 and again in 1799
caught the interest of an artillery expert, Colonel William Congreve, who set out to design rockets
for use by the British military.
The Congreve rockets were highly successful in battle.
Used by British ships to pound Fort McHenry in the War of
1812, they inspiredFrancis Scott Key to write about “the
rockets’ red glare” in his poem, “The Siege of Fort
McHenry,” which we know today as “The Star-Spangled Banner.”
Even with Congreve’s work, the accuracy of rockets still had not improved much from the early days.
The devastating nature of war rockets during this era was not their accuracy or power, but their
numbers. During a typical siege, thousands of them might be fired at the enemy. The effects of such
a rain of rockets could be devastating! All over the world, rocket researchers experimented with
ways to improve accuracy. The Englishman William Hale developed a technique called spin
stabilization, in which the escaping exhaust gases expanded through small vanes at the bottom of
the rocket, causing it to spin much as a bullet does in flight. Variations of this principle are still used
Rockets continued to be used with success in battles all over the
European continent. However, in a war with Prussia, the Austrian
rocket brigades met their match against newly designed artillery
pieces. Breech-loading cannon with rifled barrels and exploding
warheads were far more effective weapons of war than the best
rockets. Once again, rockets were relegated to peacetime uses.

20th Century and Beyond
Modern Rocketry
In 1898, a Russian schoolteacher, Konstantin Tsiolkovsky (18571935), proposed the idea of space exploration by rocket. In 1903,
Tsiolkovsky published a report entitled Exploration of the Universe
with Rocket Propelled Vehicles. In it, he suggested the use of liquid
propellants for rockets in order to achieve greater range. Tsiolkovsky

stated that the speed and range of a rocket were limited only by the exhaust velocity of escaping
gases. For his ideas, careful research, and great vision, Tsiolkovsky has been called the Father of
Modern Astronautics.
Early in the 20th century, an American, Robert H. Goddard (1882-1945), conducted a variety of
practical experiments in rocketry. He was interested in a way of achieving higher altitudes than were
possible for lighter-than-air balloons. He published a pamphlet in 1919 entitled A Method of
Reaching Extreme Altitudes, a mathematical analysis of what is today called the meteorological
sounding rocket.
Goddard’s earliest experiments were with solid-propellant
rockets. In 1915, he began to try various types of solid fuels
and to measure the exhaust velocities of the burning gases.
While working on solid-propellant rockets, Goddard became
convinced that rocket efficiency would be greatly improved
by using liquid fuel. No one had ever built a successful
liquid-propellant rocket before. Doing so was much more
difficult than
rockets. Fuel
and oxygen
turbines, and
chambers all
would be
In spite of
achieved the first successful flight with a liquidpropellant rocket on March 16, 1926. Fueled by liquid
oxygen and gasoline, the rocket flew for only two and
a half seconds, climbed 12.5 meters, and landed 56
meters away in a cabbage patch. By today’s
standards, the flight was unimpressive. But like the
first powered airplane flight by the Wright brothers in
1903, Goddard’s gasoline rocket was the forerunner
of a whole new era in rocket flight. Goddard’s
experiments in liquid-propellant rockets continued for
many years. His rockets became bigger and flew
higher. He developed a gyroscope system for flight
control and a payload compartment for scientific
instruments.Parachute recovery systems were
employed to return rockets and instruments safely.
Goddard, for his achievements, has been called
the Father of Modern Rocketry.
In 1923 a third great space pioneer, Hermann
Oberth (1894-1989), published a book entitled The
Rocket into Interplanetary Space. His book became

