|Physics of Propulsion
To better understand specific propulsion
technologies, it is important to have a basic foundation of physics
concept and terms. This helps in understanding how movement
can be produced in space.
Nearly all propulsion technologies that
scientists have developed so far rely on Newton's third law, which
states that for every action, there is an equal and opposite reaction.
Basically, what this law is saying is that if you push on an object
with a certain amount of force, that object pushes back on you with
the same amount of force but in the opposite direction. To understand
the consequences of this law, we have to examine Newton's first two
laws and the concepts of force, impulse, and momentum.
Newton's first law says that if no external forces act on an object, the object will stay at rest if it is at rest or move in a straight line with a constant velocity if it's in motion.
Newton's second law says that if a constant force is applied to an object, it will accelerate in the direction of the force and the rate of acceleration will be directly proportional to the force applied and inversely proportional to the mass of the object. Mathematically, this can be stated as F=ma, where F is the force, m is the mass, and a is the acceleration.
Momentum is the product of the mass of an object and its velocity (P=mv, P is momentum, m is mass, v is velocity). The momentum of an object is what keeps it moving. The amount of momentum an object has determines how hard it is to stop. Something like a penny (low mass) dropping to your hand from a short distance (low velocity) is easy to stop because it doesn't have much momentum, while a train (high mass) or a bullet (low mass, high velocity) is hard to stop because it has high momentum.
Newton's first law can be restated in terms of momentum: If no external forces act on an object, it's momentum remains constant.
Force, defined in Newton's second law as the product of mass and acceleration, can also be defined in terms of momentum. Force is the rate at which momentum changes in a given period of time.
Impulse measures the change in momentum. If the average force acting on an object over a period of t seconds is F, then the resulting impulse would be J=Ft.
RATES OF CHANGE
Frequently in physics, we must deal with rates of change on some quantity. For example, force is the rate of change of momentum, velocity is the rate of change of position, and acceleration is the rate of change of velocity. There are two ways of measuring quantities such as these. I will use velocity as an example, but the same holds for the other rates of change.
First, you could record the position of an object at time t1 (call it s1) and at time t2 (call it s2). Then you divide the change in position by the change in time. So, the velocity would be
Velocity measured this way is called average velocity, because it's not the velocity at a specific point in time. Instead, the velocity is averaged over a period of time.
The second way of measuring velocity is based on the first. If you move t2 closer and closer to t1, the period of time between them comes closer to being a single point in time. If we consider the position of an object to be a function of time such that s(t1)=s1 and s(t2)=s2, we can write this new method of finding velocity as
Velocity found this way is known as the instantaneous velocity or simply the velocity.
The expression ds/dt is a common notation for limits such as these. The "d" basically means "change in". So something like a force F, which is the change in momentum, P, divided by the change in time, t, could be written as dP/dt. A general name of quantities like ds/dt and dP/dt is the derivative. So velocity is the derivative of position and force is the derivative of momentum.
Here's a quick reference table that might help with the next few sections.
||Velocity (v = ds/dt)
||Acceleration (a = dv/dt)
||Force (F = dP/dt)
SYSTEMS OF PARTICLES
In an isolated system of particles, momentum is constant. What does this mean? Well, a system of particles just means some group of objects. It could be a bag of rocks, two cars on a street, or bunch of atoms. An isolated system is a system where the overall forces on that system is zero. For example, a person with a backpack standing on Earth. The only forces acting on this person is gravity and the ground. However, since the force of the ground pushing up on the person's feet cancels out the downward force of gravity, there is no overall force acting on the person (otherwise, he'd be accelerating in some direction). If the person were falling from the sky, on the other hand, the system would no longer be isolated because there's an overall downward force causing the person to fall faster and faster to the ground.
The next question is, what is the momentum of a system? To figure this out, we need to start out with the center of mass. You probably have some idea of what the center of mass of a solid object is, although you might not know it by that name. If you've ever tried to balance something on your hand, you were finding the center of mass of that object. The center of mass, however, doesn't have to be a part of the object. For example, suppose to have a plate. The center of mass would be at the center of the plate. If you took a drill and drilled out the center of the plate, the center of mass would still be at the center of the plate, but now, the point is no longer a part of the plate.
The center of mass of a system of particles is pretty much the same thing, except now you could have something like a plate broken into pieces. Imagine you have a bunch of objects that make up a system of particles, then link them together with rods that are so lightweight that they contribute almost nothing to the mass of the system. Then you can think of the system as a single object, and the center of mass is a little easier to understand.
