LABORATORY I: DESCRIPTION OF MOTION IN ONE …visual.physics.tamu.edu/LabManual/Lab_1.pdf · ... velocity, and acceleration) ... • Describe completely the motion of any object moving

Lab I - 1

LABORATORY I: DESCRIPTION OF MOTION IN ONE DIMENSION

In this laboratory you will study the motion of objects that can go in only one dimension; that is, along a straight line. You will be able to measure the position of these objects by capturing video images on a computer. Through these meas-urements and their analysis you can investigate the relationship of quantities that are useful to describe the motion of objects. Determining these kinematic quanti-ties (position, time, velocity, and acceleration) under different conditions allows you to improve your intuition about their quantitative relationships. In particu-lar, you should be able to determine which relationships depend on specific situa-tions and which apply to all situations.

There are many possibilities for the motion of an object. It might either moves at a constant speed, speed up, slow down, or have some combination of these mo-tions. In making measurements in the real world, you need to be able to quickly understand your data so that you can tell if your results make sense to you. If your results don't make sense, then either you have not set up the situation prop-erly to explore the physics you desire, you are making your measurements incor-rectly, or your ideas about the behavior of objects in the physical world are incor-rect. In any of the above cases, it is a waste of time to continue making measure-ments. You must stop, determine what is wrong and fix it. If it is your ideas that are wrong, this is the time to correct them by discussing the inconsistencies with your partners, rereading your text, or talking with your instructor. Remember, one of the reasons for doing physics in a laboratory setting is to help you confront and overcome your incorrect preconceptions about physics, measurements, calcu-lations, and technical communications

LAB I: INTRODUCTION

Lab I - 2

OBJECTIVES: After you successfully complete this laboratory, you should be able to:

• Describe completely the motion of any object moving in one dimension using the concepts of position, time, velocity, and acceleration.

• Distinguish between average quantities and instantaneous quantities when describ-ing the motion of an object.

• Express mathematically the relationships among position, time, velocity, average velocity, acceleration, and average acceleration for different situations.

• Graphically analyze the motion of an object.

• Begin using technical communication skills such as keeping a laboratory journal and writing a laboratory report.

PREPARATION: Read Resnick/Halliday/Krane, Chapter 2. Also read Appendix D, the instructions for doing video analysis. Before coming to the lab you should be able to:

• Define and recognize the differences among these concepts:

- Position, displacement, and distance.

- Instantaneous velocity and average velocity.

- Instantaneous acceleration and average acceleration.

• Find the slope and intercept of a straight-line graph. If you need help, see Appendix C.

• Determine the slope of a curve at any point on that curve. If you need help, see Appendix C.

• Determine the derivative of a quantity from the appropriate graph.

• Use the definitions of sin θ, cos θ, and tan θ with a right triangle.

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PROBLEM #1: CONSTANT VELOCITY MOTION

These laboratory instructions may be unlike any you have seen before. You will not find work-sheets or step-by-step instructions. Instead, each laboratory consists of a set of problems that you solve before coming to the laboratory by making an organized set of decisions (a prob-lem solving strategy) based on your initial knowledge. The instructions are designed to help you examine your thoughts about physics. These labs are your opportunity to compare your ideas about what "should" happen with what really happens. The labs will have little value in helping you learn physics unless you take time to predict what will happen before you do something. While in the laboratory, take your time and try to answer all the questions in this lab manual. In particular, the exploration questions are important to answer before you make measurements. Make sure you complete the laboratory problem, including all analysis and conclusions, before moving on to the next one.

Since this design may be new to you, this first problem contains both the instructions to explore constant velocity motion and an explanation of the various parts of the instructions. The explana-tion of the instructions is in this font and is preceded by the double, vertical lines seen to the left.

Why are we doing this lab problem? How is it related to the real world? In the lab instructions, the first paragraphs describe a possible situation that raises the problem you are about to solve. This emphasizes the application of physics in solving real-world problems.

To earn some extra money, you have taken a job as a camera operator for the TexasWorld Speedway car race. Since the race will be simulcast on the Internet, you will be using a digital video camera that stores the images directly on a com-puter. You notice that the image is distorted near the edges of the picture and wonder if this affects the measurement of a car’s speed from the video image. You decide to model the situation using a toy car, which moves at a constant ve-locity.

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Does the measured speed of a car moving with a constant velocity depend on the position of the car in a video picture?

The question, framed in a box and preceded by a question mark, defines the experimental prob-lem you are trying to solve. You should keep the question in mind as you work through the problem.


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EQUIPMENT

To make a prediction about what you expect to happen, you need to have a general understand-ing of the apparatus you will use before you begin. This section contains a brief description of the apparatus and the kind of measurements you can make to solve the laboratory problem. The details should become clear to you as you use the equipment.

For this problem, you will use a toy car, which moves with a constant velocity on an aluminum air track. You will also have a stopwatch, a meter stick, a video camera and a computer with a video analysis application written in LabVIEW (described in Appendix D) to help you analyze the motion.

PREDICTION

Everyone has his/her own "personal theories" about the way the world works. One purpose of this lab is to help you clarify your conceptions of the physical world by testing the predictions of your personal theory against what really happens. For this reason, you will always predict what will happen before collecting and analyzing the data. Your prediction should be completed and written in your lab journal before you come to lab. The “Method Questions” in the next section are designed to help you determine your prediction and should also be completed before you come to lab. This may seem a little backwards. Although the prediction question is given before the method questions, you should complete the method questions before making the prediction. The prediction question is given first so you know your goal.

