Notes for 8/31 and 9/2

In the last class we talked about the earliest use of rocketry, dating from about 2000 BC, and extending up to 1900. The use of rockets in various battles, wars, and skirmishes in the 18^th and 19^th centuries was quite common-note that the Star Spangled Banner mentions "the rocket's red glare" when describing the battle at Fort McHenry (Baltimore, Maryland during the War of 1812, September 13^th and 14^th, 1814, for more on this history go to this site).

The rockets of this time used gunpowder. The origin of gunpowder dates back more than four thousand years. Gunpowder is composed of 75% Potassium Nitrate (KNO₃), 15% Charcoal (carbon), and 10% Sulfur. To make a firecracker, you pack this mixture into a tightly rolled tube of cardboard, or paper, insert a fuse, and seal the ends. Why would anyone have discovered such a mixture?

Well, charcoal ( carbon) is known to be a laxative, so it was, and has been a commonly used product throughout history. Sulfur is one of the essential minerals of life, and has long been prescribed for various ailments. But potassium nitrate? It is possible you know it by its common name, "saltpeter" (also sometimes called "niter"). It has a wide variety of uses, but one of its main uses in past times was to "calm" certain urges. You can imagine that somewhere in time, an early pharmacist had an accident, and a candle or spark ignited a bowl of this mixture... and gunpowder was born.

What causes the explosive nature of gunpowder? As we noted, saltpeter is made up of one atom of potassium, one atom of nitrogen, and three of oxygen. In the process of "burning" something, basically what is happening is that the chemical structure is being changed--in most cases it is the fuel that is being "oxidized". When wood burns, the complex molecules (such as cellulose) are broken down. The carbon becomes attached to oxygen (usually creating carbon monoxide, CO, or carbon dioxide, CO₂). The key ingredient to oxidation is oxygen. Potassium nitrate is a very rich source of oxygen. In gunpowder, the carbon and sulfur are "burned". The products are compounds containing carbon and sulfur, including molecules like hydrogen sulfide (that give fireworks their distinctive smell).

In gunpowder, due to the abundance of oxygen, the carbon and sulfur burn very quickly. Because the potassium nitrate supplies the oxygen, gunpowder will burn in a vacuum, or inside a tightly wound tube of cardboard. The burning reaction releases hot gases that try to expand. If the cardboard or paper is wound tightly, the pressure builds up to be eventually released with a bang (a shockwave), creating the sound of the firecracker. If one end of the tube is not tightly sealed, then the gases will exit from this end creating a rocket or cone-type firework. [For more on modern fireworks go here.]

Gunpowder got its name from the time of flintlocks and muskets. It was packed into a rifle or cannon barrel, a projectile ("ball") was pushed on top of the powder charge, and the powder was ignited by a fuse or spark (an excellent site discussing the types of ammunition used during the Civil War can be found here). This type of gunpowder is not used in the bullets/shells of modern guns. The gunpowder in this case is sensitive to a sharp shock, and ignites when hit by the pin of the trigger hammer.

The early rockets were identical to what we call bottle rockets nowadays. It was simply a tube (usually bamboo) that was completely sealed at one end, with a small hole at the other. The fuse ran into this hole, and ignited the gunpowder mixture. The tube was attached to long stick to provide balance. Later editions included fins, and looked more like the rockets we are familiar with.

The Physics of Rocketry

To understand how rockets work, we must discuss the concepts of elementary physics.

1) Mass

2) Velocity

3) Acceleration

4) Force

Mass:

The amount of matter in an object is its mass. Mass is measured in kilograms. Mass is not the same as "weight". Weight results from the mass of the earth pulling on the mass of your body (or some other object). Your weight depends on where you are located, but your mass is independent of your location.

Velocity:

Velocity is a measure of speed, defined as the distance traveled each second (or unit of time, such as hour). For example, miles per hour (mph) is a velocity. If you travel at constant velocity, than the distance traveled is:

Distance = velocity x time

If your velocity is 100 mph, you will travel 100 miles in one hour. The physical sciences use the metric system, based on the kilogram, the meter and the second. So velocities are designated in meters per second (m/s): 100 mph = 45 m/s (1 m/s = 2.237 mph)

Acceleration:

But what if your velocity is changing with time? Like when you push the gas pedal to the floor and keep it there. This is called accelerating. Given a long enough straight-away, eventually your car would reach its top speed and stop accelerating. If you completely let-off of the gas pedal, you experience a negative acceleration, called "deceleration" (this deceleration is caused by friction).

