Physics 301 – Analysis of Motion
VO
Motion is all around us, and studying it is a major part of physics. To describe the motion of objects,
from planes to cars to baseballs, we use words, graphs, and equations.
Instructor
In all the programs you’ve seen so far this year, we have been preparing you for the study of physics.
By now you should have all the math skills necessary to begin our study of mechanics, which deals
with objects and their motion. We introduced the idea of relative motion when we talked about the
chicken crossing the road or just standing in one place. We set some ground rules then, and they are
still important. So let’s go over them again.
First, during this unit, we’ll just describe motion, not talk about its causes. We’ll get to that later.
Second, we will consider objects to be single points, not three-dimensional bodies. And third, all
motion will be considered relative to the surface of the earth as a frame of reference.
Now that we’ve gone over the ground rules, let’s get started. We’ve already defined motion as a
change in position relative to a frame of reference. But it’s not enough to say that an object is
moving. To describe its motion we need to know how fast.
(green chalkboard on screen)
VO
To answer the question, “How fast?” we need to describe the rate of motion. What word do you
think of when you hear the word “rate?” Tell your teacher.
If you said that rate involves time, you’re right. We call speed the time rate of motion. Anytime you
see the word, “rate,” you should take it as a clue that something will be divided by time.
So speed is distance divided by time.
Instructor
There’s another term for describing the rate of motion. It’s similar to speed but includes a direction.
What is it?
I hope you said “velocity.”
(green chalkboard on screen)
VO
Velocity is speed in a certain direction.
So if speed is defined as distance divided by time, what would be the formula for calculating velocity?
Since speed and velocity are so closely related, we’ll use this general formula. Just know that the “v”
could represent speed or velocity.
(car speedometer on screen)
VO
We can measure and calculate velocity in two ways. Instantaneous velocity is measured by a car
speedometer. It changes as the car speeds up or slows down.
The word, “instant,” is the clue to remembering what instantaneous velocity is. It is the displacement
of the car in an instant, or a very, very short period of time.
Even though a car changes its instantaneous velocity many times during a trip, when the trip is
considered as a whole, we calculate average velocity.
(green chalkboard on screen)
Average velocity is total displacement divided by total time.
Instructor
Now that you have the terminology you need, it’s time to collect some data and analyze the motion of
this cart. We’ll do two separate experiments and determine both the average and instantaneous
velocities of the cart for each one.
What will we need to measure in order to calculate velocity?
You’re right if you said displacement and time. Displacement or distance will be easy. We’ll use a
meter stick. Time is a little harder. We could use a stopwatch for a total time of a few seconds, but
measuring shorter periods of time is too difficult. We just can’t start and stop the stopwatch that fast.
So we’ll use a device called a tape timer. There are several kinds of these timers, but they’re all
electric devices that make marks at regular intervals, like the tick of a clock. Some leave burn marks
on a tape, while others have metal points on a vibrating arm, which press on carbon paper to leave
marks. Watch how we set up this one for our experiments.
(students on screen)
VO
A fresh carbon paper disk is pinned in position, carbon-side-down on the timer. Then a strip of paper
tape is threaded through the timer, under the carbon disk. Watch what happens when the timer is
turned on and the tape is pulled through.
Look at the tape. The advantage of a tape timer is that we can use the tape to measure both time and
distance or displacement. Remember that the tape is moving and the timer is making marks at the
same time. Measuring the distance the tape moved between any two dots is easy. We just use a meter
stick.
Instructor
But how do we measure the time using the tape timer? Well, the timer is set so that the arm vibrates
up and down a certain number of times each second. This happens no matter what we do to the tape.
So the time intervals between dots are constant. The timer used in our experiment is set to make a
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mark every 0.020 (zero point zero two zero) seconds. So timing is just a matter of counting dots and
multiplying by 0.020 seconds per dot.
(students on screen)
VO
Watch what happens when the tape is pulled though the timer faster than before.
