KINEMATICS

Graphs of One-Dimensional Motion

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Using a Motion Detector

The ultrasonic motion detector acts like a bat when hooked up with a microcomputer 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 returns 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

1. Do not get closer that 0.5 meters 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 15°. 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.

 

Distance vs. Time Graphs of Your Motion

POSITION

Position is measured from an origin (at which the position is said to be 0), using a defined direction and units.

Positions can be positive or negative according to which side of the origin they are on. Positions in which the position is positive are said to be in the positive region; those in which it is negative are said to be in the negative region.

Differences in position are called displacements.

A displacement from A to B is described as the position of B to a position of A in the direction from A to B.

Quantities like this that have value and direction are called VECTORS

TIME

Time just has a value not a direction. We say it is a SCALAR

The origin for time is the time at which you decide to start measuring.

Time can be positive or negative. Negative just means times before you started measuring.

Position-Time Graphs (often called Distance -Time graphs)

To do the activity and those that follow you should double check to see that: a Laboratory Interface is connected to the computer, is turned on and has a motion detector plugged into Port A, and that the Motion program on your computer has been opened.

Using the equipment make and sketch position-time graphs for a member of your group in each of the following cases. Note that the motion detector is at the origin and the positive direction is AWAY from the detector. Each person should be the subject of several of the graphs.

In most experiments we should sketch exactly what we see, but in this experiment when you make your sketches, don't do them as you see them; smooth them off and draw them as you think they would look like in the ideal case.

Also, DON'T sketch parts of the motion that don't correspond to the motion you are doing; just sketch the part of the graph that applies.

Leave a space underneath each graph for a graph of velocity vs time, which we will do later.

1A1(a) Stand still close to the origin (maybe 1 metre away)

1A1(b) Stand still far from the origin (maybe 6 metre away)

1A1(c) Walk away slowly

1A1(d) Walk towards origin slowly

1A1(e) Walk away quickly

1A1(f) Walk towards origin quickly

1A1(g) Walk away from origin speeding up

1A1(h) Walk towards origin speeding up

1A1(i) Walk away slowing down [how will you do this?]

1A1(j) Walk towards origin slowing down

1A1(k) (Optional) If you are keen, you can make another 10 sketches, the same as the 10 you have just done, but sketching them how they would look if the positive direction was TOWARDS the detector

A1(l) (Optional) Matching Position vs. Time Graphs

To do the first matching activity you should Open the experiment file called match_d. A position graph like that shown on the following graph should appear on the screen. (This graph is stored so you will be able to move in front of the detector and see your graph (as latest run) traced out on top of the trace you are to match.)

Move to match the 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 try his or her own match.) You will not learn very much by just watching! Show your groups best graph to the instructor so he can initial your activity guide.

Velocity vs. Time Graphs of Your Motion

You have already plotted your position as a function of time. Another way to represent your motion during an interval 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 quantity 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. The positive direction is the direction in which the position is increasing.[moving towards origin if in negative region or away if in positive] The negative direction is the direction in which the position is decreasing. [moving towards origin if in positive region or away if in negative]

Graphs of velocity over time are more challenging to create and interpret than those for position.

To do the next few activities you should Open the motion software V_T.exp it is set to graph velocity Set the Velocity axis from -1.0 to +1.0 m/s. Also change the Time axis to read 0 to 5 s.

Make velocity-time graphs of a person moving in the same ten ways as you did for the position - time graphs. Sketch each as before underneath the corresponding position - time graphs

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 proportional 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 arrow representing your velocity vector would look like:

while moving twice as fast toward the left would be represented by the following arrow:

What is the relationship between a one-dimensional velocity vector and the sign of velocity? This depends on the way you choose to set the positive x-axis.

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 toward the left. In the left diagram this is negative velocity, and in the right diagram it is positive velocity.

Velocity Graph Matching (Optional)

In the next activity, you will try to move to match a velocity 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 challenge 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

Open The file Match_V. A velocity graph similar to the one below should appear on the screen.

