Air Powered Rockets: Supplemental Information

The primary goal of science is to understand the universe around us. This might mean trying to understand what a star is made of or what keeps the moon in orbit around the earth. It might mean figuring out which chemical elements found on earth give off heat when they're mixed with another specific element and why that heat is given off. It could be striving to learn how the heart pumps blood and what role that blood has in keeping people alive. Such knowledge is gained through measurement and careful experimentation. The results of a given experiment can be used to build a model of how that system works. More experiments might show whether or not that model is accurate. If it is, scientists attempt to understand the important parts of the model and apply these to our understanding of how nature works in general. That means we try to apply what we've learned for one object to a different object and a different set of circumstances. We predict the outcome of a new experiment. If we do enough experiments and our predictions are always right, our theory of how things work becomes a physical law.

Some of the most useful laws that physicists have discovered are conservation laws. These say that however much of a certain quantity you have at the beginning of an experiment, that's how much you will have at the end. Two of the most important of such laws are the conservation of energy and the conservation of momentum. The conservation of energy says that energy cannot be created or destroyed, only transformed from one type of energy to another. This law has passed the test of experiment so often that physicists have deep faith in it. This faith is so strong, in fact, that when new experiments suggest that energy is not conserved, scientists have hypothesized new forms of energy or new types of unseen particles that carry away the "missing" energy. In every case, further experimentation has shown that the theory of energy conservation is valid. This is how the neutrino was first detected, as missing energy.

Conservation of momentum (= mass multiplied by velocity) is another such law. This says that in the absence of external forces the total momentum of a system is conserved. Newton's second law tells us that forces change an object's momentum, hence we need the no external forces qualifier. Pool balls are a commonly used example. If one ball strikes another ball, the momentum of each ball is changed because each felt the force of the collision. The sum of the two balls' momenta does not change because our "system" includes both balls, and therefore the force is internal to the system, not external. If we drop one of the pool balls, its momentum increases since its velocity increases. The force of gravity was external to our two pool ball system. If we included the earth in our system then total momentum would have once again been conserved as the pool balls' momentum change would have been countered by the earth's momentum change. Here is an important point. Since momentum is mass times velocity, and not mass times speed, direction matters. Two objects of the same mass moving in opposite directions with the same speed don't have the same momentum. They have opposite momenta and the total for the system of the two objects is zero.

Now we can see how our air rocket experiment demonstrates conservation of momentum. At the beginning, everything is stationary so the system has zero momentum. Since the initial momentum is zero, the final momentum must also be zero. The mass of the ejected air times the speed of the ejected air is equal to the mass of the rocket times its speed. Remember that the masses times speeds are equal but the masses times velocities have the same absolute values but different signs, thus the sum of the two is zero. In the air rocket experiment, external forces can play important roles. Addition of flaps to the air rocket makes air resistance slow the rocket's progress. Comparison of the rocket's speed when the flaps are taped down to when they are extended shows that the slowing is due to air resistance and not the added mass of the flaps. The rocket experiment is also a beautiful demonstration of the importance of conservation laws. The air leaves the rocket in a complicated manner but we don't have to worry about the details of how it is exhausted. We only need to compare the mass and velocity of the exhausted air to the mass of the balloon to predict the velocity of the balloon.

By no means is the concept of conservation of momentum and its use for propulsion limited to simple laboratory physics demonstrations. Anyone who has ever fired a shotgun without having it pressed against his shoulder learns quickly not to do that again. The recoiling shotgun can leave quite a bruise. Even if the gun is firmly pressed against the shoulder, it still must absorb that recoil and a day of shooting can leave one with a sore shoulder. Film of World War II combat shows large cannons jumping in recoil as they loft shells with high speeds and early warship designers had to take care that firing the ship's armaments did not cause the ship to roll over in reaction. Jet airplanes and rocket ships work on the same principle as our air powered rocket. In fact many jets and rockets carry more mass in fuel than the mass of the rocket itself. In addition engines are developed which eject this mass at high speed so that the total momentum of the ejected fuel is as high as possible. To determine the final speed of the plane or rocket (neglecting air friction), we start with the conservation of momentum:

Dividing by the ship's mass gives the ship's velocity:

Thus the final speed of the plane or rocket is equal to the mass of ejecta times the speed of the ejecta divided by the mass of the ship, neglecting air friction. This implies that for a high exhaust velocity, only a small fuel mass is necessary, but air friction affects this dramatically. Friction is typically a speed dependent force so that higher speeds lead to higher frictional forces. In addition one likes the fuel to be exhausted uniformly throughout the flight to keep the speed as constant as possible. If all the fuel were ejected at once it might just snap your head back a little. The atmosphere thins with altitude so that air friction is less of a problem at higher altitudes. Rockets that rise above the atmosphere no longer need to overcome this air resistance so their fuel is all spent early in the flight. Jet airplanes burn fuel throughout the flight to overcome air resistance.

One example of a jet airplane is the SR71 blackbird, a plane designed to fly at altitudes of greater than 80,000 feet and speeds greater than Mach 3.2 (>2000 mph). The weight of the plane itself is 60,000 pounds while the fuel weight at takeoff is 80,000 pounds. One could calculate what speed this fuel would need to be ejected at to give the plane a speed of 2000 mph in the absence of air friction (and gravity) using the above formula. If all the fuel is exhausted at 2700 mph the ship (completely exhausted of fuel) would be traveling at about 2000 mph. Clearly, this simplistic view doesn't describe the real situation where the plane needs to counter the gravitational pull of the earth and must fight air friction.

The space shuttle is another example of a system that uses a propulsion system. The shuttle gains orbit through the use of two engines: the main engines which use fuels stored in the large external tank that the orbiter sits on and the small solid rocket boosters that are on either side of the external fuel tank. The external tank contains 1.6 million pounds of liquid oxygen and 226,000 pounds of liquid hydrogen fuel. The main engine burns this fuel and fires for a mere eight and a half minutes at the beginning of each flight. During the first two minutes of flight, the main engine is assisted by the solid rocket boosters which use aluminum powder as fuel. Each motor contains more than one million pounds of propellant. These motors are used to help the very heavy space shuttle near the flight's beginning when air friction is the strongest and the earth exerts a large gravitational pull trying to bring the big ship back to the ground.

As discussed above, once the shuttle is above most of the atmosphere, it loses little energy to air friction and doesn't need engines to stay in orbit like a jet plane needs engines to stay in flight. However, the shuttle does have engines known as the Orbital Maneuvering System (OMS) and others known as the Reaction Control System (RCS). They both use nitrogen tetroxide and monomethyl hydrazine for fuel. The OMS is used to place the orbiter in its final orbit and for extended maneuvering in space, as well as to slow the orbiter down at the end of the mission. The RCS is used to point the shuttle a certain way or to roll it as necessary for the crew to carry out designated tasks. The orbital speed of the shuttle is 17,322 mph and the orbital height can range from 155 miles to 600 miles above the earth's surface. When all the accounting is done we find that the shuttle launch weight is about 4.5 million pounds but it can only carry a payload of 65, 000 pounds, about 0.15% of the takeoff weight. The orbiter itself weighs about 200,000 pounds and the external fuel tank weighs 78,100 pounds. This means that more than 4.1 million pounds, or more than 90% of the shuttle's 4.5 million pound takeoff weight, is in fuel used to propel the ship in just the same way as the air in our rocket propelled it.