Name _________________________ Date (YY/MM/DD) ______/_________/_______ St.No. __ __ __ __ __-__ __ __ __ Section__________
INSTRUCTOR VERSION
UNIT 3: ONE-DIMENSIONAL MOTION IA Graphical Description
Approximate Classroom Time: Three 110 minute sessions
A picture is worth a thousand words!
OBJECTIVES 1. To learn about three ways that physicists can describe motion in one dimension — words, pictures and graphs.
2. To acquire an intuitive understanding of speed, velocity, and acceleration in one dimension.
3. To learn to recognize the pattern of position vs. time, ve-locity vs. time, and acceleration vs. time for the motion of objects which speed up and/or slow down at a constant rate and to recognize this type of motion as constantly acceler-ated motion.
In Unit 4 you will turn your attention to a more formal mathematical description of one dimensional motion.
We are interested in learning how to describe one-dimensional motion both graphically and mathematically. The study of how to describe motion using mathematical and graphical representations is known as kinematics.
Describing the motion of real objects is not always easy. An odd shaped cloud scooting along in the sky could be chang-ing its size and shape as it moves. Physicists often start investigations by using simplifying assumptions and ide-alizations. Thus, we begin with the study of objects that are small compared to the distances they move so we can treat them as mathematical points.
We will begin the study of motion by studying motion along a line. Such motion is said to be one dimensional. The key to the description of observed motions in one di-mension is the ability to measure the distance of an object from a reference point and the time at which it is at each distance. Physicists define distance from a reference point as position. Position and time are indeed the fundamental measurements in the study of motion. You will begin your motion studies using an ultrasonic mo-tion detector attached to a computer, to study the motion of your own body as well as that of a cart that increases and decreases its velocity at a steady rate. Since the actual rules for calculating velocity and acceleration from dis-tance and time measurements are programmed into the motion software, you have a unique opportunity to "dis-cover" intuitively what velocity and acceleration mean and how they are represented graphically.
The motion detector investigations used in this Unit are based on activities developed initially by Dr. Ronald Thornton of Tufts University and Dr. David Sokoloff of the University of Oregon.
SESSION ONE: DESCRIBING MOTION WITH WORDS AND GRAPHS 10 min
Session one will be skipped. This material is typically covered in BC grade 12 physics or SFU Physics 100 and will not be done in Physics 140. You should read this section for review. Note especially the discussion of vec-tors.
The focus in this session on kinematics is to be able to de-scribe your position and velocity over time using words and graphs. You will use a motion detector attached to a com-puter in the laboratory to learn to describe one-dimensional motion.
For the investigations in this session you will need:
! ! • a computer! ! • an laboratory interface! ! • an ultrasonic motion detector! ! • LoggerPro software! ! • OPTIONAL: a number line on the floor
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-3Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
The ultrasonic motion detector acts like a stupid bat when hooked up with a computer system. It sends out a series of sound pulses that are too high frequency to hear. These pulses reflect from objects in the vicinity of the motion detector and some of the sound energy re-turns to the detector. The computer is able to record the time it takes for reflected sound waves to return to the detector and then, by knowing the speed of sound in air, figure out how far away the reflecting object is. There are several things to watch out for when using a motion detector.
When Using a Motion Detector
1. Do not get closer that 0.2 metres from the detector because it cannot record reflected pulses which come back too soon.
2. The ultrasonic waves come out in a cone of about 15o. It will see the closest object. Be sure there is a clear path between the object whose motion you want to track and the motion detector.
3. The motion detector is very sensitive and will detect slight motions. You can try to glide smoothly along the floor, but don't be surprised to see small bumps in velocity graphs and larger ones in acceleration graphs.
4. Some objects like bulky sweaters are good sound absorbers and may not be "seen" well by a motion detector. You may want to hold a book in front of you if you have loose clothing on.
30 min
Distance vs. Time Graphs of Your MotionIn this session you will examine two different ways that the motion of an object can be represented graphically: first you will use a motion detector to create distance (posi-tion) vs. time graphs of the motion of your own body and then you will create velocity vs. time graphs.
Figure 1: Walking in front of a motion detector attached to a Computer-based Laboratory (or CBL) system.
The purpose of the first activity in this session is to learn how to relate graphs of distance as a function of time to the motions they represent.
How does a distance vs. time graph look when you move slowly? Quickly? What happens when you move toward the motion detector? Away? After completing the next few ac-tivities, you should be able to look at a distance vs. time graph and describe the motion of an object. You should also be able to to look at the motion of an object and sketch a graph representing that motion.
Note:•"Distance" means "distance from the motion detector." • The motion detector is located at the origin of each graph.
To do the activity and those that follow you should double check to see that: (1) a LabPro is connected to the com-puter, is turned on and has a motion detector plugged in, and (2) that the Logger Pro program on your computer has been opened.
✍Activity 3-1: Making Distance vs. Time Graphs Make distance-time graphs for different walking speeds and di-rections by clicking on the start button at the bottom of the screen and walking in front of the motion detector at distances that are no closer than 0.5 m. Try the following motions and sketch the graph you observe in each case:
(a) Start at 0.3 m walk away from the origin (i.e., the detector) slowly and steadily.
(b) Walk away from the origin me-dium fast and steadily.
(c) Walk toward the detector (ori-gin) slowly and steadily. Sketch the graph.