Relating Position and Velocity Graphs

 

• You 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 traveled (change in position) from a velocity vs. time graph.

To complete the next Activity, you'll need to set up the Motion Software to display both position vs. time and velocity vs. time simultaneously. To do this open the file d&v_t.exp:

1. Set the top graph to display Position from 0 to 4 m for 5 s.
2. Set the bottom graph to display Velocity from -1 to 1 m/sec for 5 s.

(I)Predicting Velocity Graphs from Position Graphs

 

1. 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.
   
   
2. After each person in your group has sketched a prediction, test your prediction by matching the position vs. time graph shown.
3. When you have made a good duplicate of the position graph, sketch your actual graph over the existing position vs. time graph.
4. (c ) Use a solid line to draw the actual velocity graph on the same graph with your prediction. (Do not erase your prediction).
5. How would the position graph be different if you moved faster? Slower?
6. How would the velocity graph be different if you moved faster? Slower?

   (II) Finding VELOCITY from Velocity-Time and Distance-Time Graphs

Finding VELOCITY from Velocity-Time Graph

(a)Find your average velocity from your velocity graph in the previous activity. Select Examine in the Analyze menu, read a few values from the portion of your velocity graph where your velocity is relatively constant, and use them to calculate the average (mean) velocity.

Velocity values from graph (m/s)

Average value of the velocity: ________m/s

(b)Then select the portion of the velocity graph where your velocity is relatively constant and use Analyze-Statistics (click the mouse on the graph at the start of the values you wish to average, then drag it across and release the mouse) to find the average velocity. Average value of the velocity: ________m/s. In the future you should use the latter method to find AVERAGE.

Finding Velocity from Position-Time Graph

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.

(a) Use the method just described in the note to calculate your average velocity from the slope of your position graph. Use Analyze -Examine to read the position and time coordinates 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 above.)

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 as below.

(b) Is the average velocity positive or negative? Is this what you expected?

(c) Does the average velocity you just calculated from the position 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?

(d) As a last method to find velocity, from the Analyze menu select tangent. Use the mouse to move the tangent line until it appears to be parallel to most of the position time graph where the velocity was constant. Record the value of the slope or tangent. How does this value compare with the other two values you found above.

(III)Predicting Position Graphs from Velocity Graphs

The final challenge is to be able to produce position vs. time graphs from velocity graphs. In order to do this successfully, you need to know the position of the person or object of interest at least one of the times.

Finding Position from a Velocity Graph

 

1. 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-meter mark.)

(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 sec 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 moving 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 motion is steady (motion at a constant velocity)?

(h) How can you tell from a velocity vs. time graph that your motion is steady (constant velocity)?

ACCELERATION

Acceleration is rate of change of velocity. It describes how quickly velocity is changing.

Velocity can change in three ways

An object can (i) speed up

(ii) slow down

(iii) change direction

For motion in a straight line acceleration can be positive or negative.

For an object speeding up the acceleration has the same sign as the velocity.

For an object slowing down the acceleration has the opposite sign as the velocity.

In order to learn to describe motion in more detail for some simple situations, you will be asked to observe and describe the motion of a glider on a flat air track.

In the last session, you looked at position vs. time and velocity 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."

 

• In order to get a feeling for acceleration, it is helpful to create and learn to interpret velocity vs. time and acceleration vs. time graphs for some relatively simple motions of a glider on an air track. You will be observing the glider with the motion detector as it moves at a constant velocity and as it changes its velocity at a constant rate.

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 sessions. For some activities you need to save the graph sets or/and associated data on your data disk. We recommend that you identify your files by the lab number and activity number for later reference along with your group initials. Thus, if Joe Smith, Alice Chen, and Hsin Wang work together on the LAB 2 activities, the file names might be, for example, Lab2_JS&AC&HW. If you use Windows 3.1 you'll have to make shorter names.

