It's professor Dave, let's talk about
horizontal motion.
As we learn classical physics, a big topic of
study will be mechanics. This is a branch
of physics that can be divided into two
smaller topics: kinematics and dynamics.
Kinematics, which was developed largely
by Galileo in the early 1600s, deals with
equations that describe the motion of
objects without reference to forces of
any kind,
whereas dynamics is the study of the
effect that forces have on the motion of
objects. These topics together comprise
mechanics. We are going to focus on
kinematics over the next few tutorials
so that we can familiarize ourselves
with the ways that simple equations will
govern the motion of objects in one and
two dimensions. These equations are
revolutionary, because from Aristotle
until Galileo we thought that mathematics
could only describe the perfect motion
of divine celestial objects, and that the
motion of objects on earth was too
imperfect and unpredictable to calculate.
But we soon found that the same
equations governing the motion of all
objects, whether on earth or in space, it
is simply that on earth we must make
approximations since there are a greater
number of variables like friction and
atmosphere that affect motion in various
ways. The kinematic equations include
variables for displacement, velocity,
acceleration, and time, and in the context
of kinematics acceleration will always
have a constant value, whether positive,
negative, or zero, since we won't look at
forces that could cause acceleration to
change over time. When you see a
subscript of zero after velocity or
displacement it indicates initial
conditions which will have some
implication depending on the problem we
are looking at. Here are the three
fundamental kinematic equations we will
be using. The first one says that the
velocity of an object at any time T is
equal to the initial velocity plus the
acceleration times time. The next one
says that the position of an object with
respect to a point of origin will be
equal to its initial position plus the
initial velocity times time plus
one-half the acceleration times x
squared.
Lastly, this one says that velocity
squared is equal to the initial velocity
squared plus twice the acceleration
times the displacement. Other
supplemental equations include these two,
which are easily derived from simple
definitions, which state that position is
equal to the average velocity times the
time interval and that the average
velocity is equal to final velocity plus
initial velocity over 2, which is the
definition for any average. Now that we
have these equations and know what all
the variables mean, we are ready to apply
them to real examples of motion. Say you
get in your car to drive to the
supermarket. While at rest, you place your
foot on the gas and apply a constant
acceleration of 2.5 meters per second
squared.
What will your velocity be after 10
seconds and how far will you have
traveled in that time? We can use these
two equations to find the answers, we
just have to plug in what we know. For
the velocity, we know that the initial
velocity was zero because we were at
rest, so we just multiply acceleration by
time and we get 25 meters per second.
That is the velocity of the car after 10
seconds. Now to find how far you will
have traveled, you will use this equation.
Once again, initial velocity is 0 so this
entire term can be ignored.
Then we have one-half times the
acceleration times 10 seconds squared
and we should get a hundred and
twenty-five meters traveled over this
time span. So it really is this simple.
You just choose the equation that is
appropriate for what you are solving for
and plug in what you know. Let's now
consider a car that is already in motion
with a velocity of 27 meters per second.
Let's say you need to stop suddenly so
you press on the brakes,
initiating a rapid deceleration of
-8.4 meters per
second squared. How long will
it take the car to come to a stop and
how far will it travel while your foot
is on the brake? Once again let's use
this equation to solve for time. It must
be this equation because we know
everything in it except for time. For
velocity let's plug in 0 because we are
curious about the time elapsed at the
moment that the car stops moving, and the
velocity when it has stopped moving will
be 0. The initial velocity is the 27
meters per second we mentioned, and we
can plug in the acceleration, solve for
time, and get 3.2 seconds. Now that we
know the time associated with this event
we can use this other equation to find
the braking distance. We plug in the
initial velocity and acceleration we
mentioned before, as well as the 3.2
seconds we just calculated, and solve for
x which will be about 43 meters traveled
from the moment you applied the brakes
to the moment that the car stops moving.
These equations work for any other
object just as they do for cars so let's check comprehension.
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