the handbook for amateur rocketeers. Because of Oberth’s work, many small rocket societies
sprang up around the world. In Germany, the formation of one such society, the Verein fur
Raumschiffahrt (Society for Space Travel), led to the development of the V-2 rocket, which was
used against London during World War II. In 1937 Oberth and other German engineers and
scientists assembled in Peenemunde on the shores of the Baltic Sea. There the most advanced
rocket of its time was built and flown under the direction of Wernher von Braun. For his
achievements, Oberth has been called the Father of Space Flight.
The V-2 rocket (in Germany called the A-4) was small by comparison to today’s rockets. It achieved
its great thrust by burning a mixture of liquid oxygen and alcohol and was able to lob a one-ton
warhead 50 miles high and hundreds of miles down range. The rocket fuselage was made of thin,
collapsible metal that was inflated with the introduction of fuel into the tanks. Once launched, the V-2
was a formidable weapon that could devastate entire city blocks.
Fortunately for London and the Allied forces, the V-2 came too late in the War to change its
outcome. Nevertheless, by the War’s end, German rocket scientists and engineers had already laid
plans for advanced missiles capable of spanning the Atlantic Ocean and landing in the United
States. These missiles would have had winged upper stages but very small payload capacities.
With the fall of Germany, many unused V-2 rockets and components were captured by the Allies.
Many German rocket scientists came to the United States, while others went to the Soviet Union.
The German scientists who came to the U.S., including Wernher von Braun and Georg von
Tiesenhausen, were amazed at the progress Goddard had made.
Both the United States and the Soviet Union realized the potential of the rocket as a military weapon
and began a variety of experimental programs. The U.S. site chosen by von Braun and his
colleagues, was Redstone Arsenal in Huntsville, Alabama, the site at which NASA’s Marshall Space
Flight Center stands today. The United States first began developing its space program with highaltitude atmospheric sounding rockets, one of Goddard’s early ideas. Later, a variety of mediumand long-range intercontinental ballistic missiles were developed. These became the starting point
of the U.S. space program. Missiles such as the Redstone, Atlas, andTitan would eventually
launch astronauts into space.
On October 4, 1957, the world was stunned by the news of the world’s
first Earth-orbiting artificial satellite launched by the Soviet Union.
Called Sputnik I, the satellite was about the size of a basketball,
weighed about 183 pounds, and had an orbital period of 98 minutes. It
was the first successful entry in a race for space
between the two superpower nations. Less than a
month later, the Soviets followed with the launch
of Sputnik 2 carrying a dog named Laika on
board. Laika survived in space for seven days
before being put to sleep before the oxygen
supply ran out.
A few months after the first Sputnik, the United
States followed the Soviet Union with a satellite of its own. Explorer I was
launched by the U.S. Army on January 31, 1958. In October of that year, the
United States formally organized its space program by creating the National
Aeronautics and Space Administration (NASA). NASA is a civilian agency
with the goal of peaceful exploration of space for the benefit of all humankind.
The Soviet Union led the Space Race in the early days. But the U.S. persisted and gradually
captured the lead, culminating with its Apollo Program to the Moon, which captured the imagination

of the entire world. Who can forget John F. Kennedy’s daring pronouncement, “We will go to the
Moon during this decade…not because it is easy but because it is hard…” or Neil Armstrong’s words
from the Moon’s Tranquility Base, “That’s one small step for man, one giant leap for mankind.”
The Apollo moon rocket is among the largest rockets ever designed to fly into space. Standing as
high as a skyscraper, the vehicle literally made the ground shake underfoot when the engines were
ignited for liftoff. And they lit the skies as Apollo ascended from Cape Canaveral toward Earth orbit.
America continued its flights to the Moon throughout the decade of the 1970’s, developing, with each
new mission, new confidence and new technology. Perhaps the most spectacular mission of all was
Apollo 13, always to be remembered for the outstanding courage and persistence displayed by all
involved in what could have been one of America’s darkest hours.
Rockets have been used to launch many post-Apollo piloted missions, including Skylab, and the
many STS missions. Rockets have also launched unpiloted military satellites, communications’
satellites, weather satellites, Earth observing satellites, planetary spacecraft, planetary surface
rovers, the Hubble Space Telescope, and so on.
Since the earliest days of discovery and experimentation, rockets have evolved from simple
gunpowder devices into gigantic vehicles capable of traveling into interplanetary space. It might be
interesting to hear the thoughts of those earliest rocket pioneers, with their fire arrows and spinning
spheres, if they could be brought through time and shown where their discoveries have led.
Rockets have certainly opened an important door to the universe.