(d) Describe the difference between the graph you made by walk-ing away slowly and the one made by walking away more quickly.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-5Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
(e) Describe the difference between the graph made by walking toward and the one made walking away from the motion detec-tor.
Note: It is common to refer to the distance of an object from some origin as the position of the object. Since the motion detec-tor is at the origin of the co-ordinate system, it is better to refer to the graphs you have made as position vs. time graphs.
Predicting the Graph of a Motion Described in Words A good way to double check that you understand how to interpret position vs. time graphs is to predict the shape of a graph that would result if a motion that can be described in words and then carry out the motion.
✍Activity 3-2: Predicting a Position vs. Time Graph(a) Suppose your were to start 1.0 m in front of the detector and walk away slowly and steadily for 4 seconds, stops for 4 seconds, and then walk toward the detector quickly. Sketch your predic-tion on the axes below using a dashed line.
(b)Test your prediction by opening the RTP experiment file L1A1-1 (Away and Back). (Select Open on the File menu, and double click with the mouse pointing to this experiment.) Move in the way described and use the graph above to sketch the ac-tual trace of your actual motion with a solid line.
(c) Is your prediction the same as the final result? If not, describe how you would move to make a graph that looks like your predic-tion.
Matching Position vs. Time GraphsCan you turn the activity you just did inside out? We'd like you to be able to look at a graph and then be able to de-scribe the motion it depicts in words. The real test is to be able to reproduce the motion correctly.
✍Activity 3-3: Matching Position vs. Time Graphs(a) To do the first matching activity you should Open the ex-periment file called L1A1-2 (Position Match). A position graph like that shown on the following graph should appear on the screen. (This graph is stored in the background as data B so you will be able to move in front of the detector and see your graph (as data A) traced out on top of the trace you are to match.)
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Time (seconds)
(b) Describe in your own words how you plan to move in order to match this graph.
(c) Move to match the Position Match graph on the computer screen. You may try a number of times. It helps to work in a team. Get the times right. Get the positions right. Do this for yourself. (Each person in your group should do his or her own match.) You will not learn very much by just watching!
(d) What was the difference in the way you moved to produce the two differently sloped parts of the graph you just matched?
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-7Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
(e) Make curved position vs. time graphs like those shown below.Note: Before trying to reproduce the shapes shown below, get rid of the graph you just matched by choosing "Hide Data B" from the data menu in the Motion software.
T i m e !
T i m e
Graph 1 Graph 2
(f) Describe how you must move to produce a position vs. time graph with each of the shapes shown.
Graph 1 answer:
Graph 2 answer:
(g) What is the general difference between motions which result in a straight line position vs. time graph and those that result in a curved-line position vs. time graph?
30 min
Velocity vs. Time Graphs of Your MotionYou have already plotted your position as a function of time. Another way to represent your motion during an in-terval of time is with a graph which describes how fast and in what direction you are moving from moment to moment. How fast you move is known as your speed. It is the rate of change of position with respect to time. Velocity is a quan-
tity which takes into account your speed and the direction you are moving. Thus, when you examine the motion of an object moving along a line, its velocity can be positive or negative meaning the velocity is in the positive or negative direction.
Graphs of velocity over time are more challenging to create and interpret than those for position. A good way to learn to interpret them is to create and examine velocity vs. time graphs of your own body motions, as you will do in the next few activities. You will need the following:
! • Logger Pro software ! • Motion detector! • LabPro Interface ! • Number line on floor in metres (optional)
To do the next few activities you should Open the motion software and set it to graph velocity. To do this you can double click anywhere on the position graph to display a dialogue box. Move the mouse pointer to the Position label, hold down the button and select Velocity. Set the Velocity axis from –1.0 to +1.0 m/s. Also change the Time axis to read 0 to 5 s.
✍Activity 3-4: Making Velocity vs. Time Graphs
Note: To change the scale of your graph so the trace fills the screen better, double click on the original graph and use the dia-logue box that pops up to choose a different maximum and mini-mum velocity
(a) Make a velocity graph by walking away from the detector slowly and steadily. Try again until you get a graph you're satis-fied with and then sketch your result on the graph that follows. (We suggest you draw smooth patterns by ignoring smaller bumps that are mostly due to your steps.).
(b) Make a velocity graph, walking away from the detector stead-ily at a medium speed. Sketch your graph below.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-9Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
Time (seconds)(c) Make a velocity graph, walking toward the detector slowly and steadily. Sketch your graph below.
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(d) What is the most important difference between the graph made by slowly walking away from the detector and the one made by walking away more quickly?
(e) How are the velocity vs. time graphs different for motion away and motion toward the detector?
Predicting Velocity vs. Time Graphs Based on WordsSuppose you were to undergo the following sequence of mo-tions:1. walk away from the detector slowly and steadily for 6 seconds
2. stand still for 6 seconds
3. walk toward the detector steadily about twice as fast as before
✍Activity 3-5: Predicting a Velocity vs. Time Graph(a) Use a dashed line in the graph that follows to record your prediction of the shape of the velocity graph that will result from the motion described above.
(b) Compare predictions with your partner(s) and see if you can all agree. Use a solid line to sketch your group prediction in the graph above.
(c) Adjust the time scale to 15 s in the Motion Software and then test your prediction. Repeat your motion until you are confident that it matches the description in words and then draw the ac-tual graph on the axes below. Be sure the 6-second stop shows clearly.
Final Result
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1. (d) Did your prediction match your real motion? If not what misunderstanding of what the elements of the graph repre-sent did you have?