The dots are spaced farther apart, but remember that the time between dots is still the same. Our tape
just traveled farther between vibrations of the timer arm.
Now we’ll show you the first experiment. After you see it, your local teacher will give you your own
tape to measure and analyze.
(students on screen)
VO
For part A of the lab, a string with a paper clip on one end is attached to a cart, and the string is hung
over a pulley. A small washer is hung on the clip. The washer serves as a counterweight to act against
friction, which tends to make the cart slow down and stop.
The cart is given a little push to get it started. Then our students watch it move down the track.
Since the cart slowed down, we need to add another washer to the paperclip, to counteract the force of
friction. After some trial and error, our students are satisfied that the cart moves smoothly down the
track.
Next, they back the cart up to the timer and measure the distance from the bottom of the washers to
the floor. They tear off this length of tape to use in the experiment. This is important because we
want the tape to run completely through the timer before the washers hit the floor.
They thread the tape through the timer and tape it to the back of the cart. Now they’re ready for a trial
run. One student will give the cart a push and his partner will turn on the timer.
Instructor
The tape from this trial looks like this. Your local teacher will give you a tape with the same
markings as the one our students produced in the lab. When you have the tape, a meter stick, and a
calculator, and you have copied the data table your teacher puts on the board, come back and we’ll
collect and analyze our data.
(Pause Tape Now graphic)
(student on screen)
VO
We’ll walk you through the measurements. Watch what we do first, and then do the same thing to
your tape.
Lay your tape out on your desk or lab table, and start on the end that was taped to the timer. Your
teacher has written “Part A” on this end of your sample tape. On the actual trial tape, we find the first
clear dot, circle it, and label it “dot zero.” Since the timer made so many dots, we don’t want to
measure each one. So we count to ten and circle the tenth dot, labeling it “one.” Keep going, labeling
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every tenth dot, two, three, four, etcetera. You should come up with six or seven circled dots.
Under your data table, draw a piece of timer tape, and show how the dots are spaced on the tape. This
will help you study the lab later.
Nest, we want to record the time that elapsed when each circled dot was made by the timer.
Remember that our timer was set to make a mark every 0.020 seconds. So the dot we labeled number
one was made after ten time intervals of 0.020 seconds each. That means that when dot number one
was made, ten times 0.020, or 0.20 seconds had passed. We’ll record this in our data table under total
time.
How much total time had passed when circled dot number two was made? That would be two times
10 times 0.020 seconds, or 0.40 seconds. Now you keep going to fill in the total time column of your
data table.
(Pause Tape Now graphic)
Your teacher will cut off the tape and give you some time. Stop when you complete the time column.
Don’t try to get ahead before you know what to do next.
(student on screen)
VO
The next thing we want to measure is the total displacement of the cart at each numbered dot. This
means that we need to start with the zero on the meter stick on dot zero and simply read the
displacement, to the nearest tenth of a centimeter, off the meter stick for each circled dot. We call this
distance, displacement, because the direction is understood to be forward. If the cart had backed up,
we would state the direction or use a negative sign.
Now don’t move the tape or meter stick while you’re making your measurements. Read the total
displacement, from dot zero to each circled dot, and record it under “total displacement” in the data
table.
(Pause Tape Now graphic)
(student on screen)
VO
Next, we want to see how fast the cart was going when each of the numbered dots was made. That
means we want to calculate the instantaneous velocity of the cart, which means we need to know how
far the car moves in a very short time interval.
To do this, we go to circled dot number one and mark off one small time interval from number one to
the very next un-numbered dot. Measure this displacement to the nearest tenth of a centimeter and
record it in the data table under instantaneous displacement. I get 1.1 centimeters. Then to find
instantaneous velocity, all we have to do is divide by this the very short time interval between dots.
For our timer, that’s 0.020 seconds, the closest thing to an instant we can measure with our timer.
Since my displacement was 1.1 centimeters, when I divide by 0.020 seconds, I get an instantaneous
velocity of 55 centimeters per second. Remember to round off to two significant digits.