Spend the first few minutes at the beginning of the lab session comparing your prediction with those of your partners. Discuss the reasons for any differences in opinion. It is not necessary that your predictions are correct, but it is necessary that you understand the basis of your predic-tion.

How would each of the graphs of position-versus-time, velocity-versus-time, and acceleration-versus-time show a distortion of the position measurement? Sketch these graphs to illustrate your answer. How would you determine the speed of the car from each of the graphs? Which method would be the most sensi-tive technique for determining any distortions? Appendix B might help you an-swer this question. Sometimes, as with this problem, your prediction is an "educated guess" based on your knowl-edge of the physical world. There is no way to calculate an exact answer to this problem. For other problems, you will be asked to use your knowledge of the concepts and principles of physics to calculate a mathematical relationship between quantities in the experimental problem.


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METHOD QUESTIONS

Method Questions are a series of questions intended to help you solve the experimental problem. They either help you make the prediction or help you plan how to analyze data. Method Ques-tions should be answered and written in your lab journal before you come to lab.

To determine if the measured speed is affected by distortion, you need to think about how to measure and represent the motion of an object. The following questions should help with the analysis of your data.

1. How would you expect an instantaneous velocity-versus-time graph to look for an object moving with a constant velocity? Make a rough sketch and explain your reasoning. Write down the equation that describes this graph. If this equation has any constant quantities in it, what are the units of those constant quanti-ties? What parts of the motion of the object does each of these constant quanti-ties represent? For a toy car, what do you estimate should be the magnitude of those quantities? How would a distortion affect this graph? How would it af-fect the equation that describes the graph? How will the uncertainty of your position measurements affect this graph? How might you tell the difference between uncertainty and distortion?

2. How would you expect a position-versus-time graph to look for an object moving with a constant velocity? Make a rough sketch and explain your reasoning. What is the relationship between this graph and the instantaneous velocity versus time graph? Write down the equation that describes this graph. If this equation has any constant quantities in it, what are the units of those constant quantities? What parts of the motion of the object does each of these constant quantities represent? For a toy car, what do you estimate should be the magni-tude of those quantities? How would a distortion affect this graph? How would it affect the equation that describes the graph? How will the uncer-tainty of your position measurements affect this graph? How might you tell the difference between uncertainty and distortion?

3. How would you expect an instantaneous acceleration-versus-time graph to look for an object moving with a constant velocity? Make a rough sketch and explain your reasoning. Write down the equation that describes this graph. If this equation has any constant quantities in it, what are the units of those constant quantities? What parts of the motion of the object does each of these constant quantities represent? For a toy car, what do you estimate should be the magni-tude of those quantities? How would a distortion affect this graph? How would it affect the equation that describes the graph? How will the uncer-tainty of your position measurements affect this graph? How might you tell the difference?


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EXPLORATION

This section is extremely important—many instructions will not make sense, or you may be lead astray, if you do not take the time to carefully explore your experimental plan.

In this section you practice with the apparatus before you make time-consuming measurements which may not be valid. This is where you carefully observe the behavior of your physical sys-tem, before you begin making measurements. You will also need to explore the range over which your apparatus is reliable. Remember to always treat the apparatus with care and re-spect. Your fellow students in the next lab section will need to use the equipment after you are finished with it. If you are unsure about how the apparatus works, ask your lab instructor.

Most apparatus has a range in which its operation is simple and straightforward. This is its range of reliability. Outside of that range, complicated corrections need to be applied. You can quickly determine the range of reliability by making qualitative observations at what you con-sider to be the extreme ranges of your measurements. Record your observations in your lab journal. If you observe that the apparatus does not function properly for the range of quantities you were considering measuring, you can modify your experimental plan before you have wasted time taking an invalid set of measurements.

The result of the exploration should be a plan for doing the measurements that you need. Re-cord your measurement plan in your journal.

Level the air track. Place the toy car on the air track. Launch the car with a gentle velocity. Determine if the car actually moves with a constant velocity. Use the meter stick and stopwatch to determine the speed of the car.

Turn on the video camera and look at the motion as seen by the camera on the computer screen. Go to Appendix F for instructions about using the video re-corder.

Do you need to focus the camera to get a clean image? How do the room lights affect the image? Which controls help sharpen the image? Record your camera adjustments in your lab journal.

The Zoom control on the camera enables you to change the angular width of the field of view. Examine the effect of zoom setting on the precision with which you can measure points on a meter stick.

The camera optics is optimized to produce a flat image transfer: the image of a 1 cm displacement occupies the same number of pixels independent of where in the overall image it is located (center or edge). The fractional distortion is the ra-tio of the difference between apparent image size in the center and the edge of the field of view, divided by the apparent image size. Measure the fractional distor-


Lab I - 7

tion of an image, for close-in zoom and for far-out zoom. Decide where you want to place the camera to minimize the distortion.

Practice taking a video of the car. Looking at the video frame by frame allows you to check whether the computer has missed any frames (the motion should be smooth). When you have the best movie possible, save it and open the video analysis application.

Make sure everyone in your group gets the chance to operate the camera and the com-puter.