Comments about Velocity Graphs: (1) Velocity implies both speed and direction. How fast you move is your speed, the rate of change of position with re-spect to time. As you have seen, for motion along a line (e.g., the positive x-axis) the sign (+ or -) of the velocity in-
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-11Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
dicates the direction. If you move away from the detector (origin), your velocity is positive, and if you move toward the detector, your velocity is negative.
(2) The faster you move away from the origin, the larger positive number your velocity is. The faster you move to-ward the origin, the "larger" negative number your velocity is. That is -4 m/s is twice as fast as -2 m/s and both motions are toward the origin.
Velocity Vectors:These two ideas of speed and direction can be combined and represented by vectors.
A velocity vector is represented by an arrow pointing in the direction of motion. The length of the arrow is drawn pro-portional to the speed; the longer the arrow, the larger the speed. If you are moving toward the right, your velocity vector can be represented by the arrow shown below.
If you were moving twice as fast toward the right, the ar-row representing your velocity vector would look like:
while moving twice as fast toward the left would be repre-sented by the following arrow:
What is the relationship between a one-dimensional veloc-ity vector and the sign of velocity? This depends on the way you choose to set the positive x-axis.
0 +
Positive velocity
Negative velocity0+
Positive velocity
Negative velocity
In both diagrams the top vectors represent velocity toward the right. In the left diagram, the x-axis has been drawn so that the positive x-direction is toward the right. Thus the top arrow represents positive velocity. However, in the right diagram, the positive x-direction is toward the left. Thus the top arrow represents negative velocity. Likewise, in both diagrams the bottom arrows represent velocity to-ward the left. In the left diagram this is negative velocity, and in the right diagram it is positive velocity.
✍Activity 3-6: Sketching Velocity Vectors Sketch below velocity vectors representing the three parts of the motion described in the prediction you made in Activity 3-5.
(a) Walking slowly away from the detector:
(b) Standing still:
(c) Walking rapidly toward the detector:
Velocity Graph MatchingIn the next activity, you will try to move to match a veloc-ity graph shown on the computer screen. This is often much harder than matching a position graph as you did in the previous investigation. Most people find it quite a chal-lenge at first to move so as to match a velocity graph. In fact, some velocity graphs that can be invented cannot be matched! To do this activity pull down the File Menu and select Open. Then double click on L1A2-2 (Velocity Match). The following velocity graph should appear on the screen.
Vel
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Time (seconds)4 8 12 16 20
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✍Activity 3-7: Matching a Velocity Graph(a) Describe how you think you will have to move in order to match the given velocity graph.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-13Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
(b) Move in such a way that you can reproduce the graph shown. You may have to practice a number of times to get the move-ments right. Work as a team and plan your movements. Get the times right. Get the velocities right. You and each person in your group should take a turn. Then draw in your group's best match on the axes that follow.
Vel
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Time (seconds)4 8 12 16 20
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(c) Describe how you moved to match each part of the graph.
(d) Is it possible for an object to move so that it produces an abso-lutely vertical line on a velocity time graph? Explain.
(e) Did you run into the motion detector on your return trip? If so, why did this happen? How did you solve the problem? Does a velocity graph tell you where to start? Explain.
Relating Position and Velocity GraphsYou have looked at position and velocity vs. time graphs separately. Since position vs. time and velocity vs. time graphs are different ways to represent the same motion, it ought to be possible to figure out the velocity at which someone is moving by examining her/his position vs. time graph. Conversely, you ought to be able to figure out how far someone has travelled (change in position) from a veloc-ity vs. time graph.
To explore how position vs. time and velocity vs. time graphs are related, you will need the following:
! • Logger Pro software ! • Motion detector! • LabPro Interface ! • Number line on floor in metres (optional)
To complete the next Activity, you'll need to set up the Mo-tion Software to display both position vs. time and velocity vs. time simultaneously. To do this:
1. Clear data from any previous graphs
2. Pull down the Display Menu and select Two Graphs.
3. Set the top graph to display Position from 0 to 4 m for 5 s.
4. Set the bottom graph to display Velocity from -1 to 1 m/s for
5 s.
✍Activity 3-8: Predicting Velocity Graphs from Po-sition Graphs(a) Carefully study the position graph shown below and predict the velocity vs. time graph that would result from the motion. Using a dashed line, sketch your prediction of the corresponding velocity vs. time graph on the velocity axes.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-15Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
(b) After each person in your group has sketched a prediction, test your prediction by matching the position vs. time graph shown.When you have made a good duplicate of the position graph, sketch your actual graph over the existing position vs. time graph.
(c)Use a solid line to draw the actual velocity graph on the same graph with your prediction. (Do not erase your prediction).
(d) How would the position graph be different if you moved faster? Slower?
(e) How would the velocity graph be different if you moved faster? Slower?
Estimating and Calculating VelocityIn this activity, you will estimate an average velocity from the velocity graph you created in the previous Activity and then calculate an average velocity using your position graph.
✍Activity 3-9: Average Velocity Calculations(a) Find your average velocity from your velocity graph in the previous activity. Select Examine in the Analyze menu, read a
number of values (say ten) from the portion of your velocity graph where your velocity is relatively constant, and use them to calcu-late the average (mean) velocity.