Close Window

Water (or Bottle) Rockets
Bottle rockets or water rockets, what are they?
When someone mentions bottle rockets, do you envision placing a
firecracker attached to a stick into a glass bottle and launching it?
Water rockets have been a source of entertainment and education
for many years. They are usually made with an empty two-liter
plastic soda bottle by adding water and pressurizing it with air for
launching (like the image to the right).
Soda companies began using plastic bottles in 1970. The
Polyethylene Terephthalate (PET) material used in most plastic soda

bottles today was introduced in 1973.
Water rockets are used in schools to help students understand the principles of aeronautics. The
Science Olympiads provide challenges of bottle rocket design and flight, including altitudes and
distances reached. Many interesting designs and additional information on bottle rockets can be
found with a simple Web search.
Teachers and students provide the following feedback to the Secondary Science Education
Department at the University of Nebraska:

“Two-Liter Pop Bottle Rockets may well be the GREATEST PHYSICAL SCIENCE
TEACHING TOOL EVER CREATED!!” Middle grades students can manipulate and control
variables, see their hypotheses verified or refuted, and graph their findings. High school
students experience the nature of science at its best. They can document their abilities
with the following concepts: inertia, gravity, air resistance, Newton’s laws of motion,
acceleration, relationships between work and energy or impulse and momentum,
projectile motion, freefall calculations, internal and external ballistics, and the practice
of true engineering.
How could something that sounds so simple be so complex? Open your mind to the
science and mathematics behind this educational “toy.” Below are links to a brief
history timeline of rocketry, a comparison between water rockets and a NASA rocket,
and additional information on the parts of a water rocket.

British Broadcasting Corporation
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Hands-on science: Water bottle rockets
Putting science and technology to the test


Hands-on Science

Water Bottle Rockets

Wet Jet Go!
This stunt has lots going on: air pressure, a satisfying ‘whoosh’ noise, a high speed departure
and a good soaking if you are standing close. It’s classic science that’s fun to do as a family or a
team. Practise and challenge your friends!
Download the Water bottle Rockets PDF(378 Kb). Adobe Acrobat is required.

Step by step
Yan’s video guide
Dr Yan shows you how to construct and launch your water bottle rocket safely

Difficulty: moderate

Some adult help needed

Time/effort: moderate

Half an hour or so to make. Launching takes

Hazard level: slight

Sharp tools while building. Take care at

SAFETY: This rocket uses air pressure to fling a bottle at high speed 20–50 metres.
The bottle must be made entirely of plastic, it must have no sharp points and it must be for a
fizzy drink, so that the plastic is designed to hold pressure inside it.
Making a hole through the cork using a drill, punch or awl is a job for an adult.
The rocket launches with little or no warning and can fly in a random direction. Never fire it
towards or over anyone.

Before you start pumping, make sure the area you are firing into is empty of people, animals
and breakable objects.

You need

One 2 litre plastic fizzy drink bottle

A wine cork that fits snugly in the bottle spout

A valve from a bicycle inner tube, the longer the valve the better. Cycle shops will often give
away old punctured ones

A pump that fits the valve. Hand pumps or foot pumps are fine

Something that can hold the bottle, neck down, at an angle to the ground – the handle of a
garden fork works well but branches, bits of wood or plant pots will all work as a launch pad

A few litres of tapwater
Alternative: see the Football pump method below

What you do

If the valve is too short, cut the cork. Wrap tape around the cork if it doesn’t fit snugly in the
Check the cork will be a good tight fit in the bottle. You can wrap tape around it to make it a bit
Check the valve is long enough to poke through the cork and still let you attach the pump. You
can shorten the cork by slicing it through if you need to.
Safety: Adult help required for the next step.
Make a hole through the length of the cork so that the valve can let air in. Use a drill, or a punch
and a small screwdriver. The hole needs to be as small as it can be while still letting the valve go
Safety: Choose your launch area carefully so you there’s no hazard to you or to anyone else.

The handle of a garden fork makes a good launch holder.
Make a launchpad that holds the bottle with the neck downwards, lets you attach the pump and
stand behind the bottle. One good method is to plant a spade or garden fork in the ground at a
low angle and poke the bottle through its handle.
Into the bottle put roughly 500ml of water (a quarter of its capacity). Seal the bottle with the
cork and valve. Open the valve and attach the pump.
Safety: Check the launch area is clear before you start pumping. Get anyone watching to stand
at least 3 metres back.
Start pumping gently and steadily, keeping the bottle lined up as best you can.
Keep pumping until the cork pops and the bottle flies away.
Football pump method
Here’s a simpler design using a pump that has a needle for inflating things like rugby or soccer
balls. You don’t need the bicycle inner tube valve and you don’t need to make a hole in the cork.
Instead carefully drive the pump’s needle through the cork so you can add air. You will probably
get even wetter using this technique.

