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[MUSIC] . To understand and to predict motion, we
first need to learn how to describe motion.
So let's say we see some object in our surroundings whose motion attracts our
interest. We know from experience that motion in
the real world can be complicated, including perhaps shaking and rolling,
spinning, twisting, and turning. So to make any progress, we will pick out
and study a single key feature of the total motion before we tackle other
aspects. So, here's how we'll start.
Take the object that we're observing and let's imagine in our mind's eye that we
crush the object down to a point, placed at the average location of the original
object. It's the motion of this point that we're
going to focus on first. This average location has a special name.
It's called the center of mass. Later, we'll learn how to calculate the
center of mass for different objects. However, for now, if we use our intuition
to pick a good point to represent the average location of the object then that
will be fine. The motion of this average location has a
special name too. Its called translation.
Now even with our attention centered on the average location, the corresponding
translational motion may be a little too complex for us to start with.
So we're going to look first at cases where our object moves along a straight
line. We'll call such case, one-dimensional, or
1D motion. 1D motion happens in the real-world, so
it's useful to start here. We now will identify the physics concepts
in quantities, that we'll use to describe translation.
Let's start with two concepts, the location, where the object is, and the
time when it is at that location. To describe the object location ,or
equivalently, the objects position for the 1D case, we can use a single
cartisian cordinate access. Recall that we are free to chose the
origin orientation title of our access, but once we make such a choice, we'll
stick with it. For example, we are going to title the
axis, the x - axis, but we could have chosen any other name.
The y-axis. The z-axis.
The Spongebob-axis. The choice of label as well as the choice
or origin and of axis or intation doesn't matter as long as we are consistent.
So lets say when we first observe our object the objects position is at x1 and
the corresponding time is say T1. The position has the dimensions of length
and the time has the dimensions of time. We will often use the meter as the unit
for position and the second as the unit for time.
However, we will allow ourselves some flexibility here, as we'll try to choose
units that are best suited for the situation we're studying.
Motion often involves change for example, lets say we see our object progress from
x one to x two as time advances from t one to t two.
So we need to describe both a change in position and a change in time.
The change in time is described by the difference.
The final time t two minus the initial time t1, which we can represent
symbolically as delta t. The Greek letter delta is often used to
symbolize a change. Notice that Delta-t, which is a scalar,
will typically be positive, because the final time is later than the initial
time. Notice, also, Delta-t has the dimensions,
and units, as t1 and t2. The change in position is described by
the difference. The final position x2 minus the initial
position x1 which we can represent symbolically as delta x.
The change in position has a special name.
It's called the displacement. The displacement is a vector and
therefore communicates two separate pieces of information, the direction of
the position change and the magnitude or size of the position change.
Let's use an arrow to represent displacement delta x.
We do this by putting the tail of the arrow at the initial position, x1, and
drawing the arrow, so that the arrow head lands on the final position.
X two. Sometimes it's useful to draw the arrow a
bit to one side for the sake of clarity. To help me remember how to draw the arrow
correctly, I think of starting to draw the arrow at the start and finish drawing
the arrow at the finish. By drawing arrows we can see that a
change in position from x1 to x2 is not the same as a change in position from x2
to x1. The two displacement do not point in the
same direction. Drawing arrows to represent vectors is a
very good habit to develop. As a final note, we recognize delta-x has
the same dimensions and units as x1 and x2.
Now, let's introduce a single concept that describes both how fast and in what
direction an object is moving. Let's take the ratio of delta-x to
delta-t. If we do that, we obtain a quantity we
call the Average Velocity, Vx, where the bar over Vx, symbolizes the average.
The dimensions of the average velocity, are length divided by time, with units,
typically, Meters divided by seconds or meters per second.
The average velocity is a vector which describes 2 distinct pieces of
information, how fast the object is moving and in what direction the object
is moving. Let's draw an arrow to represent the
average velocity. Since the displacement vector is divided
by a positive scalar to yield the average velocity.
Therefore the arrow we draw for the average velocity must point in the same
direction as the arrow drawn for the displacement.
The direction of both the displacement and the average velocity tell us the same
thing, the direction of translation. However, the length of the arrow for the
average velocity tells us something very different, how fast the object is moving.
So we have to make an extra decision about how long to draw the arrow to
represent a particular magnitude of average velocity.
Once we make that decision, we can then represent the average velocity for a fast
object with a long arrow and for a slow object with a short arrow.
Now, as an object moves the object can speed up, slow down, or change direction.
This can be captured by the change in the average velocity, which we'll call here
delta vx. We see that the change in average
velocity has the same dimensions and units as the average velocity.
To compute the change in average velocity we need two average velocities.
We computed one already vx1 Notice we added a subscript one because we're no
keeping track of multiple velocities. And that this required a pair of
positions and corresponding times. We need to measure at least on more
position, and time, to obtain a new displacement, and change in time So that
we can compute another average velocity, VX2.
Then, the change in average velocity, Delta Vx, will be given by the
difference. The final average velocity, Vx2, minus
the initial avarage velocity, Vx1. The change in average velocity is a
vector, so let's represent it with an arrow.
If we already have the arrows representing the velocities Vx1 and Vx2,
then we can draw the arrow representing delta Vx.
First, we start by putting the arrow tails for Vx1 and Vx2 at the same place.
And then use the rule, start drawing the arrow at the start and finish drawing the
arrow at the finish. This is the same procedure we applied a
little earlier to draw the arrow representing the displacement and is a
general way to describe the subtraction of two vectors.
As a vector, the change in average velocity carries two pieces of
information, how much the velocity is changing and the direction in which the
velocity is changing. We'll explore the meaning of the
magnitude and direction of delta v x later, when we consider specific
examples. So let's summarize.
We now know how to describe motion, specifically how to describe the
translation of a single object along a straight line.
To do this, we need to say where the object is and when the object is at that
location. We need to say how fast and in what
direction the object is moving. which depends on our being able to
characterize the objects change in position during a change in time.
And finally we need to say whether or not the object is speeding up or slowing
down, and or changing direction. The description of motion is called
kinematics. The kinematic quantities we discussed
here can provide a pretty complete description of 1D translation.
But they're not sufficient for general understanding and prediction of one D
translation. Description is necessary, but not
sufficient for understanding and prediction.
So there's more to the motion story. We'll continue with that in our next
segment.