MEASUREMENT

Now that you have predicted the result of your measurement and have explored how your appa-ratus behaves, you are ready to make careful measurements. To avoid wasting time and effort, make the minimal measurements necessary to convince yourself and others that you have solved the laboratory problem.

Measure the speed of the car using a stopwatch as it travels a known distance. How many measurements should you take to determine the car’s speed? (Too few measurements may not be convincing to others, too many and you may waste time and effort.) How much accuracy do you need from your meter stick and stopwatch to determine a speed to at least two significant figures? Make the number of measurements you need and record them in a neat and organized manner so that you can understand them a month from now if you must. Also make sure to record precisely how you make these measurements. Some future lab problems will require results from earlier ones.

Take a video of the motion of the car to determine its speed. Measure some object you can see in the video so that you can tell the analysis program the real size of the video images when it asks you to calibrate. The best object to measure is the car itself. When you digitize the video, why is it important to click on the same point on the car’s image? Estimate your accuracy in doing so. Be sure to take measurements of the motion of the car in the distorted regions (edges) of the video.

Make sure you set the scale for the axes of your graph so that you can see the data points as you take them. Use your measurements of total distance the car travels and total time to determine the maximum and minimum value for each axis before taking data.

Are any points missing from the position versus time graph? Missing points result from more data being transmitted from the camera than the computer can write to its memory. If too many points are missing, make sure that the size of


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your video frame is optimal (see Appendix D). It may also be that your back-ground is too busy. Try positioning your apparatus so that the background has fewer visual features.

Note: Be sure to record your measurements with the appropriate number of significant fig-ures (see Appendix A) and with your estimated uncertainty (see Appendix B). Otherwise, the data is nearly meaningless.

ANALYSIS

Data by itself is of very limited use. Most interesting quantities are those derived from the data, not direct measurements themselves. Your predictions may be qualitatively correct but quantita-tively very wrong. To see this you must process your data. Always complete your data processing (analysis) before you take your next set of data. If something is going wrong, you shouldn't waste time taking a lot of useless data. After analyzing the first data, you may need to modify your measurement plan and re-do the measurements. If you do, be sure to record the changes in your plan in your journal.

Calculate the average speed of the car from your stopwatch and meter stick measurements. Determine if the speed is constant within your measurement un-certainties. Can you determine the instantaneous speed of your car as a function of time?

Analyze your video to find the instantaneous speed of the car as a function of time. Determine if the speed is constant within your measurement uncertainties. See Appendix D for instructions on how to do video analysis.

CONCLUSIONS

After you have analyzed your data, you are ready to answer the experimental problem. State your result in the most general terms supported by your analysis. This should all be recorded in your journal in one place before moving on to the next problem assigned by your lab instructor. Make sure you compare your result to your prediction.

Compare the car’s speed measured with video analysis to the measurement using a stopwatch. How do they compare? Did your measurements and graphs agree with your answers to the Method Questions? If not, why? What are the limitations on the accuracy of your measurements and analysis?

Do measurements near the edges of the video give the same speed as that as found in the center of the image within the uncertainties of your measurement? What will you do for future measurements?

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PROBLEM #2: MOTION DOWN AN INCLINE

You have a summer job working with a team investigating accidents for the state safety board. To decide on the cause of one accident, your team needs to de-termine the acceleration of a car rolling down a hill without any brakes. Everyone agrees that the car’s velocity increases as it rolls down the hill. Your team’s su-pervisor believes that the car's acceleration also increases as it rolls down the hill. Do you agree? To resolve the issue, you decide to measure the acceleration of a cart moving down an inclined track in the laboratory.

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How does an object accelerate as it moves down a ramp?

EQUIPMENT

For this problem you will have a stopwatch, a meter stick, an end stop, a wood block, a video camera and a computer with a video analysis application written in LabVIEW . You will also have a cart to roll down the inclined track.

PREDICTION

Make a rough sketch of how your supervisor expects the acceleration-versus-time graph to look for a cart released from rest near the top of an inclined track. If you have a different idea, make a rough sketch of how you expect the acceleration-versus-time graph to look.

Do you think the cart's acceleration changes as it moves down the track? If so, how does the acceleration change (increase or decrease)? Does the acceleration change uniformly, is it greater near the top of the track or near the bottom of the track? Or do you think the acceleration is constant (does not change) as the cart moves down the track? Make your best guess and explain your reasoning.


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METHOD QUESTIONS

The following questions should help with your prediction and the analysis of your data.

1. How would you expect an instantaneous acceleration-versus-time graph to look for a cart moving with a constant acceleration? With an increasing acceleration? With a decreasing acceleration? Make a rough sketch of the graph for each and explain your reasoning. To make the comparison easier, it is useful to draw these graphs next to each other. Write down the equation that best represents each of these graphs. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph?

2. Write down the relationship between the acceleration and the velocity of the cart. Use that relationship to construct an instantaneous velocity versus time graph just below each of your acceleration versus time graphs from question 1. Use the same scale for your time axis. Write down the equation that best repre-sents each of these graphs. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation representing the acceleration versus time graphs? Which graph do you think best represents how velocity of the cart changes with time? Change your prediction if necessary.