What should happen
As you pump, you can see air bubbling into the bottle. When the pressure is too great for the
cork to contain, the cork is pushed out.
The bottle flies off, leaving a trail of water behind it and probably soaking the person pumping
Once the bottle is empty of water, its flight starts to slow.

If it doesn’t work for you
If the rocket doesn’t fly at all, check two things:

Is water escaping from the bottle? Look around the cork to check if it’s sealing reasonably well.

Is air getting into the bottle? Look for air bubbles in the water when you pump. If no, make sure
the valve can open to let air through from the pump. It may be damaged or screwed shut.
If the rocket does fly but not very well, there are things you can adjust.

The amount of water makes a big difference. Start with the bottle one quarter full (500ml water

in a 2l bottle) and adjust it up or down for best flying distance.

Check that your launchpad allows the bottle to fly cleanly away. You may need to try a few
methods depending on your pump and the shape of your bottle.

Flying further
This design is a basic one to show how the principle works. Using just the friction of a cork to
hold in the air and water means the pressure can’t get very high. And most bottles aren’t a great
shape for smooth flight.
There are many other, more advanced, rocket designs. Some people compete in national

Explore the Internet for inspiration and see how you can improve your rocket. There are a few
suggested links top right on this page.

Bear in mind that if you change the design, you also need to consider extra safety precautions.

What’s going on?
Backwards and forwards forces
Rockets work by ejecting something out of the back and a so-called ‘reaction force’ then pushes
the body of the rocket forward.
Here, water and air are shoved out the back. The water is heavier so that’s what gives the bottle
the main kick forwards.
The energy to force the water out is stored as air pressure inside the bottle. You supply the
energy as you pump air into the bottle.
The air pressure inside builds up and pushes on the water. But friction holds the cork in place and
that pushes back on the water, so for a while nothing moves.
Once the friction force can no longer contain the pressure, the cork is shoved out and the
pressure then acts on the water to eject it from the bottle.
Compared to the bottle, the water is heavy. So pushing water out at a moderate speed
backwards gives the bottle a lot of forward speed.

Sir Isaac Newton
Perhaps you have heard of Newton’s laws of motion. Isaac Newton was the 17th century British
scientist whose ideas about gravity and other forces transformed science. Oh, and an apple may
have fallen on his head.
The water bottle rocket demonstrates two of his laws.

Every action has an equal and opposite reaction. The water moves one way and the bottle goes
the opposite way.

The same amount of force can accelerate a heavy object slowly or a light object more quickly.
The water is much heavier than the empty plastic bottle.



Water Rocket Launch Pad
Instructions for Building the Launcher
The launcher is simple and inexpensive to construct. Most needed parts are available
from hardware stores. In addition you will need a tire valve from an auto parts store and
a rubber bottle stopper from a school science experiment. The most difficult task is to drill
a 3/8-inch hole in the mending plate. An electric drill is a common household tool. If you
do not have access to a drill or do not wish to drill the holes in the metal mending plate,
find someone who can do the job for you. Ask a teacher or student in your school’s
industrial arts shop or the parent of a student to help.
Materials Needed

4 5-inch corner irons with 12 3/4-inch wood screws to fit
1 5-inch mounting plate
2 6-inch spikes
2 10-inch spikes or metal tent stakes
2 5-inch by 1/4-inch carriage bolts with 6 1/4-inch nuts
1 3-inch eyebolt with 2 nuts and washers
4 3/4-inch diameter washers to fit bolts
1 #3 rubber stopper with a single hole
1 Snap-in Tubeless Tire valve (small 0.453 inch hole, 2 inches long)
Wood board 12 x 18 x 3/4 inches
1 2-liter plastic bottle
Electric drill and bits including a 3/8-inch bit
Screw driver
Pliers or open-end wrench to fit nuts
12 feet of 1/4-inch cord