Velocity values ____ ____ ____ ____ ____
from graph (m/s) ____ ____ ____ ____ ____
Average value of the velocity: ________m/s
Note: Average velocity during a particular time interval can also be calculated as the change in position divided by the change in time. (The change in position is often called the displacement. ) By definition, this is also the slope of the position vs. time graph for that time period. As you have observed, the faster you move, the more inclined is your position vs. time graph. The slope of a position vs. time graph is a quantitative measure of this incline, and therefore it tells you the velocity of the object.
(b) Use the method just described in the note to calculate your average velocity from the slope of your position graph in Activity 3-7. Use Examine (under the Analyze menu) to read the posi-tion and time co-ordinates for two typical points while you were moving. (For a more accurate answer, use two points as far apart as possible but still typical of the motion, and within the time interval over which you took velocity readings in Activity 3-8.)
Position (m) Time (sec)
Point 1
Point 2
(c) Calculate the change in position (displacement) between points 1 and 2. Also calculate the corresponding change in time (time interval). Divide the change in position by the change in time to calculate the average velocity. Summarize the results of your calculations below.
Change in position (m)
Time interval (sec)
Average velocity (m/s)
(d) Is the average velocity positive or negative? Is this what you expected?
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-17Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
(e) Does the average velocity you just calculated from the posi-tion graph agree with the average velocity you estimated from the velocity graph? Do you expect them to agree? How would you account for any differences?
Predicting Position Graphs from Velocity GraphsThe final challenge is to be able to produce position vs. time graphs from velocity graphs. In order to do this suc-cessfully, you need to know the position of the person or object of interest at least one of the times.
✍Activity 3-10: Finding Position from a Velocity Graph(a) Carefully study the velocity graph that follows. Using a dashed line, sketch your prediction of the corresponding position graph on the bottom set of axes. (Assume that you started at the 1-metre mark.)
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(b) After each person has sketched a prediction, do your group's best to duplicate the top (velocity vs. time) graph by walking. (Reset the Time axis to 0 to 10 seconds before you start.) When you have made a good duplicate of the velocity vs. time graph, draw your actual result over the existing velocity vs. time graph.
(c) Use a solid line to draw the actual position vs. time graph on the same axes with your prediction. Do not erase your prediction.
(d) How can you tell from a velocity vs. time graph that the mov-ing object has changed direction?
(e) What is the velocity at the moment the direction changes?
(f) Is it possible to actually move your body (or an object) to make vertical lines on a position vs. time graph? Why or why not? What would the velocity be for a vertical section of a position vs. time graph?
(g) How can you tell from a position vs. time graph that your mo-tion is steady (motion at a constant velocity)?
(h) How can you tell from a velocity vs. time graph that your mo-tion is steady (constant velocity)?
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-19Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
Discussion of the Motion HomeworkYou should share your observations on motion and conclu-sions you drew from them with your classmates.
50 minVelocity and Acceleration GraphsBody motions can be very jerky and irregular. We are in-terested in having you learn to describe some simple mo-tions in which the velocity of an object is changing. In or-der to learn to describe motion in more detail for some simple situations, you will be asked to observe and de-scribe the motion of a cart on a flat track. Although, graphs and words are still important representations of these mo-tions, you will also be asked to draw velocity vectors, which are arrows that indicate both the direction and speed of a moving object. Thus, you will also learn how to represent simple motions with velocity diagrams .
Describing Motion With Pictures . . .
t = 0s t = 1s t = 2s
Ave
rage
Joe
Sch
moe
Phys
ics
Stud
ent
1.1 m 1.4 m2.1 m
In the last session, you looked at position vs. time and ve-locity vs. time graphs of the motion of your body as you moved at a "constant" velocity. The data for the graphs were collected using a motion detector. Your goal in this session is to learn how to describe various kinds of motion in more detail. It is not enough when studying motion in
physics to simply say that "the object is moving toward the right" or "it is standing still." You have probably realized that a velocity vs. time graph is better than a position vs. time graph when you want to know how fast and in what direction you are moving at each instant in time as you walk. When the velocity of an object is changing, it is also important to know how it is changing. The rate of change of velocity is known as the acceleration.
In order to get a feeling for acceleration, it is helpful to create and learn to interpret velocity vs. time and accelera-tion vs. time graphs for some relatively simple motions of a cart on a track. You will be observing the cart with the mo-tion detector as it moves at a constant velocity and as it changes its velocity at a constant rate. For the activities in this session you will need:
! •a computer! •a laboratory interface (LabPro)! • an ultrasonic motion detector! • motion software (Logger Pro)! • a cart or toy car with very little friction! • a smooth, level track (Force and Motion Track)! • a battery-operated fan cart ! • 4 fresh AA cells and 2 aluminum dummy cells
IMPORTANT NOTE ON SAVING YOUR FILES: You may be asked to use the data collected using the motion software in some of the Activities for mathematical analysis in future ses-sions. For each activity please save the graph sets and associated data on the SFU fileserver, or on your own USB drive, as in-structed in Appendix A. We recommend that you identify your files by the lab number and activity number for later reference along with your group initials. Thus, if Stern, Ricci, and Pers-inger work together on the LAB 2 activities, the file names might be, for example, L2A1-1(SRP), L2A1-2 (SRP), etc.
Graphing a Constant Velocity Cart MotionLet's start by giving a frictionless cart a push along a smooth track and graphing its motion.
0.3 m
To do this you should:
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-21Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
1. Set up the motion detector at the end of a level track, If the cart has a friction pad, move it out of contact with the track so that the cart can move freely.