If you accelerate at a constant rate for a specified amount of time, you can calculate your velocity from:

Velocity = acceleration x time (V= a x t, or a = V/t)

Example: Drag racers can run a quarter mile (416 meters) in a time of 6 seconds or less. What is the (average) acceleration they undergo?

First calculate the car's average speed: D = v x t, or v = D/t = 416/6 = 69 m/s

So: 69 m/s = a x t

which we rearrange to be: a = 69/t = 11.56 m/s²

What is the car's velocity after one second? (11.56 m/s = 26 mph)

What would be our velocity if we accelerated at this rate for 100 seconds?

(1156 m/s, or about 2600 mph!)

Newton's Laws

#1: An object at rest will remain at rest, and an object in motion will stay in motion (on a straight line), unless acted upon by a force.

#2: The acceleration of an object of mass "m" is proportional to the force applied to it, or:

#3: "For every action, there is an equal and opposite reaction"

Let's examine our drag car. Assume it has a mass of 1000 kg (equivalent to 2,200 lbs on Earth). What is the force applied to it in our earlier example?

F = 1000 (kg) x 11.56 m/s² = 11,560 kg-m/s² = 11,560 Newtons (N)

If we could halve the mass of our car, how fast could we run the quarter mile if our engine produced 11,560 N? We have to do some rearranging to solve the equations for the unknown "t", since we only know the distance D, the mass m, and the force F. But we can calculate "a" from Newton's second law.

Remember that v = a x t, or a =v/t, and D = v x t, or v= D/t.

Let's write a = F/m = v/t = D/t², so we have:

F/m = D/t². We know D (416 m), and we know F and m:

11,560/500 = 416/t², or t² = 416 x 500/11,560 = 17.99 s²

Taking the square root, our time to run the quarter mile is: t = 4.24 seconds. If you will notice 4.24 = 6/(square root of 2), we reduced the weight by 1/2, but only improved the time by the 1/sqrt(2). What was our velocity? V = a x t = 98.0 m/s (219 mph).

If you are going to go fast, it is very important to keep the mass of your car low, while keeping the force exerted by the engine as high as possible. The same rules apply to rockets.

The Rocket Equation

Earlier we discussed mass versus weight. What is weight? Weight is a force caused by the pull of the earth's gravity:

W = m x g

where "g" is the acceleration due to the mass of the earth pulling on some object of mass "m". In the metric system weight is measured in Newtons, in the English system it is measured in pounds. At the surface of the Earth, the acceleration due to gravity has a value of g = 9.8 m/s². This is the force that keeps you confined to the Earth's surface. This force is always directed downwards toward the Earth's center. To launch a rocket, or jump over a fence, you must supply a larger force upward.

If we want to go upwards (the "positive" direction), we can think of the force of gravity pulling us downward, so we write:

F_grav = -mg.

our rocket motor (to be discussed later) provides a force upward: F_motor. The net force applied to our rocket F = F_motor - F_grav = ma - mg. To get our rocket to go somewhere, it has to be accelerated at a rate that is higher than the force of gravity (a > g).

In our drag racing car we achieved an acceleration of 11 m/s² without too much difficulty--but obviously, we can't launch the car, since it produces this motion by interaction with the racetrack. Let's assume we had a rocket engine that produced an acceleration of 11 m/s, for 100 s, how fast is a 1000 kg rocket going when the engine cuts off?

The net force is F = ma - mg = m(a - g) = 1.2 m/s² = a. Since v = at, v = 120 m/s.

How high the rocket goes is much more complicated, because after the engine shuts off, the rocket still is traveling at high speed. It is the force of gravity, and the drag forces due to the interaction with air, that determine exactly how far a rocket goes.

To get a rocket into orbit, or to visit another planet requires very high velocities. The equation (which we will not derive here) is:

v = 7.86 x sqrt[2h/(R_earth + h)] (km/s)

For example, the velocity of an object in orbit at a height of h = 200 km above the Earth's surface (R_earth = 6378 km) is:

v = 7.86 x sqrt[400/(6378 + 200)] (km/s) = 1.94 km/s = 4,400 mph

To escape the gravity of the Earth, we must reach a velocity of 24,000 mph (= 10.73 km/s).