Construction of the Launcher
1. Prepare the rubber stopper by enlarging the hole with a drill. Grip the stopper
lightly with a vice and gently enlarge the hole with a 3/8 inch bit and electric drill.
The rubber will stretch during cutting, making the finished hole somewhat less
than 3/8 inches.
2. Remove the stopper from the vice and push the needle valve end of the tire stem
through the stopper from the narrow end to the wide end.
3. Prepare the mounting plate by drilling a 1-3/8 inch hole through the center of the
plate. (As safety precautions, hold the plate with a vice during drilling and
wear eye protection.) Using a drill bit slightly larger than the holes, enlarge the
holes at the opposite ends of the plates. The holes must be large enough to pass
the carriage bolts through them.
4. Lay the mending plate in the center of the wood base and mark the centers of the
two outside holes that you enlarged. Drill holes through the wood big enough to
pass the carriage bolts through.
5. Push and twist the tire stem into the hole you drilled in the center of the
mounting plate. The fat end of the stopper should rest on the plate.
6. Insert the carriage bolts through the wood base from the bottom up. Place a hex
nut over each bolt and tighten the nut so that the bolt head pulls into the wood.
7. Screw a second nut over each bolt and spin it about halfway down the bolt. Place
a washer over each nut and slip the mounting plate over the two bolts.
8. Press the neck of a 2-liter plastic bottle over the stopper. You will be using the
bottle’s wide-neck lip for measuring in the next step.
9. Set up two corner irons so that they look like bookends. Insert a spike through
the top hole of each iron. Slide the irons near the bottleneck so that the spike
rests immediately above the wide neck lip. The spike will hold the bottle in place
while you pump up the rocket. If the bottle is too low, adjust the nuts beneath
the mounting plate on both sides to raise it.
10. Set up the other two corner irons as you did in the previous step. Place them on
the opposite side of the bottle. When you have the irons aligned so that the
spikes rest above and hold the bottle lip, mark the centers of the holes on the
wood base. (For more precise screwing, drill small pilot holes for each screw and
then screw the corner irons tightly to the base.)
11. Install an eyebolt to the edge of the opposite holes for the hold-down spikes. Drill
a hole and hold the bolt in place with washers and nuts on top and bottom.
12. Attach the launch “pull cord” to the head end of each spike. Run the cord through
the eyebolt.
13. Make final adjustments to the launcher by attaching the pump to the tire stem
and pumping up the bottle. Refer to the launching instructions for safety notes. If

the air seeps out around the stopper, the stopper is too loose. Use a pair of pliers
or a wrench to raise each side of the mounting plate in turn to press the stopper
with slightly more force to the bottleneck. When satisfied with the position, thread
the remaining hex nuts over the mounting plate and tighten them to hold the
plate in position.
14. Drill two holes through the wood base along one side. The holes should be large
enough to accommodate large spikes (metal tent stakes). When the launch pad is
set up on a grassy field, the stakes will hold the launcher in place as you yank the
pull cord to launch the rocket.
15. The launcher is now complete.

In this section, we will learn why it is so expensive to launch payloads into
space. In particular, we will discuss the various components of mass on a
rocket and use these quantities to derive the Rocket Equation in free space
(also known as the Ideal Rocket Equation or simply the Rocket Equation).

Rockets have a limited size.
You can only put a certain amount of propellants on board.
And so there’s a real question of how much performance can you
get out of a given amount of propellant
that you can load on board.
This is where we get into a little bit of mathematics.

If you don’t want to follow the math,
you can skip ahead a little bit but the conclusions
of this section are really critical.

So even if you don’t want to follow the math,
please do come back and join us when we get to the conclusions
because this famous rocket equation, which we’re
going to derive and then show how it’s used,
is really critical in answering some of the questions
about why space flight is so difficult.
Why is it so hard to get into space?
It takes a lot of energy to get up there.
How much energy do you actually need
to put 1 kilogram in orbit?
Well if you remember your high school physics,
there’s two forms the energy can take.
There’s potential energy that you
need to actually raise 1 kilogram
off the surface of the Earth.
And let’s say we’re going to about 400 kilometers
and and we’re going to neglect the fact that the Earth’s
gravity is an inverse square law and the gravity gets slightly
weaker as you go away, but let’s just
assume that it’s 9.8 meters per second squared, which
is what it is here on the surface of the Earth.
So if we take 1 kilogram, the potential energy
that you need to raise it a certain distance
is the mass times the gravitational acceleration
times the height, mgh.