2. Set up the position and velocity axes as shown on the next page by opening the experiment L2A1-1 (Pos and Vel).
✍Activity 3-11: Position, Velocity and Acceleration Graphs of Constant Velocity(a) How should the position and velocity graphs look if you move the cart at a constant velocity away from the motion detector starting at the 0.3 metre mark? Sketch your predictions with dashed lines on the axes that follow.
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(b) Test your prediction for the velocity graph. (Single click on the velocity graph to plot velocity first.) Be sure that the cart is never closer than 0.2 metre from the motion detector. Try several times until you get a fairly constant velocity. Sketch your results with solid lines on the axes shown.
(c) Did your position vs. time and velocity vs. time graphs agree with your predictions? What characterizes constant velocity mo-tion on a position vs. time graph?
(d) What characterizes constant velocity motion on a velocity vs. time graph?
(e) Acceleration is defined as the time rate of change of velocity. Sketch your prediction of the cart acceleration (for the motion you just observed) on the axes that follow using a dashed line.
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(f) Display and the acceleration graph of the cart and use a solid line to sketch it using the axes above. Note: To display an accel-eration graph in the motion software the mouse button down on the Position label on your graph and selecting Acceleration. You may wish to change the minimum on the Acceleration axis by clicking on 0.0 and changing it to -2.0.
Finding Accelerations To find the average acceleration of the cart during some time interval (the average time rate of change of its veloc-ity), you must measure its velocity at two different times, calculate the difference between the final value and the initial value and divide by the time interval.
To find the acceleration vector from two velocity vectors, you must first find the vector representing the change in velocity by subtracting the initial velocity vector from the final one. Then you divide this vector by the time interval.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-23Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
✍Activity 3-12: Representing Acceleration(a) Does the acceleration vs. time graph you observed agree with this method of calculating acceleration? Explain. Does it agree with your prediction?
(b) The diagram below shows the positions of the cart at equal time intervals. (This is like taking snapshots of the cart at equal time intervals.) At each indicated time, sketch a vector above the cart which might represent the velocity of the cart at that time while it is moving at a constant velocity away from the motion detector.
(c) Explain how you would find the vector representing the change in velocity between the times 1.0 s and 2.0 s in the dia-gram above. From this vector, what value would you calculate for the acceleration? Explain. Is this value in agreement with the acceleration graph you obtained in Activity 3-11?
Speeding Up at a Moderate RateIn the next activity you will look at velocity and accelera-tion graphs of the motion of a cart when its velocity is changing. You will be able to see how these two represen-tations of the motion are related to each other when the cart is speeding up.
In order to get your cart speeding up smoothly you can use a propeller driven by an electric motor to accelerate the cart. Your task in the next activity is to create nice smooth
graphs of position and velocity vs. time of a fan cart with a moderate thrust that shows the cart starting from rest.
You should set up the cart, track, fan attachment and mo-tion detector as shown in the following diagrams.
0.3 m
• Use two batteries and two dummy cells in the fan assembly so that the cart does not speed up too fast.)
• Always catch the fan at the end of a run before it crashes!
• To keep the motion detector from collecting bad data from the rotation of the propeller, be sure that the fan blade does not ex-tend beyond the front end of the cart.
• Use the motion software experiment file L2A1-2 (Speeding Up) to display Position from 0 to 2.0 m and Velocity from -1.0 to 1.0 m/sec for a total time of 3.0 sec.
• Scale the graphs so that the traces fill the screen
• Practise several times before completing the tasks described below.!
✍Activity 3-13: Graphs Depicting Speeding Up(a) Create position vs. time and velocity vs. time graphs of your fan cart as it moves away from the detector and speeds up. Sketch the graphs neatly on the following axes.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-25Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
(c ). Keep your data for analysis in the next activity by selecting Store Latest Run . . . on the Experiment Menu. Then Save the file with the name SPEEDUP1.XXX, where XXX are your initials.
(d) How does your position graph differ from the position graphs for steady (constant velocity) motion?
(e) What feature of your velocity graph signifies that the motion was away from the detector?
(f) What feature of your velocity graph signifies that the cart was speeding up? How would a graph of motion with a constant veloc-ity differ?
(g) Sketch the acceleration graph on the axes that follow. (Adjust the axes if necessary).
(h) During the time that the cart is speeding up, is the accelera-tion positive or negative? How does speeding up while moving away from the detector result in this sign of acceleration? Hint: Remember that acceleration is the rate of change of velocity. Look at how the velocity is changing.
(i) How does the velocity vary in time as the cart speeds up? Does it increase at a steady rate or in some other way?
(j) How does the acceleration vary in time as the cart speeds up? Is this what you expect based on the velocity graph? Explain.
Using Vectors to Describe the AccelerationLet's return to the Vector Diagram representation and use it to describe the acceleration.
✍Activity 3-14: Acceleration Vectors (a) The diagram that follows shows the positions of the cart at equal time intervals. At each indicated time, sketch a vector above the cart which might represent the velocity of the cart at that time while it is moving away from the motion detector and speeding up.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-27Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
(b) Show below how you would find the approximate length and direction of the vector representing the change in velocity be-tween the times 1.0 s and 2.0 s using the diagram above. No quantitative calculations are needed. Based on the direction of this vector and the direction of the positive x-axis, what is the sign of the acceleration? Does this agree with your answer to Ac-tivity 3-13 (h)?