3. Write down the relationship between the velocity and the position of the cart. Use that relationship to construct a position versus time graph just below each of your velocity versus time graphs from question 2. Use the same scale for your time axis. Write down the equation that best represents each of these graphs. If there are constants in your equation, what kinematic quantities do they repre-sent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation representing the velocity versus time graphs? Which graph do you think best represents how posi-tion of the cart changes with time? Change your prediction if necessary.


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EXPLORATION

What is the best way to change the angle of the incline in a reproducible way? How are you going to measure this angle with respect to the table? Hint: Think trigonometry!

Start with a small angle and with the cart at rest near the top of the track. Ob-serve the cart as it moves down the inclined track. Try a range of angles. BE SURE TO CATCH THE CART BEFORE IT HITS THE END STOP! If the angle is too large, you may not get enough video frames, and thus enough position and time measurements, to measure the acceleration accurately. If the angle is too small the acceleration may be too small to measure accurately with the precision of your measuring instruments. Select the best angle for this measurement.

Where is the best place to put the camera? Is it important to have most of the motion in the center of the picture? Which part of the motion do you wish to cap-ture? Try some different camera positions.

Where is the best place to release the cart so it does not damage the equip-ment but has enough of its motion captured on video? Be sure to catch the cart before it collides with the end stop. Take a few practice videos and play them back to make sure you have captured the motion you want.

What is the total distance through which the cart rolls? How much time does it take? These measurements will help you set up the graphs for your computer data taking.

Write down your measurement plan.

MEASUREMENT

Choose one angle, which will give you the clearest measurement of the differ-ence between this motion and that for constant velocity motion. Make a video of the cart moving down the track at that angle. Don't forget to measure and record the angle (with estimated uncertainty). You may use it for later labs.

Choose an object in your picture for calibration.

Why is it important to click on the same point on the car’s image to record its position? Estimate your accuracy in doing so.

Are any points missing from the position versus time graph? Missing points result from more data being transmitted from the camera than the computer can


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write to its memory. If too many points are missing, make sure that the size of your video frame is optimal (see Appendix F). It may also be that your back-ground is too busy. Try positioning your apparatus so that the background has fewer visual features.

Use a meter stick and a stopwatch to determine the average acceleration of the cart. Under what condition will this average acceleration be equal to the instanta-neous acceleration of the cart? From your video data can you make that assump-tion? As a check, comparing the acceleration measured with a meter stick and stopwatch with that measured by video analysis. How much accuracy from the meter stick and stopwatch is necessary to determine the acceleration with at least two significant figures?

Note: Be sure to record your measurements with the appropriate number of significant figures (see Appendix A) and with your estimated uncertainty (see Appendix B). Other-wise, the data is nearly meaningless.

ANALYSIS

Choose a function to represent the position-versus-time graph. How can you estimate the values of the constants of the function from the graph? You can waste a lot of time if you just try to guess the constants. What kinematic quantities do these constants represent?

Choose a function to represent the velocity-versus-time graph. How can you calculate the values of the constants of this function from the function representing the position-versus-time graph? Check how well this works. You can also esti-mate the values of the constants from the graph. Just trying to guess the constants can waste a lot of your time. What kinematic quantities do these constants repre-sent?

From the velocity versus time graph, determine if the acceleration is constant, increasing, or decreasing as the cart goes down the ramp. Use the function repre-senting the velocity versus time graph to calculate the acceleration of the cart as a function of time. Make a graph of that function. Is the average acceleration of the cart equal to its instantaneous acceleration in this case?

Calculate the average acceleration of the cart from your stopwatch and meter stick measurements.

Compare the accelerations for the cart you found with your video analysis to your acceleration measurement using a stopwatch. How do they compare?


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As you analyze your video, make sure everyone in your group gets the chance to operate the computer.


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CONCLUSION

How do the graphs of your measurements compare to your predictions?

Was your friend right about how a cart accelerates down a hill? If yes, state your result in the most general terms supported by your analysis. If no, describe how you would convince your friend. What are the limitations on the accuracy of your measurements and analysis?

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PROBLEM #3: MOTION UP AND DOWN AN INCLINE

A proposed ride at the Valley Fair amusement park launches a roller coaster car up an inclined track. Near the top of the track, the car reverses direction and rolls backwards into the station. As a member of the safety committee, you have been asked to compute the acceleration of the car throughout the ride. To check your results, you decide to build a laboratory model of the ride.

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What is the acceleration of an object moving up and down a ramp at all times during its motion?

EQUIPMENT

For this problem you will have a stopwatch, a meter stick, an end stop, a wood block, a video camera and a computer with a video analysis application written in LabVIEW . You will also have a cart to roll up the inclined track.

PREDICTION

Make a rough sketch of how you expect the acceleration-versus-time graph to look for a cart given an initial velocity up along an inclined track. The graph should be for the entire motion of going up the track, reaching the highest point, and then coming down the track. Do you think the acceleration of the cart moving up an inclined track will be greater than, less than, or the same as the acceleration of the cart moving down the track? What is the acceleration of the cart at its highest point? Explain your reasoning.


