| Class
notes by RE-SEED Leader - Alex
Vanderburgh
Motion brings
in TIME as an important parameter. We are interested in the speed of an
object - a.k.a the rate of change of"displacement", and the
"acceleration",(Rate of Change of Speed.) But how about force?
Well Newton proposed three "laws":
1. Inertia - If there is no net force acting on an object it will stay
put if at rest, or continue to move in a straight line at constant speed
if moving.
2. F = MA - The force needed to accelerate an object is proportional to
the mass and to the acceleration.
3. Newton's third law. - "You must have somthing to push against."
- "Forces come in pairs."
"Impulse" and "Momentum"
We can rewrite F=MA in an interesting way. Assume that "F" acts
for a time "t". The mass "M" will be accelerated to
a velocity "v" in that time. The acceleration is equal to v/t
i.e. the rate at which velocity changes. So we can say F=Mv/t or Ft =
Mv. It turns out that Mv is an interesting quantity. In ideal collisions,
none of it is lost! Mv is called Momentum. The Ft is also interesting.
It is called an "impulse". A high force for a short time (like
a blow with a hammer) can have the same effect as a low push for a long
time. [Neglecting friction, of course!] Did you notice that Ft is the
"area" under the F vs. t curve? So if F is an arbitrary function
of t, can we say that the area under it is equal to Mv ? [Alas, you have
to wait a few years - High School or College - to say that.] [Note also
that it was Newton who brought up this idea.]
In ramps and free fall, we assume that the acceleration comes from gravity
and is therefore the same for all masses. The forces will be different.
A more massive car is harder to push on the level, but would fall off
a cliff with the same velocity as a small one! A big car has more momentum
than a small one at the same velocity. It will take a larger force to
stop it in the same time!
Newton's third law is often called "action" and "reaction".
However, "action" and "reaction" implies cause and
effect. This is misleading, for both occur at the same time. We prefer
to say "Forces occur in pairs." [For some reason, it is not
obvious that without something to push against...You can't push at all!]
Did you notice that our version is one dimensional? That's what a "straight
line" means! Once we talk about motion on a plane surface, "speed"
becomes a "vector" (It has direction as well as size) and we
start calling it "velocity". A force is needed to change direction!
"F=ma" is still true, but acceleration and force are vectors
too! In space, we must go to three dimensions, and in some cases "time",
which makes it four dimensions for some people. [Personally, I have trouble
with calling time a fourth dimension. I just call it another parameter.
Who knows, we may need a fourth ordinary dimension some day.] In any event,
we probably can stay one dimensional in middle school. If the basic stuff
looks old hat when our students get to high school - so much the better!
It will give us
a chance to keep the line between our mathematical model and the real
world clearly understood! It has a tendency to get fuzzy!
5. "Kinematics", and "Dynamics" 6-2
Kinematics describes the motion in terms of speed and location with no
concern about the forces that were needed to get started, or to make
things change along the way. [Or, for that matter, to stop things in motion.]
You need to know that distance equals rate times time, (as we did in fifth
grade). We were expected to answer such questions as "How long will
it take?, How far did we go?, How fast do we need to go?". [Questions
my Dad would ask me when we were on a trip and I would keep asking "Are
we almost there, Daddy?" ]
Dynamics relate to the accelerations and forces. [ F = ma, etc.] (Still
in one dimension.) Of course, they are related. We usually need to consider
both.
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- Alex
Vanderburgh , RE-SEED Leader Volunteer |