Newton spoke of mass as the quantity of matter, and momentum as the quantity of motion. Both these quantities lead to important results. We look first at momentum:

Typically the letter *p* is used to
represent the momentum of an object. We write *p = m v*. (The
momentum of an object is, as defined by Newton, its mass times its velocity).
Starting from Newton's second law we can write

*F = ma = m )v/)**t,
*since acceleration a is just a measure of how velocity changes with time. As
was implied in the page on Newton's laws, this form of the second law assumes
the mass remains constant. The more general statement of the second law moves
the mass back along side the velocity, providing *F* = *)(m
v)/**)**t,
or F = **)p/**)**t.
*This form reduces to the familiar form *F = m a* in the case of
constant mass, but allows for a time varying mass if the occasion demands (as,
for example, in the case of the launch of the space shuttle, where the fuel
outweighs the shuttle).

Suppose two objects (we like to call them
particles) collide head on, one coming in
from the left, the other from the right. According to Newton's third law, the
force that the first particle exerts on the second is equal and opposite to the
force the second exerts on the first. As an equation, we write *F _{1-2}
= - F_{2-1}* where the subscript 1-2 means the force Particle #1
exerts on #2, and 2-1 the force #2 exerts on #1. In other words, the forces are
equal and opposite. Using Newton's second law we can rewrite this as

In the game of billiards the cue ball strikes an (at rest) object ball, and then stops. How fast does the object ball move after the collision? Answer: Just as fast as the cue ball was moving. Why? Conservation of momentum.

Conservation of Energy

What is energy? Everyone knows what energy is - we use it all the time, but how is it defined? Write down your definition before you read further. Chances are you had a difficult time defining energy. Here's a definition from Webster's New World Dictionary, Third College Edition: "the capacity for doing work" (along with many other, non-scientific definitions). OK, then what is work? "the means by which energy is transferred" (again, along with many other definitions). Here's how we define one kind of energy:

We start with a definition of work: Work
is the product of the force applied to an object times the displacement of the
object due to the force. We are careful to count only the force which is applied
in the direction of the motion. Here's an example: You push a box along the
floor, pushing exactly in the direction the box moves. If your force is 10
Newtons and the box moves 3 meters you have done 30 Newton-meters of work. (A
Newton in the SI system of units is equal to about 0.225 pounds; a Newton-meter
is more usually called a Joule). Now suppose you push the same box the same
distance, but this time you push downward at an angle of, say 60^{0 }from
the floor. Now some of your effort is trying to push the box into the
floor. To get the same 10 Newtons of force in the direction the box moves you
have to exert a total force of 20 Newtons. You do the same work - 10N times 3 m,
but you may be a bit more tired if you move the box this way. (Actually, it's a
bit more involved, because the downward part of your applied force has the
effect of adding to the weight of the box, but the point's the same: only the
portion of the force in the direction of the motion counts towards the work
done).

So: Work = Force X distance (remembering
the its the force in the direction of movement that counts). Suppose a force *F*
causes a movement *)x.
*The work done if *F**
)x.
*Since* F = m a, *it follows that* **F**
)x
= m a **)x.
*Remember Galileo's equations of
motion? I reproduce them here for convenience (I've changed y to x and x to x -
x_{0} to make things a bit more general):

A little algebra should convince you that,
if you eliminate time from these two equations you arrive at an expression
relating velocity, position and acceleration. It turns out to be (you should
verify the result for yourself)

.

Now *x - x _{0}* is just

.

*F**
)x
*is the work done on the mass*
m. *The expression on the right is
*defined* to be the kinetic energy of the mass m. (Actually the right hand
side represents the change in kinetic energy - its initial kinetic energy was *
1/2 m v _{0}^{2} * and its final kinetic energy was

So, energy is a *concept*. It is an
idea useful in describing what happens to something when work is done on it. The
change in kinetic energy of an object is a measure of the work done on it. We
call kinetic energy "energy of motion" (for obvious reasons).

Remember when we multiplied *F* by *)t
*and eventually arrived at
conservation of momentum? It turns out that multiplying *F* by *)x
*provides another conservation
law, conservation of energy. In its most general form it states that the total
energy in the universe is conserved. We may change its form from one kind of
energy to another, but the total remains constant. Perhaps the most remarkable
thing about this law is that it deals with our *invented* concept of
energy. It is actually a convenient bookkeeping scheme upon which the very
foundations of physics are built.

I speak of conservation of energy, but
its easy to find a problem with what I have said so far: Think of two identical
pucks on an air table coming toward each other at some speed v. They each have
kinetic energy 1/2 m v^{2}. After they collide they are each moving away
from each other and it's easy to show that they will each have kinetic energy 1/2 m v^{2}
as before - energy is conserved. But for a brief time both pucks have stopped,
at the instant they collide. Where was the energy then? Well, as they collide
the two pucks distort somewhat and then regain their original shape. In doing so
the pucks push away from each other, thereby getting their energy back. We say
that there is energy stored in the distorted pucks (as if the pucks were made of
tiny springs that compress and then release). This form of stored energy is
named "potential energy". An example in a different context may be
useful: Suppose you have a weight (give it mass *m*) that you lift a height
*h* above the floor. How do you do his? You apply a force that overcomes
Earth's gravity and exert that force over a distance *h* - you do work!
This work moves the weight upwards. Once at the height *h* you can hold it
there (or put it on a shelf) and, whenever you want, let it fall to the ground.
As it falls, it converts that stored potential energy into energy of motion -
kinetic energy. So it is the combined energy that is conserved - kinetic plus
potential (in this case gravitational potential). Carry the process one step
further: On the floor under the weight place a coiled spring. The weight falls,
lands on and compresses the spring and then rebounds up again, gaining
(gravitational) potential energy, reaches the height *h*, falls, rebounds,
on and on forever. This is great! You do work raising a ball once and, since
energy is conserved, it bounces forever as it converts gravitational energy into
kinetic energy into spring energy (another form of potential energy), back to
kinetic energy, etc. This is a perpetual motion machine! Try it! It won't work.
Why? Because we forgot friction. Eventually the weight will stop bouncing. Well,
there goes conservation of energy. Not quite. Let's just include the frictional
heat as part of the energy. Run another thought experiment: Give a weight a push
across the floor: You push (apply a force) over a distance to get the weight
going (and so do work equal to *F**
)x*)
on the weight. This changes its kinetic energy from zero (the weight was at rest
to begin with) to * 1/2 m v ^{2}*. After a while the weight stops,
as friction opposes its motion and brings it to a stop. What has happened to the
energy? The work done was converted to the heat due to friction. As was argued
by Julius Robert von Meyer(5), (with some paraphrasing by
me) "If energy, once in existence, cannot be annihilated but only change
its form. what other forms is it capable of assuming?" He then argues
"If work can be converted to heat, then heat must be a form of energy, and
must naturally be equivalent to kinetic and potential energy." The direct
measurement of the "mechanical equivalent of heat" was done by James
Prescott Joule in his famous Paddle Wheel experiment. This continually
expanding role of energy was finally set into comprehensive form by Hermann
Helmholtz who concluded that energy was the sum total of kinetic, potential,
heat, electrical and "other forms to be discovered or invented".

So we have it: Energy has many forms (but only one form of
kinetic energy, 1/2 m v^{2}).
If in our searching we find a process that seems to violate the conservation of
energy, we simply invent another form to take up the slack.