You’ll continue this procedure to find the instantaneous velocity of the cart at each numbered dot.
Measure the displacement from the circled dot to the very next dot and then divide by 0.020seconds
each time.
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(table on screen)
VO
When your teacher stops the tape this time, we want you to finish your measurements and calculations
and then construct two graphs.
The first graph will have total time plotted on the x-axis and total displacement on the y-axis.
The second graph will have total time plotted on the x-axis again, and instantaneous velocity plotted
on the y-axis. Make a best-fit graph, keeping in mind experimental errors. We’ll come back and talk
about your graphs when everyone has finished.
(Pause Tape Now graphic)
(table on screen)
VO
Let’s analyze the data we collected in Part A of the lab. Here’s a sample set of total time and total
displacement data. Remember that we can call these numbers displacement because the direction is
understood to be forward. Yours may be a little different because of estimation, but not much.
(graph on screen)
To see the relationship between our two variables, we graphed total time on the x-axis and total
displacement on the y-axis. Here’s what your graph should have looked like. Now you may have
noticed different wording on the y-axis. Because displacement is defined as change in position in a
certain direction, many physics books label the y-axis, “position” instead of displacement. They still
use the letter “d” as the abbreviation. It’s just a matter of terminology.
(meter stick on screen)
For example, you used the meter stick to find the position of a circled dot. This one is at the 78
centimeter mark. The displacement of the cart was the distance the cart traveled forward to get to the
dot. It’s still 78 centimeters. It’s just a matter of definitions.
(graph on screen)
Now, back to our graph. The straight line shows that displacement and time are directly proportional.
You know how to change a proportional into an equation. It’s by inserting a constant. And what does
the constant tell us about our line? (Brief Pause) If you said the slope, you’re right!
(graph on screen)
VO
To find the slope of the line, we draw a rise, run triangle. Watch me do it and then your teacher will
give you time to find the slope of your line. No two graphs will be the same, so everyone’s will be a
little different. First, find two points on the line that are relatively far apart and easy to read. Don’t
use actual data points, since they may not fit on the line.
Where the line crosses the corner of a block on the grid is a good point. Here are two on my graph
that look good.
Once we’ve selected two points, we draw vertical and horizontal lines to form a right triangle. The
vertical side of the triangle will give us the rise and the horizontal will be the run. Since rise is change
in position, we use the abbreviation, “delta d.” The delta symbol means “change in.” So “delta d” is
change in position or displacement. In this example, our rise is from five to 60 centimeters. So “delta
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d” is the difference, 60 minus five, or 55 centimeters.
The run goes from 0.1 to 1.1seconds on the x-axis. We sometimes call this “delta t” to represent a
time interval. Don’t forget the units. They’re very important. The run, “delta t,” is 1.1 minus 0.1 or
1.0 seconds.
Now we just do the math on the calculator and we round our answer off to two significant digits. Rise
over run equals 55 divided by 1.0 or 55 centimeters per second.
(graph on screen)
VO
Since the graph of displacement versus time for part A of our lab is a straight line, the slope is
constant. No matter where you draw a slope triangle, you’d get the same answer for the slope.
Remember that slope means something. In this case, rise over run is displacement divided by time.
What does that give us?
If you said velocity, you’re right. The velocity of our cart was a constant 55 centimeters per second.
Let’s confirm this a couple of ways.
(table on screen)
First, look at your data table again. In the last column, you calculated instantaneous velocity. Within
bounds of experimental error, my instantaneous velocity stayed right around 55 centimeters per
second.
(graph on screen)
Now look at the graph of total time versus instantaneous velocity. Within bounds of experimental
error again, the best-fit graph is a flat line, showing that instantaneous velocity is constant.