30 minMeasuring Acceleration In this investigation you will analyse the motion of your accelerated cart quantitatively. This analysis will be quan-titative in the sense that your results will consist of num-bers. You will determine the cart's acceleration from the slope your velocity vs. time graph and compare it to the average acceleration read from the acceleration vs. time graph. You can display actual values for your acceleration and velocity data using the motion software.
At any rate, to do this next activity you will need the Log-ger Pro software and your experiment file, Speedup1.xxx, from Activity 3-13.
✍Activity 3-15: Calculating Accelerations (a) Re sketch the velocity and acceleration graphs you found in Activity 3-13 using the axes that follow. Correct the scales if nec-essary.
(b) List 10 of the typical recorded accelerations of the cart. To find these values select Examine from the Analyze Menu. Scroll along with the mouse to display the values at different times. (Only use values from the portion of the graph after the cart was released and before you stopped it.)
Accel. Accel.1 62 73 84 95 10
Select Examine again to turn off the display of values.
(c) Calculate the average value of the acceleration.
Average acceleration (mean): _________m/s2
(d) Since the average acceleration during a particular time pe-riod is defined as the change in velocity divided by the change in time. This is the average rate of change of velocity. By definition, the rate of change of a quantity graphed with respect to time is also the slope of the curve. Thus the (average) slope of an object's velocity vs. time graph is the (average) acceleration of the object.
Find the data needs to calculate the approximate slope of your velocity graph. Use Examine to read the velocity and time co-ordinates for two typical points on the velocity graph. For a more
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-29Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
accurate answer, use two points as far apart in time as possible but still during the time the cart was speeding up.
Velocity (m/s) Time (s)Point 1Point 2
(e) Calculate the change in velocity between points 1 and 2. Also calculate the corresponding change in time (time interval). Di-vide the change in velocity by the change in time. This is the av-erage acceleration. Show and then summarize your calculations below.
Speeding Up
Change in Velocity (m/s)
Time Interval (s)
Avg. Acceleration (m/s2)
(f) Is the acceleration positive or negative? Is this what you ex-pected?
(g) Does the average acceleration you just calculated agree with the average acceleration you calculated from the acceleration graph? Do you expect them to agree? How would you account for any differences?
Speeding Up at a Faster RateSuppose that you accelerate your cart at a faster rate by putting four batteries instead of two in it? How would your velocity and acceleration graphs change?
✍Activity 3-16: Calculating Accelerations (a) Re sketch the velocity and acceleration graphs you found in Activity 3-13 once more using the axes that follow.
-1
+1
0
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-
-- - - - -
-1
+1
0
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.6 1.2 1.8 2.4 3.00
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Acc
eler
atio
n (m
/s/s)
Vel
ocity
(m/s)
Time (seconds)
(b) In the previous set of axes, use a dashed line or another col-our to sketch your predictions for the general graphs that depict a fan cart running under more power (i.e., the four batteries). Exact predictions are not expected. We just want to know how you think the general shapes of the graphs will change.
(c) Test your predictions by accelerating the cart with four AA cells in the fan assembly battery compartment. Repeat if neces-sary to get nice graphs and then sketch the results in the axes that follow. Use the same scale as you did for the sketch of the two battery graph.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-31Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
Time (seconds) EXPERIMENTAL RESULTS OF SPEEDING UP RAPIDLY
(d) Did the general shapes of your velocity and acceleration graphs agree with your predictions? How is the greater magni-tude (size) of acceleration represented on a velocity vs. time graph?
(e) How is the greater magnitude (size) of acceleration repre-sented on an acceleration vs. time graph?
SESSION THREE: SLOWING DOWN, SPEEDING UP, AND TURNING30 min
Discussion of the Changing Motion HomeworkYou should share your observations on motion and conclu-sions you drew from them with your classmates.
5 minAbout Slowing Down, Speeding Up and TurningIn the previous sessions in this unit, you explored the characteristics of position vs. time, velocity vs. time and acceleration vs. time graphs of the motion of a cart. In the cases examined, the cart was always moving away from a motion detector, either at a constant velocity or with a con-stant acceleration. Under these conditions, the velocity and acceleration are both positive. You also learned how to find the magnitude of the accel-eration from velocity vs. time and acceleration vs. time graphs, and how to represent the velocity and acceleration using vectors.
In the motions you studied in the last sessions the velocity and acceleration vectors representing the motion of the cart both pointed in the same direction.In order to get a better feeling for acceleration, it will be helpful to examine velocity vs. time and acceleration vs. time graphs for some slightly more complicated motions of a cart on a track. Again you will use the motion detector to observe the cart as it changes its velocity at a constant rate. Only this time the motion may be toward the detec-tor, and the cart may be speeding up or slowing down.
In order to complete the activities in this session you will need the same apparatus you used last time. This includes:
! • a computer! • a Lab Pro interface! • an ultrasonic motion detector! • Logger Pro software! • a cart or toy car with very little friction! • a smooth, level track! • a battery-operated fan cart
45 min Slowing down and Speeding UpIn this activity you will look at a cart (or toy car) moving along a track and slowing down. A car being brought to rest by the steady action of brakes is a good example of this type of motion. Later you will examine the motion of the cart toward the motion detector and speeding up. In both cases, we are interested in the shapes of the velocity vs.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-33Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
time and acceleration vs. time graphs, as well as the vec-tors representing velocity and acceleration. Let's start with the creation of velocity and acceleration graphs of when it is moving away from the motion detector and slowing down. To do this activity, you should set up the cart, track, and motion detector as shown below. If the cart has a friction pad, move it out of contact with the track so that the cart can move freely. Use the same two AA cells and two dummy cells as you used for the Speeding Up 1 activity in the last session.