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METHOD QUESTIONS


1. Sketch a graph of how you expect an instantaneous acceleration-versus-time graph to look if the cart moved down the track with the direction of a constant ac-celeration always down along the track. Sketch a graph of how you expect an in-stantaneous acceleration-versus-time graph to look if the cart moved up the track with the direction of a constant acceleration always down along the track after an initial push. When the cart moves up the track and then down the track, graph how would you expect an instantaneous acceleration-versus-time graph to look for the en-tire motion after an initial push? Explain your reasoning for each graph. To make the comparison easier, it is useful to draw these graphs next to each other. Write down the equation(s) that best represents each of these graphs. If there are con-stants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph?

2. Write down the relationship between the acceleration and the velocity of the cart. Use that relationship to construct an instantaneous velocity versus time graph just below each of your acceleration versus time graphs from question 1. The connection between the derivative of a function and the slope of its graph will be useful. Use the same scale for your time axes. Write down the equation that best represents each of these graphs. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these con-stants from your graph? Can any of these constants be determined from the con-stants in the equation representing the acceleration versus time graphs? Which graph do you think best represents how velocity of the cart changes with time? Change your prediction if necessary.

3. Write down the relationship between the velocity and the position of the cart. Use that relationship to construct a position versus time graph just below each of your velocity versus time graphs from question 2. The connection be-tween the derivative of a function and the slope of its graph will be useful. Use the same scale for your time axes. Write down the equation that best represents each of these graphs. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation representing the velocity versus time graphs? Which graph do you think best represents how position of the cart changes with time? Change your prediction if necessary.


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EXPLORATION

What is the best way to change the angle of the inclined track in a reproduci-ble way? How are you going to measure this angle with respect to the table? Think about trigonometry. How steep of an incline do you want to use?

Start the cart up the track with a gentle push. BE SURE TO CATCH THE CART BEFORE IT HITS THE END STOP ON ITS WAY DOWN! Observe the cart as it moves up the inclined track. At the instant the cart reverses direction, what is its velocity? Its acceleration? Observe the cart as it moves down the in-clined track. Do your observations agree with your prediction? If not, this is a good time to change your prediction.

Where is the best place to put the camera? Is it important to have most of the motion in the center of the picture? Which part of the motion do you wish to cap-ture?

Try several different angles. Be sure to catch the cart before it collides with the end stop at the bottom of the track. If the angle is too large, the cart may not go up very far and give you too few video frames for the measurement. If the an-gle is too small it will be difficult to measure the acceleration. Determine the use-ful range of angles for your track. Take a few practice videos and play them back to make sure you have captured the motion you want.

Choose the angle that gives you the best video record.



Measure the thickness of the shim that you use to create the angle, and the distance between the support point at the shim and the support point at the far end of the track. From this you should be able to calculate the angle of the track with respect to horizontal. Use this to extract an estimate of the acceleration of gravity from your measurement of the acceleration of the cart.


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MEASUREMENT

Using the plan you devised in the exploration section, make a video of the cart moving up and then down the track at your chosen angle. Make sure you get enough points for each part of the motion to determine the behavior of the accel-eration. Don't forget to measure and record the angle (with estimated uncertainty).



Are any points missing from the position versus time graph? Missing points result from more data being transmitted from the camera than the computer can write to its memory. If too many points are missing, make sure that the size of your video frame is optimal (see Appendix D). It may also be that your back-ground is too busy. Try positioning your apparatus so that the background has fewer visual features.

ANALYSIS

Choose a function to represent the position versus time graph. How can you estimate the values of the constants of the function from the graph? You can waste a lot of time if you just try to guess the constants. What kinematic quantities do these constants represent? Can you tell from your graph where the cart reaches its highest point?

Choose a function to represent the velocity versus time graph. How can you calculate the values of the constants of this function from the function representing the position versus time graph? Check how well this works. You can also esti-mate the values of the constants from the graph. Just trying to guess the constants can waste a lot of your time. What kinematic quantities do these constants repre-sent? Can you tell from your graph where the cart reaches its highest point?

From the velocity versus time graph determine if the acceleration changes as the cart goes up and then down the ramp. Use the function representing the ve-locity versus time graph to calculate the acceleration of the cart as a function of time. Make a graph of that function. Can you tell from your graph where the cart reaches its highest point? Is the average acceleration of the cart equal to its instan-taneous acceleration in this case?


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CONCLUSION

How do your position-versus-time and velocity-versus-time graphs compare with your answers to the method questions and the prediction? What are the limi-tations on the accuracy of your measurements and analysis?

Did the cart have the same acceleration throughout its motion? Did the accel-eration change direction? Was the acceleration zero at the top of its motion? Did the acceleration change direction? Describe the acceleration of the cart through its entire motion after the initial push. Justify your answer. What are the limitations on the accuracy of your measurements and analysis?

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PROBLEM #4: MOTION DOWN AN INCLINE WITH AN INITIAL VELOCITY

Because of your physics background, you have a summer job with a company that is designing a new bobsled for the U.S. team to use in the next Winter Olym-pics. You know that the success of the team depends crucially on the initial push of the team members – how fast they can push the bobsled before they jump into the sled. You need to know in more detail how that initial velocity affects the mo-tion of the bobsled. In particular, your boss wants you to determine if the initial velocity of the sled affects its acceleration down the ramp. To solve this problem, you decide to model the situation using a cart down along an inclined track.

?

Does the acceleration of an object down a ramp depend on its initial velocity?

EQUIPMENT

For this problem you will have a stopwatch, a meter stick, an end stop, a wood block, a video camera and a computer with a video analysis application written in LabVIEW . You will also have a cart to roll down an inclined track.