And we work out the numbers here.
1 kilogram, 9.8 meters per second, 400,000 meters.
We’re using the MKS system, and we get about 3.9 megajoules
of energy.
Now the other energy that we need
is the kinetic energy because if you just
put something 400 kilometers up in the air and let go of it,
it’s going to fall back down to the earth.
We’ll see later in another lecture about orbital mechanics
that in order to stay in orbit, you
need to be going at a rather high speed, in fact,
just about 8 kilometers a second to be in orbit a few kilometers
above the surface of the earth.
And we know from high school physics
that kinetic energy is 1/2 the mass times
the velocity squared, so we take the mass of 1 kilogram,
we square the velocity of 8 kilometers per second,
and we get 3.2 times 10 to the 7th joules, or 32 megajoules.
So we can see in low Earth orbit,
the kinetic energy dominates over the potential energy,
but let’s add them up together.
And we get 36 megajoules.
Now we’re going to convert the energy because 1
joule per second equals a watt.
So one joule equals 1 watt second and 36 megajoules

equals 3.6 times 10 to the 7th watt seconds.
And we can convert that into kilowatt hours, which
are the units that we use to measure electrical energy
Typically I pay maybe a little more than $0.10 a kilowatt
So 10 kilowatt hours, that costs about a dollar.
That’s the amount of energy that it takes
to get 1 kilogram into orbit.
And I hope you find that surprising
because the amount of energy, if we could just put that energy
right into that 1 kilogram, is really very inexpensive.
It would revolutionize the space business if we could do that.
But of course, we have to use rockets to get into space.
And the propellant that you need for that last little bit
of your acceleration up to orbital velocity (the speed of something in a given direction)
you have to carry all the way with you.
So you need a propellant to carry the propellant
that you’re going to use, and then you need more propellant
to carry that, and the whole thing is
kind of an exponential runaway.
And that’s what we’re going to look at when we
deal with the rocket equations.
So this is kind of a surprising result for most people.
It really shows what we sometimes

refer to as the curse of the rocket equation
because this is what we have to deal with.
We can’t get 1 kilogram to orbit for the cost of $1.
It does cost many thousands of dollars.
We can do much better.
So let’s take a look at the rocket equation.
Let’s look at a schematic of a simple rocket.
And as we saw before, we have the propellants,
we have the fuel, we have the oxidizer.
Together those are the propellant
and we can talk about the mass of the propellant.
Of course, the reason we’re launching the rocket
is we have a payload which we want to put into orbit.
So we have the mass of the payload, but then the rocket
itself has a certain amount of mass.
The structure, the skin around the outside of the rocket,
the fins, the mass of the rocket motor itself, and so on.
All that is the structure of the rocket
and we lump that together as the structural mass.
And so when you have a fully loaded rocket sitting
on the launchpad, we have what we
call the initial mass of the rocket, which
is the mass of the payload plus the mass of the structure
plus the mass of the propellant.
And we’ll see later that for typical rockets,

the mass of the propellant is completely dominant.
So now we fire the rocket engine.
And for simplicity now, let’s assume
that we’re out in free space so we
don’t have to worry about atmospheric drag.
We don’t have to worry about gravity.
There’s no force acting on the rocket except the force
of the rocket engine itself.
If we want to calculate the thrust, that is the force, what
we’re really looking at is, remember,
we have all of the gas that’s now being
directed out the back side.
And we take the amount of material that’s coming out,
that’s the mass flow rate, the time
derivative of the mass times the velocity at the exit
or the exhaust velocity.
We call it v sub e.
And that will give you, by conservation of momentum, that
gives you the forward thrust.
Now there’s another term here that
has to do with the pressure, which
we’re going to ignore for now.
Let’s just assume we’re flying in a vacuum.
So basically this is the equation
we’re going to work with.

