Energy and Momentum Conservation
Momentum Principle
In this introduction we begin
a discussion of the mathematics and physics underlying Galileo's momentum principle
for those readers who may not be familiar with vectors, energy and momenta.
We will of course return to discuss these ideas in later Chapters, particularly
in Satellite Orbits. Galileo formulated a simple and elegant law of motion
that is the starting point for his, for Newton's, and later for Einstein's investigations
of mechanics. In the Principia, Newton states as the first law of motion: "Every
body perseveres in its state of rest or of uniform motion in a right line, unless
it is compelled to change that state by forces impressed thereon."
This is the principle of
momentum. Momentum is the measure of the quantity of motion of a body. It is
not simply the speed, but it is the quantity which is directly changed (as Newton
says) by a force. The amount of momentum is the "product" of the speed
and the mass of a moving object. Momentum has direction also, and so it is often
represented as what is called a "vector". Vectors arise in many places
in these lectures because they are so useful in the study of motion (See especially
the later derivation of Kepler's Laws.)
So we will take a close look
at them here. According to Galileo and Newton, momentum is preserved as a body
moves undisturbed by "forces". What is a force?
Forces as Vectors
It is a thing that changes
momentum: it is manifested as a rate of change, in time, of the velocity of
an object. Einstein takes a slightly different (and more elegant) point of view,
and says that a new quantity which he calls "energy-momentum" is what
is preserved. We cannot enter into that discussion here because it entails a
different view of time, its relationship to energy, and therefore, since they
are the same thing, matter. But in order to understand this, it is certainly
necessary first to follow Galileo and Newton.
Let us start with vectors.
A vector in the plane represents a "displacement" or a translation
from one place to another. When used to represent velocity, it is an drawn as
an arrow whose starting point is where the particle is, and whose ending point
is where the particle will be if it travels with uniform (equable) motion for
one unit of time with its present motion. This is not to say that the particle
is moving equably. The amount and direction of its motion may change from moment
to moment (That is, it may be acted upon by "forces") but, at each
instant, we may abstract a velocity which represents its state of motion at
that instant, and we may thus imagine that it would move uniformly to a new
place after a unit of time with that (frozen) state of motion. The length of
the vector is the speed, and its direction tells which way the particle is heading.
The following picture may be helpful.
In this picture, there are
3 vectors associated with an object orbiting a center. The dark red curve is
the orbit. The blue vector represents the vector connecting the center, where
the dark lines cross, to the object. This is called a "position vector".
Superimposed on that, but pointing in the other direction (towards the center)
is a red vector, the vector of acceleration. This represents the central (Newton
calls it centripetal) force pulling the object back towards the center. Finally,
the green vector, departing from the object along a tangent to the curve, is
the vector of velocity. It terminates at a point where the object would go if
flung from the orbit (in uniform motion with its present velocity) after one
unit of time.
Conservation of Linear Momentum
In this exercise, we will
work, not with velocity or acceleration vectors but with momenta, to illustrate
two of the great conservation principles of physics. The first is Galileo's
principle stated above, that momentum is conserved, either for a single object,
or for a system of objects. And the second is that energy of motion (Kinetic
energy) isconserved when a system of particles enter into interactions (such
as elastic collisions) in which no energy of other type is lost in the individual
encounters. We used a "hockey puck" to represent this exercise for
reasons that will be clear soon.
Suppose given two hockey
pucks sliding without friction on the ice. Each has a momentum ( 
)
that can be represented by a vector. In this case, we may assume the pucks are
moving uniformly so the velocity really does represent the displacement in a
unit time. The momentum is simply a magnification of the velocity according
to the mass. The following picture represents two pucks moving with equal speed
for which the blue one has twice the mass of the red one.
The momentum vector of the blue puck is twice as long as that of the red puck.
Now we call a pair of moving
objects a system of objects. While each object has at any instant a position
and a mass, the system also has, in a sense, a position and a mass. Suppose
the first object has mass 
and position 
, and the second object has mass 
and position 
. We say that the system has mass: 
.
Center of Mass
And what is its position? It is what we call the "center of mass"
When you multiply a vector
by a positive number, you simply change its scale. Further, if the velocity
of the first object is 
and
that of the second object is 
, then we say the momentum of the system is the sum of the individual momenta.
We will explain what it means to add vectors shortly.
Thus it is 
and the "velocity" of the system is
that is, the momentum of the system divided by its mass.
Now according to an interpretion
of Galileo's principle that extends it to systems of moving objects, the momentum
of a system of objects is constant, even if the objects interact among themselves
by collision. Newton explains this in the Principia, Corollary 4.