For this problem you will slant the aluminum track at the same angle you used in Problem #2 (Motion Down an Incline) and give the cart a gentle push down the inclined track instead of releasing it from rest.

PREDICTION

Based on your results for Problem #2, make a sketch of how you think the ac-celeration-versus-time graph for a cart released from rest on an inclined track will look. On the same graph sketch how you think the acceleration-versus-time graph will look when the cart is given an initial velocity down the track.

Do you think the cart launched down the inclined track will have a larger accelera-tion, smaller acceleration, or the same acceleration as the cart released from rest? Ex-plain your reasoning.


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METHOD QUESTIONS


1. Sketch a graph of how would you expect an instantaneous acceleration-versus-time graph to look when the cart moves down the track after an initial push. How does this graph compare to an instantaneous acceleration-versus-time graph for a cart released from rest? Explain your reasoning for each graph. To make the compari-son easier, it is useful to draw these graphs next to each other. Write down the equation(s) that best represents each of these graphs. If there are constants in your equation, what kinematic quantities do they represent? How would you deter-mine these constants from your graph?

2. Write down the relationship between the acceleration and the velocity of the cart. Use that relationship to construct an instantaneous velocity versus time graph, after an initial push, just below each of your acceleration versus time graphs from question 1. The connection between the derivative of a function and the slope of its graph will be useful. Use the same scale for your time axes. Write down the equation that best represents each of these graphs. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be deter-mined from the constants in the equation representing the acceleration versus time graphs? Which graph do you think best represents the velocity of the cart? Change your prediction if necessary.

3. Write down the relationship between the velocity and the position of the cart. Use that relationship to construct a position versus time graph, after an ini-tial push, just below each of your velocity versus time graphs from question 2. The connection between the derivative of a function and the slope of its graph will be useful. Use the same scale for your time axes. Write down the equation that best represents each of these graphs. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these con-stants from your graph? Can any of these constants be determined from the con-stants in the equation representing the velocity versus time graphs? Which graph do you think best represents the position of the cart? Change your prediction if necessary.


Lab I - 23

EXPLORATION

Slant the track at the same angle you used in Problem #2: Motion Down an In-cline.

Determine the best way to gently launch the cart down the track in a consis-tent way without breaking the equipment. BE SURE TO CATCH THE CART BEFORE IT HITS THE END STOP!

Where is the best place to put the camera? Is it important to have most of the motion in the center of the picture? Which part of the motion do you wish to cap-ture? Try taking some videos before making any measurements. Be sure to catch the cart before it collides with the end stop at the bottom of the track.



Make sure everyone in your group gets the chance to operate the camera and the computer.

MEASUREMENT

Using the plan you devised in the exploration section, make a video of the cart moving down the track at your chosen angle. Make sure you get enough points for each part of the motion to determine the behavior of the acceleration. Don't forget to measure and record the angle (with estimated uncertainty).

Choose an object in your picture for calibration. Why is it important to click on the same point on the car’s image to record its position? Estimate your accu-racy in doing so.



Lab I - 24

ANALYSIS

Choose a function to represent the position versus time graph. How can you estimate the values of the constants of the function from the graph? You can waste a lot of time if you just try to guess the constants. What kinematic quantities do these constants represent?

Choose a function to represent the velocity versus time graph. How can you calculate the values of the constants of this function from the function representing the position versus time graph? Check how well this works. You can also esti-mate the values of the constants from the graph. Just trying to guess the constants can waste a lot of your time. What kinematic quantities do these constants repre-sent?

From the velocity versus time graph, determine the acceleration as the cart goes down the ramp after the initial push. Use the function representing the ve-locity versus time graph to calculate the acceleration of the cart as a function of time. Make a graph of that function.


CONCLUSIONS

Look at the graphs you produced with your video analysis. How do they compare to your answers to the method questions and your predictions? Explain any differences. What are the limitations on the accuracy of your measurements and analysis?

What will you tell your boss? Does the acceleration of the bobsled down the track depend on the initial velocity the team can give it? Does the velocity of the bobsled down the track depend on the initial velocity the team can give it? State your result in the most general terms supported by your analysis.

Lab I - 25

PROBLEM #5: MASS AND MOTION DOWN AN INCLINE

Your neighbors' child has asked for your help in constructing a soapbox derby car. In the soapbox derby, two cars are released from rest at the top of a ramp. The one that reaches the bottom first wins. The child wants to make the car as heavy as possible to give it the largest acceleration. Is this plan reasonable?

?

How does the acceleration of an object down a ramp depend on its mass?

EQUIPMENT

For this problem you will have a stopwatch, a meter stick, an end stop, a wood block, a video camera and a computer with a video analysis application written in LabVIEW . You will also have a cart to roll down an inclined track and additional cart masses to add to the cart.

For this problem you will slant the aluminum track at the same angle you used in Problem #2 (Motion Down an Incline) and release the cart from rest.

PREDICTION

Make a sketch of how you think the acceleration-versus-mass graph will look for carts with different mass released from rest from the top of an inclined track.

Do you think the acceleration of the cart increases, decreases, or stays the same as the mass of the cart increases? Make your best guess and explain your reasoning.