(table on screen)
Now let’s do one more thing. Look at your data again. Let’s calculate average velocity, which is
total displacement divided by total time. You’ll find these totals right here, at the last circled dot. My
total displacement is 76.8 centimeters and my total time is 1.4 seconds. Rounding to two significant
digits, my average velocity for the trip is 55 centimeters per second. All of this information describes
a particular type of motion, uniform motion.
(green chalkboard on screen) .
VO
When the motion of an object is uniform the object moves at a constant speed in a straight line. This
means that the velocity of the object is constant. When motion is uniform the instantaneous velocity
of the object at any time will equal the average velocity for the trip.
A graph of displacement versus time will be a straight line for uniform motion. Its slope equals the
velocity of the object.
And a graph of instantaneous velocity versus time will be a flat line.
Save a little space in your notes. We’ll come back after we analyze Part B of the lab and show you
one more thing about uniform motion and this last graph. Make another bullet under the graph to
remind you.
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(students on screen)
VO
Now it’s time to see part B of the lab. In this part, the apparatus is set up again, with the same length
of timer tape attached to the back of the cart and threaded through the timer. This time, a hanging
mass replaces the washers on the string. One partner will hold the cart in place, the timer will be
started, and then the cart will be released. Watch what happens.
Watch again, in slow motion.
Now the students remove the timer tape. Look at the pattern made by this motion of the cart.
Remember that the dots are made each 0.020 (zero point zero two zero) seconds, no matter what the
cart does.
(student on screen)
VO
We’re going to go over the measurements you’ll need to make for Part B, and then your teacher will
give you a sample tape and you will make all the measurements. So watch carefully. The data table
for Part B is the same as the one for part A, with one exception. In Part B of the lab, we started the
timer before the cart started moving. That’s because we wanted to collect data for the entire trip. So
your data table will include dot zero this time. All the data for dot zero will be zeros, since the cart
was not moving at this time. Your teacher will remind you of this when you start recording your data.
As in Part A, you will circle every tenth dot, starting with dot zero, which is the first clear dot that was
made by the timer. You should have about five circled dots for this part. Sketch a piece of tape to
show the pattern for this part, as you did before.
Then record total time. It will be the same as it was for each circled dot in Part A, so you can just
copy this column.
Next, measure total displacement at each numbered dot. Remember to lay the tape on your meter
stick and read the position on the meter stick. That will give you the displacement from zero
centimeters to this point.
Next, we want to determine the instantaneous velocity of the cart at each dot. Mark off one time
interval from your numbered dot to the very next un-circled dot and divide by 0.020 seconds to
calculate instantaneous velocity.
Before you return, we want you to make two graphs. First, graph total displacement on the y-axis
versus total time on the x-axis, and then plot instantaneous velocity on the y-axis versus total time on
the x-axis, just like before. Your teacher will give you a data sheet or put all these instructions on the
board for you. And, of course, you can always rewind this tape and see my instructions again. You
have lots of work to do. When you return, we’ll analyze the data for this type of motion.
(Pause Tape Now graphic)
(tape on screen)
VO
Your tape for Part B of the lab should look like this.
Notice that the dots start out close together and then spread out more and more during the trip. What
does this tell you about the motion of the cart?
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If you said that the speed or velocity of the cart is increasing, you’re right.
(table on screen)
Look at your total displacement and total time data. Yours should be similar. And it shows that over
each successive time period, the cart is covering more ground than before.
(graph on screen)
To show the relationship between displacement and time, look at the first graph we made. Notice that
the graph is a curve, not a straight line as in Part A. So the slope of the graph is changing. In this
case, it’s getting steeper and steeper. And what does the slope tell us about the motion of the cart?
Remember that to find what slope represents, just look at rise over run. Rise is displacement and run
is time, so slope equals displacement over time or velocity. In Part B of the lab, the velocity of the
cart changes.
(table on screen)
Now look at your data for instantaneous displacement and instantaneous velocity. Notice that the
instantaneous velocity is increasing as time goes by, not staying the same, as in Part A.