Direction of hand push
Direction of fan thrust
0.3 m
The fan in the picture is exerting a thrust on the cart to-wards the motion detector. then the force on the cart is towards the motion In this activity you will examine the velocity and acceleration of this motion.
✍Activity 3-17: Graphs Depicting Slowing Down(a) If you give the cart a push away from the motion detector and release it, will the acceleration be positive, negative or zero (after it is released)? Sketch your predictions for the velocity vs. time and acceleration vs. time graphs on the axes below.
-
- - - - -
Vel
ocity
(m/s)
- - - - -
Acc
eler
atio
n (m
/s/s)
Time (seconds)
+
0
-+
0
-
(b) To test your predictions open the experiment L3A1-1 (Slow-ing Down) to display the velocity vs. time and acceleration vs. time. Then locate the cart 0.5 m from the detector, turn on its fan, and push the cart away from the motion detector once starts clicking. Graph velocity first. Catch the cart before it turns around.
Draw the results on the axes that follow. You may have to try a few times to get a good run. Don't forget to change the scales if this will make your graphs clearer before making the sketch so change the scale on the axes shown, if needed.
-2
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0
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ccel
erat
ion
(m/s
/s)
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(m/s
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1 2 3 4 50Time (seconds)
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SLOWING DOWN MOVING AWAY (c) Keep your graphs for later comparison, by selecting Store Latest Run from the Experiment menu, and then label your graphs with— "A" at the spot where you started pushing. "B" at the spot where you stopped pushing. "C" at the spot where the cart stopped moving.Also sketch on the same axes the velocity and acceleration graphs for Speeding Up from Activity 3-13. Use the same scale for the two sketches.
(d) Did the shapes of your velocity and acceleration graphs agree with your predictions? How is the sign of the acceleration repre-sented on a velocity vs. time graph?
(e) How is the sign of the acceleration represented on an accel-eration vs. time graph?
(f) Is the sign of the acceleration what you predicted? How does slowing down while moving away from the detector result in this
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-35Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
sign of acceleration? Hint: Remember that acceleration is the rate of change of velocity. Look at how the velocity is changing.
Constructing Acceleration Vectors for Slowing DownLet's consider a diagrammatic representation of a cart which is slowing down and use vector techniques to figure out the direction of the acceleration.
✍Activity 3-18: Vector Diagrams for Slowing Down(a) The diagram that follows shows the positions of the cart at equal time intervals. (This is like taking snapshots of the cart at equal time intervals.) At each indicated time, sketch a vector above the cart which might represent the velocity of the cart at that time while it is moving away from the motion detector and slowing down.
(b) Show below how you would find the vector representing the change in velocity between the times 1s and 2 s in the diagram above. Based on the direction of this vector and the direction of the positive x-axis, what is the sign of the acceleration? Does this agree with your answer to Question (f) in Activity 3-17?
(c) Based on your observations in this activity and in the last session, state a general rule to predict the sign and direction of the acceleration if you know the sign of the velocity (i.e. the di-rection of motion) and whether the object is speeding up or slow-ing down.
Speeding Up Toward the Motion DetectorLet's investigate another common situation. Suppose the cart is allowed to speed up when travelling toward the mo-tion detector. What will the direction of the acceleration be? Positive or negative?
✍Activity 3-19: Graphs Depicting Speeding Up(a) Use the general rule that you stated in Activity 3-18 to pre-dict the shapes of the velocity and acceleration graphs. Sketch your predictions using the axes that follow.
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- - - - -
Vel
ocity
(m/s)
- - - - -
Acc
eler
atio
n (m
/s/s)
Time (seconds)
+
0
-+
0
-
(b) Test your predictions by turning on the fan and releasing the cart from rest at the end of the track after the motion detector starts clicking. Catch the cart before it gets too close to the detec-tor.
Draw the results on the axes that follow. You may have to try a few times to get a good run. Don't forget to change the scales if this will make your graphs clearer before making the sketch so change the scale on the axes shown, if needed.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-37Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
(c) How does your velocity graph show that the cart was moving toward the detector?
(d) During the time that the cart was speeding up, is the accel-eration positive or negative? Does this agree with your predic-tion? Explain how speeding up while moving toward the detector results in this sign of acceleration. Hint: Thin about how the ve-locity is changing.
Constructing Acceleration Vectors for Speeding UpLet's consider a diagrammatic representation of a cart which is speeding up and use vector techniques to figure out the direction of the acceleration.
✍Activity 3-20: Vector Diagrams for Slowing Down(a) The diagram that follows shows the positions of the cart at equal time intervals. (This is like taking snapshots of the cart at
equal time intervals.) At each indicated time, sketch a vector above the cart which might represent the velocity of the cart at that time while it is moving toward the motion detector and speeding up.
(b) Show below how you would find the vector representing the change in velocity between the times 1s and 2 s in the diagram above. Based on the direction of this vector and the direction of the positive x-axis, what is the sign of the acceleration? Does this agree with your answer to Question (d) in Activity 3-19?
(c) Was the general rule you developed in Activity 3-18 (c) cor-rect? If not, modify it and restate it here.
Moving Toward the Detector and Slowing DownThere is one more possible combination of velocity and ac-celeration for the cart, that of moving toward the detector while slowing down.