The first person to work out much of the physics involved in rocketry was Konstantin Tsiolkovsky.

Konstantin Tsiolkovsky

"KT" was born on September 5, 1857 in the small Russian town of Izhevskoye, 100 miles SW of Moscow.

His father was educated, but earned his living as a forester and they lived in poverty.

At the age of nine, KT came down with scarlet fever, and this disease left him nearly deaf.

The deafness forced him to leave school, and to educate himself. It also provided the impetus to "do something big".

By the age of sixteen he was living in Moscow and continuing his self-taught education in physics.

At this time, what was known about space?

Using Newton's laws, the motions of the planets in the solar system were now understood, and could be predicted to high accuracy (note that the planet Neptune was discovered in 1846 because the planet Uranus was not moving the way it should).

Hot air balloon flights had shown that the higher you go, the lower the density of air. This lead to the conclusion that the space between the planets was empty, it was a "vacuum".

But, how could you move around in a vacuum, with nothing to push against?

KT realized that Newton's third law (for every action there is an equal and opposite reaction) meant you could travel in space--by ejecting mass in one direction, you could move in the opposite direction.

KT also realized that by having rockets pointing in different directions, you could move in any direction you desired--this is how many modern spacecraft orient themselves once in orbit.

By 1897 KT had derived the formulae that describe the motion of a rocket, including the influence of gravity on its motion, and the effects of air resistance.

In 1903, he published a paper with the fundamental result that showed that the maximum velocity that a rocket can achieve is directly related to the velocity at which the gases are exhausted from the combustion chamber:

V_max = 2.3 x V_fuel log(1+m_f/m_R)

Because the quantity inside the parenthesis is "logged" it changes very slowly [for example, log(10) = 1.0, log(20) = 1.3]. To achieve the high velocities required to reach space means that you have to find a fuel with a high V_fuel.

Assume we had a "perfect" rocket that used gunpowder (and m_f/m_r = 9):

V_max = 2.3 x 2136 [log(1 + m_f/m_r)] => 4900 m/s (= 4.9 km/s)= 11,123 mph

(But gunpowder rockets generally had m_f/m_r = 0.25!)

Assume we have an LH2/LOX rocket:

V_max = 2.3 x 4459 [log(1 + m_f/m_r)] => 10,256 m/s (= 10.256 km/s) = 23,280 mph

The Space Shuttle uses two solid rocket motors, each containing 503,487 kg of fuel (total = 1,006,974 kg), and the main engines use 721,000 kg of LH2/LOX, or a total of 1.73 million kg of fuel. Lets say V_ex = 3000 m/s = 3.0 km/s

2.0 = 2.3 X 3.0 [log(1 + m_f/m_r)] => m_f/m_r = 0.95

The payload is 105% of the total fuel mass, so we could theoretically launch 1.8 million kg. Note that this is for perfect efficiency in a vacuum. This neglects air resistance, and the changing gravitational field. In fact, the Space Shuttle can get about 125,000 kg to such an orbit.

KT continued his work, and by 1914, he discussed the forces humans would experience, the concept of "zero gravity", the ways of steering a rocket, multiple-staged rockets.

With the overthrow of the Tsarist empire in 1917, KT's life changed due to the fact that the communist government wanted to promote the achievements of the common worker and, because of his background, this brought renewed attention to KT.

In the mid-20's, he began work on ideas about jet/rocket planes, and revised his work on multi-staged rockets. (His model was quite for this was mostly unworkable.)

Konstantin Tsiolkovsky Summary

1) Derivation of many of the fundamental equations used to describe spaceflight

2) Recognized that liquid fuels were highly desirable propellants.

3) Concept of multi-staged rockets

4) Ideas about space stations, space suits and artificial gravity

5) Ideas about going to the moon and beyond, engineering the environments of planets, populating the asteroid belt, and communicating with extraterrestrials.

Most of these ideas were decades ahead of anyone else. Though the actual application of these ideas through construction of functioning rockets was left to others.

Newton's Laws

#1: An object at rest will remain at rest, and an object in motion will stay in motion (on a straight line) unless acted upon by a force.

#2: The acceleration of an object of mass "m" is proportional to the force applied to it, or:

a = F/m or, more commonly: F = m x a

#3: "For every action, there is an equal and opposite reaction"