Let’s calculate average velocity again. Look at the last circled dot to find your total displacement and
total time for the trip. Mine was 86.0 centimeters in 1.2 (one point two) seconds, for an average
velocity of 72 centimeters per second. If you look at all the instantaneous velocities in the last
column, you’ll see that instantaneous velocity does not equal average velocity.
(graph on screen)
Look at the graph of instantaneous velocity versus total time. All this data and the two graphs leads
us to describe the motion of the cart in Part B as accelerated motion.
(green chalkboard on screen)
Narrator VO
When the motion of an object is accelerated, its velocity changes.
The graph of total displacement versus total time will be curved, and the slope will change. A steeper
slope shows that velocity is increasing.
And the instantaneous velocity of the object does not equal its average velocity.
Instructor
Let’s take a better look at the last graph you made, the one with instantaneous velocity on the y-axis
and time on the x-axis. Not only does it tell us that the motion of our cart was accelerated, it tells us
that the velocity of the cart increased and that it did so uniformly or evenly. Physicists are not content
to say that motion is accelerated. We want a more quantitative description. This means a new derived
quantity to deal with, acceleration.
(cars on screen)
VO
It is not enough to say that velocity increases or decreases. We want to know how fast it does so.
(text on screen - Take notes.)
Acceleration is the time rate of change in velocity.
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To calculate acceleration, we divide the change in velocity by the time interval.
This means that the unit for acceleration in the MKS system is meters per second per second or meters
per second squared.
Instructor
Now we can look at the slope of a velocity versus time graph. Use your rise and run to tell you what
the slope represents. Rise is change in velocity and run is the time interval or change in time. So the
slope of a velocity versus time graph is acceleration. When the graph is a straight line, like this one,
acceleration is constant and the motion is uniformly accelerated.
(green chalkboard on screen)
VO
The slope of a velocity versus time graph is change in velocity over change in time, or acceleration.
When the graph is a straight line, acceleration is constant. We call this motion uniformly accelerated.
The only way to tell if an object is uniformly or non-uniformly accelerated is to graph instantaneous
velocity versus time.
Instructor
Now let’s go back to your notes on uniform motion. Don’t get uniformly accelerated motion
confused with just plain uniform motion. Look again at the velocity versus time graph for Part A,
where the motion of the cart was uniform. What is the slope of the graph?
(graph on screen)
Since the rise is zero, the slope of this line is zero. That means when motion is uniform, acceleration
equals zero. Add these two bullets to your notes on uniform motion.
Instructor
For now, you want to become skilled in interpreting motion graphs. They can be confusing if you try
to surf your way through them, skimming the surface and trying to memorize instead of thinking and
understanding. You need to pay attention to what each graph tells you.
(graph on screen)
Here’s an example. Look at these two graphs. Surfers will look at them and say that the only
difference in the two is the slope of the line. But take a closer look. Graph number one has velocity
on the y-axis, and number two has position or displacement on the y-axis. So they show completely
different things. Can you describe the motion of the object from each graph? I’ll wait while you
make your decisions.
(Pause Tape Now graphic)
(graph on screen)
When you are using a motion graph, always look at the y-axis first to see what the graph is showing.
In graph number one, we’re looking at velocity. As time progresses, the velocity of the object
increases, so the motion is accelerated. And since the slope of this straight line is change in velocity
over time or acceleration, we know that “a” is constant and the motion is uniformly accelerated. That
is the best and most complete answer for number one.
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Now look at graph number two. Since this is a position or displacement versus time graph, it shows
that the object is moving forward as time progresses. And since rise over run is displacement over
time, or velocity, which is constant, this is a graph of uniform motion. That’s the best answer for this
one.
When this program ends, it will be your turn to describe motion from graphs that your teacher will
give you. Try to use correct terms and be as specific as you can. And always look at the y-axis to see
what is changing and what the slope will represent. When we return for the next program, we’ll go
over the graphs.
(graphs on screen)
VO
Local Teachers, turn off the tape and give students problem set number one from facilitator's guide.
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