Suppose you have a system of n objects with masses m1, ..., mn. Let them all be in a line for simplicity. Relative to some reference point, P, (doesn't matter where you choose the point, as long as it stays fixed), you can measure the distance from P to the kth, xk. Also, let xk be negative if it's to the left of P and positive if it's to the right of P. Then, the center of mass (denoted by xcm) can be calculated using this formula:
||m1x1 + ... + mnxn
m1 + ... + mn
If you find the derivative of this expression, you find the velocity of the center of mass, vcm:
||m1v1 + ... + mnvn
m1 + ... + mn
Finding the derivative of vcm, we get the acceleration of the center of mass, acm:
||m1a1 + ... + mnan
m1 + ... + mn
Also, the mass of the entire system is just
the mass of the objects in the system: M = m1 + ... + mn.
With these values defined, we can figure out what the momentum of
a system is and show that it is constant. First, recall that we are
talking about an isolated system where the overall force on the system
is zero. The overall force is just the mass of the system multiplied
by the acceleration of the center of mass of the system. In other
Foverall = Macm = m1a1 + ... + mnan = 0
Similarly, the momentum of the system is given by:
Poverall = Mvcm = m1v1 + ... + mnvn
However, since force is the derivative of momentum (specifically, in this case, Foverall is the derivative of Poverall), Poverall is a constant, i.e.:
m1v1 + ... + mnvn = constant
MOMENTUM AND PROPULSION
what does all of this have to do with propulsion? Good question! Suppose
you're floating out in space holding a rock. Your mass is about 60
kilograms and the rock's mass is about 5 kilograms. You and the rock
make up an isolated system, and so, we can use the above formula for
momentum. No matter what happens within an isolated system, the momentum
must remain constant. Initially, you and the rock aren't going anywhere,
so your momentum is zero and must always be zero (unless some outside
force changes that, but then the system wouldn't be isolated). Now,
suppose you throw the rock away from you with a velocity of 10 m/s.
Then your velocity must also change, otherwise the overall momentum
of the system would not be zero. Using the formula for the overall
momentum of the system, we can calculate your velocity:
m1v1 + m2v2 = 0
(60)(v1) + (5)(10) = 0
60v1 = -50
v1 = -0.83 m/s
The negative sign indicates that you are moving in the opposite direction of the rock. We could have
chosen the direction of the rock to be negative (so v2 = -10) and then your velocity would have been positive.
The important thing to note is that although the system's
momentum stayed the same, you managed to get yourself moving
by throwing away the rock. Had the rock been more massive
or had you thrown it faster, your resulting velocity would
also have been faster. This is basically what Newton was saying
in his third law. When you throw the rock, you exert a certain
force on it, causing it to accelerate for a period of time.
At the same time, the rock pushes back on you with the same
force, causing you to accelerate in the opposite direction.
Now, suppose you started out with 10 rocks of mass 5 kilograms.
If you threw one rock after another, you would gradually build
up your speed. This is the same principle used in most propulsion
systems to get something moving. However, instead of throwing
rocks, propulsion systems usually shoot out large amounts
of gas (known as the fuel or propellant) at very high velocities
(similar to if you inflated a balloon then let go, except
much more powerful).
MEASURES OF A PROPULSION SYSTEM
There are two important ways of measuring a propulsion system's performance: specific impulse and thrust. We already know that if you consider a propulsion system (including the fuel), there is no overall force acting on it. However, looking only at the ship itself, there is a force, which causes it to accelerate. If you allow the propulsion system to run for as long as it has fuel, then the force accelerating the ship will cause a certain amount of change in momentum, which is the total impulse of the system. If you then divide the total impulse by the weight of the fuel used by the system, you get what is known as the specific impulse, which is measured in seconds. The specific impulse measures how efficient a propulsion system is relative to how much fuel it requires. A high specific impulse means a system can create a large change in momentum using a small amount of fuel.
The other measure of performance, thrust,
is simply the force the propulsion system produces in accelerating
the ship. It basically measures how quickly the ship can accelerate,
which is very important in some cases (such as manned missions, where
the flight time is an important factor), but not so important in other
cases (such as space probe missions).
Almost all propulsion systems use the physics
described above. Such systems are often called action-reaction systems
because they rely on Newton's law about actions and reactions. The
physics explained here described how it is possible to get moving
in space because of the conversation of momentum in an isolated system.
However, propulsion systems would not work without a source of energy.
The energy source is what sends the propellant in one direction, and
therefore, sends you in the other direction. In the example where
you were in space with a rock, your muscles supplied the energy that
threw the rock. The source of energy in a propulsion system and how
it is used to propel the ship plays a huge part in determining things
such as the specific impulse and maximum thrust of the system.