Motion Graphs for Object Speeding Up

You should set up the glider, air track and motion detector as shown in the following diagram.

• use the motion software to display Position from 0 to 2.0 m and Velocity from -1.0 to 1.0 m/sec for a total time of about 3.0 sec.
• practice several times before completing the tasks described below.

Graphs Depicting Speeding Up

(a)Create position vs. time and velocity vs. time graphs of your glider as it moves away from the detector and speeds up. Sketch the graphs neatly on axes like below.

Change the Position display to Acceleration. Adjust the acceleration scale so that your graph fills the axes. Sketch your graph on the acceleration axes that follow.

(b)During the time that the glider is speeding up, is the acceleration 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.

(c) How does the velocity vary in time as the glider speeds up? Does it increase at a steady rate (ie is the v-t graph a straight line?)or in some other way?

(d) How does the acceleration vary in time as the glider speeds up? Is this what you expect based on the velocity graph? Explain.

DO NOT ERASE THE GRAPHS; YOU NEED THEM FOR THE ACTIVITY BELOW.

Calculating Accelerations from acceleration-time and velocity-time graphs

Choose sections of the velocity and acceleration graphs where the velocity graph is a straight line of constant slope and thus the acceleration is constant.

Finding Acceleration from Acceleration-Time graph

If the acceleration were exactly constant we could simply read it from the graph. However, the graph is bumpy due to the nature of the equipment and so we need to average accelerations at different times in some way.

Select the Acceleration vs. Time graph and then select Analyze-Statistics from the Data Menu. Use the mouse select the straight region.

Average acceleration (mean): _________m/s2

Finding Acceleration from Velocity-Time Graph

Since the average acceleration during a particular time period 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.

Select the data you need to calculate the approximate slope of your velocity graph. Use Analyze -Tangent to find the slope of velocity-time as nearly as you can for the straight portion.

Slope of velocity graph __________ m/s

This is the average acceleration.

Comparing Acceleration Found these two ways

Summarize the results from the two previous observations below.

(f) Is the acceleration positive or negative? Is this what you expected?

(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?

(h) How would the above graphs differ if the airtrack were raised higher?

Graphs Depicting Slowing Down

 

• This time, sketch motion graphs for an object slowing down towards the origin. To do this you will need to give the glider a push toward the origin. Make sure that you sketch only the motion when the glider was moving free of your hand.
• Change scales if it makes the graphs easier to observe

Graphs Depicting Slowing Down, Changing Direction and Speeding Up Again

Combine your observations or take more in order to be able to sketch velocity-time and acceleration-time graphs for a glider slowing down toward the origin, turning around and speeding up again.

For each of the three parts of the graphs mark the sign of velocity and acceleration

Save Data for a later Lab

Use the mouse to select the straight portions of the velocity graph (where the acceleration was constant); this should include the portions from right after you release it to just before it returned to the initial position. From the Edit menu choose copy data; this will copy the time, position, velocity, and acceleration data. Then minimize the window and open Excel. In the spreadsheet paste the data starting in row two. Put titles in row one then Save the file to your disk with an appropriate name.

Same Thing again But with Directions Reversed

Sketch position-time, velocity-time and acceleration-time graphs for an object speeding up toward the origin, turning around and slowing down away from the origin.

For each of the three parts of each graph mark the sign of velocity and acceleration.

To do this activity, you should set up the glider, air track, and motion detector as shown below.

Does Zero Velocity Mean Zero Acceleration?

Observe the points in the last two sets of graphs where the velocity is zero (ie where the glider turns around)

Is the acceleration zero at those points? Explain

(Hint: Remember that acceleration is the rate of change of velocity. When the glider is at rest at the end of the tilted air track, what will its velocity be in the next instant? Will it be positive or negative?)

Tossing a Ball

Suppose you throw a ball up into the air. It moves upward, 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?

Consider the ball toss carefully. Assume that downward is the positive direction. Indicate in the table that follows whether the velocity 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.