✍Activity 3-21: Slowing Down Toward the Detector(a) Use your general rule to predict the direction and sign of the acceleration when the cart is slowing down as it moves toward
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-39Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
the detector. Explain why the acceleration should have this di-rection and this sign in terms of the velocity and how the velocity is changing.
(b) The diagram below shows the positions of the cart at equal time intervals for slowing down while moving toward the detec-tor. At each indicated time, sketch a vector above the cart which might represent the velocity of the cart at that time while it is moving toward the motion detector and slowing down.
(c) Show below how you would find the vector representing the change in velocity between the times 1s and 2 s in the diagram above. Based on the direction of this vector and the direction of the positive x-axis, what is the sign of the acceleration? Does this agree with the prediction you made in part (a)?
40 minAcceleration and Turning AroundIn Session 2 of this unit and in the first activity in this ses-sion, you looked at velocity vs. time and acceleration vs. time graphs for a cart moving in one direction with a changing velocity. In this investigation you will look at what happens when the cart slows down, turns around and then speeds up. How is its velocity changing? What is its acceleration?
The set-up you can use shown below--the same as for the first activity you did in this session. Once again fan cart should have two AA cells and two dummy cells.
Direction of hand push
0.3 m
Direction of fan thrust
To practice this motion you should start the fan, and give the cart a push away from the motion detector. It moves toward the end of the track, slows down, reverses direction and then moves back toward the detector. Try it without activating the motion detector! Be sure to stop the cart be-fore it hits the motion detector.
✍ Activity 3-22: Reversing Direction(a) For each part of the motion — away from the detector, at the turning point, and toward the detector, predict in the table that follows whether the velocity will be positive, zero or negative. Also indicate whether the acceleration is positive, zero or nega-tive.
Moving Away Turning Around Moving Toward
Velocity
Acceleration
(b) Sketch the predicted shapes of the velocity vs. time and accel-eration vs. time graphs of this entire motion on the axes that fol-low.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-41Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
(c) To test your predictions set up the velocity and acceleration axes as shown below. (Open the experiment L3A1-1 (Slowing Down) if it is not already opened.) . Use procedures that are similar to the ones you used in the slowing down and speeding up activities. You may have to try a few times to get a good run. Don't forget to change the scales if this will make your graphs clearer. When you get a good run, sketch both graphs on the axes.
-2
+2
0
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Acc
eler
atio
n (m
/s/s
)
-1
+1
0
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Velo
city
(m/s
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1 2 3 4 50Time (seconds)
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(d) Label both graphs withA. where the cart started being pushed.B. where the push ended (where your hand left the cart).C. where the cart reached the end of the track (and is
about to reverse direction).D. where you stopped the cart.
(e) Did the cart have a zero velocity? (Hint: Look at the velocity graph. What was the velocity of the cart at the end of the track?) Does this agree with your prediction? How much time did it spend at zero velocity before it started back toward the detector? Explain.
(f) According to your acceleration graph, what is the acceleration at the instant the cart comes to rest? Is it positive, negative or zero? Does this agree with your prediction?
(g) Explain the observed sign of the acceleration at the end of the track. (Hint: Remember that acceleration is the rate of change of velocity. When the cart is at rest at the end of the track, what will its velocity be in the next instant? Will it be positive or nega-tive?)
Tossing a BallSuppose you throw a ball up into the air. It moves up-ward, reaches its highest point and then moves back down toward your hand. What can you say about the directions of its velocity and acceleration at various points?
✍ Activity 3-23: The Rise and Fall of a BallConsider the ball toss carefully. Assume that upward is the posi-tive direction. Indicate in the table that follows whether the ve-locity is positive, zero or negative during each of the three parts of the motion. Also indicate if the acceleration is positive, zero or negative. Hint: Remember that to find the acceleration, you must look at the change in velocity.
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-43Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
Before Wednesday, September 21st:• Complete the sheets entitled HOMEWORK FOR LAB 1: INTRODUCTION TO MOTION . it is a stapled collection of white papers attached to this assignment page. You may have to return to the classroom after hours to complete the activi-ties, but the homework can be done anywhere.
• Review the comment about velocity vectors on pages 3-12 and 3-13 and read the section below on how to construct simple motion diagrams.
• Construct motion diagrams described in supplemental problem SP3-1 below.
Position-Velocity Motion Diagrams
A position-velocity motion diagram can be used to sketch a quick picture of the changes in motion that an object might undergo that almost anyone can under-stand. A motion diagram represents to position and velocity of an object at several equally spaced times. At each position the object's velocity is represented by an ar-row. Two sample motion diagrams are shown in the figures below.
Figure 1: A motion diagram of a bike moving to the left with a constant velocity. The accel-eration is zero because the velocity is not changing.
Figure 2: A motion diagram for a bike moving to the right with a decreasing speed.
S3-1: (a) Construct a motion diagram for a dog running to the right with a decreasing speed, (b) Construct a motion diagram for a truck moving to the left with increasing speed, and (c) Construct a motion diagram for a rocket moving vertically downward at an increasing speed.
UNIT 3 HOMEWORK AFTER SESSION THREE
Before Friday September 23rd:• Complete the HOMEWORK FOR LAB 2: CHANGING MOTION.
• Complete Unit 3 entries for Sessions 2 and 3 in this Activity Guide
Calculus-based Workshop Physics II: Unit 3-One Dimensional Motion! Page 3-45Authors: Priscilla Laws, Ronald Thornton, & David Sokoloff