We will see this later in the exercise. What of energy? Recall that the Kinetic
Energy, say of object 1 above is defined to be: 
where 
is the length of the vector 
(the speed).
This is the "energy of motion" of object 1. If the object is moving
under the influence of gravity then 
is constant, where we have called 
the "potential or gravitational" energy. But we ignore gravity here
because it has no effect on our hockey pucks in principle.
Kinetic Energy
Now the Kinetic energy of
the system of objects is:
Here, it is important to bring up a delicate point, and to observe that positions,
velocities, momenta, and energies are defined (and measured) in coordinate systems.
Therefore their measures depend on those coordinate systems in the same way
that any measurements depend on the establishment of units.
Now if my coordinate system
is in uniform motion with respect to yours, then uniform motion in mine will
correspond to uniform motion in yours. And so, the generalized Galileo principle
of constant momentum (hence uniform motion of the center of mass) of a system
of objects may be valid for one coordinate system, called inertial, and will
then be experienced by observers using any coordinate system in uniform motion
with respect to it.
Inertial Coordinate Frames
These will all be called inertial.
Galileo's principle (and Newton's formulation of it) requires the existence
of such a coordinate system. If the new coordinate system is not in uniform
motion with respect to an inertial system, for example if it is rotating (with
respect to the fixed stars), then there will be "fictitious forces"
acting on the object that the new observer will experience, but not the inertial
observer. We will assume that our coordinate system is inertial.
Our formulation of the constancy
of the energy of a system, as we defined that energy above, should be made also
with respect to measurements in an inertial coordinate system (for example,
one, like our hockey rink, in which there are no effective gravitational forces).
It doesn't matter which one
we choose, but there is a "natural" choice for a system of particles.
That is the coordinate system whose origin is at any time, the center of mass
of the system. These are called "center of mass" coordinates. We will
measure the energy of the system with respect to this coordinate system. It
will be inertial, and its origin will "move" uniformly with respect
to the coordinate system of the rink (which we assume to be inertial also).
The motion of the center of mass of the two pucks will be indicated by the 
.
Thus, we represent momenta
as vectors with respect to the coordinate system shown and we measure energy
with respect to the center of mass coordinate system.
There is one final comment
before we give instructions for the exercise. The momentum of the system of
objects is the "vector sum" 
of the momenta of the individual objects. How do we add vectors? Newton explains
this also in the Corollary 1.
Represent two vectors as arrows
emanating from a common origin. This pair of vectors determine two adjacent
sides of a unique parallelogram. Construct the parallelogram, and then the vector
from the common origin to the vertex opposite that origin is the vector that
represents the sum. So the green vector is the sum of the red and the blue vectors
below:
Exploration:
Momentum as a vector
The metaphor for this Exploration
is a hockey puck on an ice skating rink. In this exercise, you may set
up and launch two hockey pucks on an ice skating rink. You determine their masses,
their velocities and their positions. Set the masses of the red and blue puck
by typing the values in the fields:
You may move the pucks to
any desired position by dragging them. Just press 
and then click near the desired puck and, holding the left button down, drag
it to the new position. When satisfied, press the right mouse button and the
cursor will revert to an arrow from the pointing hand. Finally, to set the velocity
(and therefore the momentum) of the puck, just press the 
button then click once on the puck to set the tail of the velocity vector at
its center, and once again away from the puck to determine the head of the vector
(where it will be after one unit of time).
Once you have determined these things, you are ready to do the simulation.
For that, press the
button!
Several things will happen.
First the system will draw the initial momenta of the pucks, and their sum (the
momentum of the system) at the origin of coordinates. The blue vector is the
momentum of the blue puck, the red vector is the momentum of the red, and the
green vector is the sum. It is this sum that should remain constant. It will
also print the energy of the system, measured in the center of mass coordinates,
in the box:
Next,
it will move the pucks, and the center of mass
until one puck reaches a boundary. Here, you will want to observe the trajectory
of the center of mass. It should move uniformly in a straight line according
to Galileo. It is difficult to observe uniform motion, but it doesn't matter.
At the end, the system draws three more vectors. The dark red one is the final
momentum of the red puck. The dark blue one is the final momentum of the blue
puck. If there is a collision, these will not coincide with the original vectors
as you will see. The dark green vector is the final momentum of the system.
If there is any justice, this will coincide with the original green vector.
See for yourself! Finally, the energy at the end will be printed in
This
should be essentially the same as the original (it may differ slightly, since
we are approximating things.) There are many interesting experiments to perform.
You may reset the system by pressing
Question 1: Suppose you arrange the two pucks
in a certain configuration of positions, masses, and velocitiesand you film
the interaction. Next, you change the masses but preserve their ratio, and film
again. You show the films to a friend. Will she be able to distinguish the cases?