Lab I - 26

METHOD QUESTIONS


1. Based on your results for Problem #2, make a sketch of the acceleration-versus-time graph for a cart released from rest on an inclined track. On the same graph sketch how you think the acceleration-versus-time will look for a cart with a much larger mass. Explain your reasoning? Write down the equation that best represents each of these accelerations. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph?

2. Write down the relationship between the acceleration and the velocity of the cart. Use that relationship to construct an instantaneous velocity versus time graph for each case. The connection between the derivative of a function and the slope of its graph will be useful. Write down the equation that best repre-sents each of these velocities. If there are constants in your equation, what ki-nematic quantities do they represent? How would you determine these con-stants from your graph? Can any of these constants be determined from the constants in the equation representing the acceleration? Which do you think best represents the velocity of the cart? Change your prediction if necessary.

3. Write down the relationship between the velocity and the position of the cart. Use that relationship to construct a position versus time graph for each case. The connection between the derivative of a function and the slope of its graph will be useful. Write down the equation that best represents each of these po-sitions. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation representing the velocity? Which do you think best represents the position of the cart? Change your prediction if necessary.


Lab I - 27

EXPLORATION

Slant the track at the same angle you used in Problem #2: Motion Down an In-cline.

Observe the motion of several carts of different mass when released from rest at the top of the track. BE SURE TO CATCH THE CART BEFORE IT HITS THE END STOP! From your estimate of the size of the effect, determine the range of mass that will give the best results in this problem. Determine the first two masses you should use for the measurement.

How do you determine how many different masses do you need to use to get a conclusive answer? How will you determine the uncertainty in your measure-ments? How many times should you repeat these measurements? Explain.



Make sure everyone in your group gets the chance to operate the camera and the computer.

MEASUREMENT

Using the plan you devised in the exploration section, make a video of the cart moving down the track at your chosen angle. Make sure you get enough points for each part of the motion to determine the behavior of the acceleration. Don't forget to measure and record the angle (with estimated uncertainty).



Make sure you set the scale for the axes of your graph so that you can see the data points as you take them. Use your measurements of total distance the cart travels and the total time to determine the maximum and minimum value for each axis before taking data.

Are any points missing from the position versus time graph? Missing points result from more data being transmitted from the camera than the computer can write to its memory. If too many points are missing, make sure that the size of your video frame is optimal (see Appendix D). It may also be that your back-


Lab I - 28

ground is too busy. Try positioning your apparatus so that the background has fewer visual features.

Make several videos with carts of different mass to check your qualitative prediction. If you analyze your data from the first two masses you use before you make the next video, you can determine which mass to use next. As usual you should minimize the number of measurements you need.

ANALYSIS


Choose a function to represent the velocity versus-time-graph. How can you calculate the values of the constants of this function from the function representing the position-versus-time graph? Check how well this works. You can also esti-mate the values of the constants from the graph. Just trying to guess the constants can waste a lot of your time. What kinematic quantities do these constants repre-sent?

From the velocity-versus-time graph determine the acceleration as the cart goes down the ramp. Use the function representing the velocity-versus-time graph to calculate the acceleration of the cart as a function of time.

Make a graph of the cart’s acceleration down the ramp as a function of the cart’s mass. Do you have enough data to convince others of your conclusion about how the acceleration of the cart depends on its mass?


CONCLUSION

Did your measurements of the cart's motion agree with your initial predic-tions? Why or why not? What are the limitations on the accuracy of your meas-urements and analysis?

What will you tell the neighbors' child? Does the acceleration of the car down its track depend on its total mass? State your result in the most general terms supported by your analysis.

Lab I - 29

PROBLEM #6: MOTION ON A LEVEL SURFACE WITH AN ELASTIC CORD

You are helping a friend design a new ride for the State Fair. In this ride, a cart is pulled along a long straight track by a stretched elastic cord like a bungee cord. Before spending money to build it, your friend wants to you to determine if this ride will be safe. Since sudden changes in velocity can lead to whiplash, you decide to find out how the acceleration of the cart changes with time. In particu-lar, you want to know if the greatest acceleration occurs when the sled is moving the fastest or at some other time. To solve this problem, you decide to model the situation in the laboratory with a cart pulled by an elastic cord along a level sur-face.

?

How does an object pulled by an elastic cord along a level surface accelerate? Is the ac-celeration greatest when the velocity is great-est?

EQUIPMENT

For this problem you will have a stopwatch, a meter stick, a scissor, a video camera and a computer with a video analysis application written in LabVIEW . You will also have a cart to roll on a level track. You can attach one end of an elastic cord to the cart and the other end of the elastic cord to an end stop on the track.

PREDICTION

Make a sketch of how you expect the acceleration-versus-time graph to look for a cart pulled by an elastic cord. Just below that graph make a graph of the ve-locity versus time on the same time scale. Identify where on your graphs the ve-locity is largest and the acceleration is largest.


Lab I - 30

METHOD QUESTIONS

The following questions should help with your prediction and the analysis of your data:

1. Make a sketch of how you expect an acceleration-versus-time graph to look for a cart pulled by an elastic cord. Explain your reasoning. For a comparison, make acceleration-versus-time graphs for a cart moving at constant velocity and a con-stant acceleration. Write down the equation that best represents each of these ac-celerations. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graphs?