The forward thrust of the rocket equals
the rate at which the mass is flowing out of the rocket.
That is the rate of the mass of the exhaust
times the exhaust velocity.
And by conservation of momentum, we’ll
get forward motion of the rocket.
All right, so let’s take the rocket on the launch pad.
And we showed the initial mass, now we’re in free space.
We look at the rocket and what’s the difference?
It’s now still has its payload, still has its structure,
but the propellant tanks are empty.
And so the final mass of the rocket
is now the initial mass of the rocket minus the mass
of the propellant.
This is just setting up the basics
for calculating how much we can get out
of that mass of propellant.
The mass of propellant is fixed.
It’s limited by the size of your rocket and the way
you store your propellants.
Now let’s look at how all these equations fit together.
So remember that we looked at the basic force
from the conservation of momentum,
which equals the mass flow rate times the exhaust velocity.
That’s the force.

But we also know from Sir Isaac Newton
that force equals mass times acceleration.
Now the mass is changing because you’re taking some of the mass
and you’re throwing it out the back of the engine.

m is the instantaneous mass at any given time.
And so the force equals the mass times the acceleration,
which is the time rate of change of the velocity.
And that’s the velocity of the rocket
now, which is not the same as the velocity of the exhaust,
which when we multiply it by the time
derivative of the mass, that is the mass flow rate,
also gives us the force.
So we have these two expressions for the force.
One from the conservation of linear momentum,
and one from Newton’s law.
And now we can work with that.
You notice we have dt in both denominators.
We can get rid of that.
And so we’re left with an expression of dv
on the left side.
That’s the increment in velocity that you
get by burning that little increment of mass,
which is the dm.
That’s the mass which you’ve just thrown

out the back of your engine.
And the equation, of course, has the v exhaust,
which we can treat as a constant,
and dm divided by the current mass on the right hand side.
Now let’s integrate from the initial state
to the final state.
And again, if you haven’t had calculus
and you can’t follow this, hang in and come back and look
at the final conclusion because it’s really quite important.
So we’re going to integrate from the initial state,
in the case it’s the initial velocity to the final velocity.
On the other side it’s from the initial mass to the final mass.
And of course the integral of dv is just v.
And so we have on the left hand side
the final velocity minus the initial velocity, which
we call the change in velocity.
And delta is the Greek symbol we use for change,
so we often refer to this as the delta v or the velocity
increment that we can get by burning
a certain amount of propellant.
And on the right side, the integral of dm over m
is just the natural log of m and so we put in the numbers
and we get the log of the final divided by the initial mass.
We have a minus sign because the exhaust velocity is going
in this direction, but the velocity of the spaceship

is it increasing in this direction,
and so we can get rid of the minus sign
and just flip the numerator and denominator in the logarithm.
And we now have, essentially, the rocket equation.
The amount of change and velocity
that you can get by burning a certain amount of propellant
is equal to the exhaust velocity of that propellant
coming out the back of the rocket
times the natural log of the ratio between the initial mass,
that’s with the propellant, and the final mass once you’ve
burned all of that propellant.
This rocket equation essentially tells you
how much you can increase the velocity of your rocket,
or decrease if you want to slow down.
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MO 2.3

The Rocket Equation was developed by the Russian scientist Konstantin
Tsiolkovsky, and uses the principle of conservation of momentum to
relate the maximum change in velocity (ΔV) to the propellant exhaust
velocity and the initial and final rocket mass (mi and mf, respectively).
Using just basic physics and math (with a little bit of calculus), the
Rocket Equation expresses one of the most fundamental relations in all
of astronautics and has significant implications for spaceflight.
As you watch the following video, pay special attention to what the
various quantities in the Rocket Equation imply for propellant fraction
and rocket performance. You will be asked to apply what you have

learned to complete several problems.

Video: Implications of the Rocket Equation
Konstantin E. Tsiolkovsky was a very famous Russian visionary
of space travel.
He was a school teacher out in rural Russia
and received essentially no government support
until very late in his life when his work
started to be recognized.
But on his own, he essentially not only
derived the rocket equation, but speculated
on many aspects of space travel what
it would be like being weightless, how you could build
a space station with artificial gravity, problems of radiation
in space.
All sorts of things.
So Tsiolkovsky, really an amazing guy.
And let’s now take a look at what his famous rocket
equation actually means for rocket science.
So here’s the rocket equation that we just derived.
And we can change it from the logarithmic form
to an exponential form.
And we see that the ratio of the initial to the final mass,
that is before and after you burn the propellant,
is the exponential of the change in velocity, which

you get from burning this propellant, divided
by the exhaust velocity.
Now, we’ll see why a statement that I made earlier
is actually true.
I said that the critical things in the performance of a rocket
engine are first of all how much material you can throw out
the back and that is the more your initial mass can
be compared to your final mass,
because it’s all the propellant and how fast
you can throw it out the back,
and that’s the exhaust velocity.
Let’s take a look at what that means.
We can look at, for instance, the mass that
is the propellant, which is just the initial
minus the final masses.
And just a little algebra and we can get a expressions for that.
This is important, because it’s often
referred to as the propellant fraction.
That is, when you start out with the rocket fueled up
on the launch pad,
what fraction of its total mass has to be propellant in order
to get you the velocity you need?
So that’s how we would derive it.
And the rest of it is inert.
You can actually do lots of algebra on all of this.

But let’s look at why the exhaust velocity
is so absolutely critical.
For typical chemical rockets, that
is all the way from solid rockets, which
are at the low end of the scale,
their exhausts goes out at about two kilometers a second.
The most efficient liquid propellant rockets
burning hydrogen and oxygen, they
could push their propellant out at about 4 and 1/2 kilometers
a second.
And by the way, just in case you ever
run across the expression specific impulse,
engineers use this for various historical reasons,
which I won’t go into.
But it is a very simple relationship.
The exhaust velocity is the specific impulse
multiplied by the acceleration of gravity at the Earth.
That is 9.8 meters per second squared.
And it doesn’t matter if you’re taking off
from Mars, or the Moon, or Jupiter,
or wherever you still use g zero, the acceleration
at the Earth’s gravity.
So chemical rockets have specific impulses, roughly
from 200 to 450 seconds.
And that’s because it’s acceleration here

and velocity this specific impulse is in seconds.
So what does this all matter?
And why do we care about having a higher or a lower exhaust
Well, orbital velocity in low Earth orbit
is about eight kilometers a second.
But if you’re launching through the atmosphere
you have atmospheric drag and what we call gravity launch
losses, which I’m not going to go into now.
We’ll talk about flying through the atmosphere a little later.
We’re also not going to talk about the fact
that the Earth is spinning.
Let’s just assume that we need to have
a velocity change of nine kilometers per second
to get to orbit.
Now, let’s put in some numbers.
Suppose we’re using a solid rocket
with a typical exhaust velocity of two kilometers a second.
So the delta V of 9 divided by the exhaust velocity of 2
gives us 4.5.
And when we put that into the rocket equation,
we see that the ratio of the final to the initial masses
is just a little over 1%.
What does that actually mean?

It means that the mass of the propellant when you’re
sitting on the launch pad has to be 99% of the entire mass.
That means that only 1% of the mass
can be devoted to the structure of the rocket itself
and the payload that’s sitting up on the top
and that’s not enough.
Frankly, we can’t build a rocket like that.
But now let’s look what happens if we just
double the exhaust velocity.
Just a factor of two.
Now that ratio is 2.25, but the critical thing and here’s
what you have to remember about the rocket equation,
that critical number is in the exponential.
We’ve only changed the exhaust velocity by a factor of two,
but look what it did to the ratio of the propellant
to the initial mass.
We now can devote up to about 10% of the mass of our rocket
to the structure and the payload, because our propellant
fraction is only 89%.
So instead of 1%, we’ve gone all the way up to 10% of mass
that we can devote to the structure in the payload
even though we only changed the exhaust velocity
by a factor of two and that’s the real essence of the rocket

You’re dealing with this exponential
and increasing the exhaust velocity
by just a little bit can really make a huge difference
in the performance you get out of the engine for a given
amount of propellant.

MOMENTUM = (All objects have mass; so if an object is moving, then it has momentum – it
has its mass in motion. The amount of momentum that an object has is dependent upon two
variables: how much stuff is moving and how fast the stuff is moving. Momentum depends
upon the variables mass and velocity).
(Momentum is a vector quantity that is the product of the mass and the velocity of an object or
particle. The standard unit of momentum magnitude is the kilogram-meter per second (kg. ·
m/s or kg.)


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