2. Write down the relationship between the acceleration and the velocity of the cart. Use that relationship to construct a velocity-versus-time graph for each case. The connection between the derivative of a function and the slope of its graph will be useful. Write down the equation that best represent each of these velocities. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation rep-resenting the acceleration? Which do you think best represents the velocity of the cart? Change your prediction if necessary.

3. Write down the relationship between the velocity and the position of the cart. Use that relationship to construct a position-versus-time graph for each case. The connection between the derivative of a function and the slope of its graph will be useful. Write down the equation that best represents each of these positions. If there are constants in your equation, what kinematic quantities do they represent? How would you determine these constants from your graph? Can any of these constants be determined from the constants in the equation representing the veloc-ity? Which do you think best represents the position of the cart? Change your prediction if necessary.

EXPLORATION

Test that the track is level by observing the motion of the cart. Attach an elastic cord to the cart and track. Gently move the cart along the track to stretch out the elastic. Be careful not to stretch the elastic too tightly. Start with a small stretch and release the cart. BE SURE TO CATCH THE CART BEFORE IT HITS THE END STOP! Slowly increase the starting stretch until the cart's motion is long enough to get enough data points on the video, but does not cause the cart to


Lab I - 31

come off the track or snap the elastic. Be sure to catch the cart before it collides with the end stop.

Practice releasing the cart smoothly and capturing videos.


Make sure everyone in your group gets the chance to operate the camera and the com-puter.

MEASUREMENT

Using the plan you devised in the exploration section, make a video of the cart’s motion. Make sure you get enough points to determine the behavior of the acceleration.




ANALYSIS


Choose a function to represent the velocity-versus-time graph. How can you calculate the values of the constants of this function from the function representing the position-versus-time graph? Check how well this works. You can also esti-mate the values of the constants from the graph. Just trying to guess the constants can waste a lot of your time. What kinematic quantities do these constants repre-sent?


Lab I - 32

From the velocity-versus-time graph determine the acceleration of the cart. Use the function representing the velocity-versus-time graph to calculate the ac-celeration of the cart as a function of time.

Make a graph of the cart’s acceleration as a function of the time. Do you have enough data to convince others of your conclusion?


CONCLUSION

How does your acceleration-versus-time graph compare with your predicted graph? Are the position-versus-time and the velocity-versus-time graphs consis-tent with this behavior of acceleration? What is the difference between the motion of the cart in this problem and its motion along an inclined track? What are the similarities? What are the limitations on the accuracy of your measurements and analysis?

What will you tell your friend? Is the acceleration of the cart greatest when the velocity is the greatest? How will a cart pulled by an elastic cord accelerate along a level surface? State your result in the most general terms supported by your analysis.

How would the acceleration-versus-time graph look if you attached another piece of elastic to the other end of the cart and to the opposite end stop of the track? How about the velocity-versus-time and position-versus-time graphs? If you have time, try it.

Lab I - 33

þ CHECK YOUR UNDERSTANDING

1. Suppose you are looking down from a helicopter at three cars traveling in the same direction along the freeway. The positions of the three cars every 2 seconds are represented by dots on the diagram below. The positive direction is to the right.

Car A

Car B

Car C

1t 2t 3t 4t 5t 6t

1t

1t

2t

2t

3t

3t

4t

4t

5t

5t 6t

a. At what clock reading (or time interval) do Car A and Car B have very nearly the same speed? Explain your reasoning.

b. At approximately what clock reading (or readings) does one car pass another car? In each instance you cite, indicate which car, A, B or C, is doing the overtaking. Explain your reasoning.

c. Suppose you calculated the average velocity for Car B between t1 and t5. Where was the car when its instantaneous velocity was equal to its average velocity? Explain your reasoning.

d. Which graph below best represents the position-versus-time graph of Car A? Of Car B? Of car C? Explain your reasoning.

e. Which graph below best represents the instantaneous velocity-versus-time graph of Car A? Of Car B? Of car C? Explain your reasoning. (HINT: Examine the distances traveled in successive time intervals.)

f. Which graph below best represents the instantaneous acceleration-versus-time graph of Car A? Of Car B? Of car C? Explain your reasoning.

t

+

(g)-

t

+

(d)

t

+

(b)

+

t

(h)-

t

+

(e)

t

+

(i)-

t

+

(c)

t

+

(f)

t

+

(a)

Lab I - 34

2. A cart starts from rest at the top of a hill, rolls down the hill, over a short flat section, then back up another hill, as shown in the diagram above. Assume that the friction between the wheels and the rails is negligible. The positive direction is to the right.

a. Which graph below best represents the position-versus-time graph for the motion along the track? Explain your reasoning. (Hint: Think of motion as one-dimensional.)

b. Which graph below best represents the instantaneous velocity-versus-time graph? Explain your reasoning. c. Which graph below best represents the instantaneous acceleration-versus-time graph? Explain your

reasoning.

t

+

t

+

t

+

t

+

(b) (c)

(d) (f)

t

+

t

+

t

+

(g) (h) (i)- --

t

+

(e)

t

+

(a)

Lab I - 35

PHYSICS 218H LABORATORY REPORT

Laboratory I

Problem # & Ti-tle:_________________________________________

Name and ID#: __________________________Student ID#__________________

Dates performed: ________________ Section #: __________________

Lab Partners' Names: ________________________________________________

__________________________________________________________________

Lab Instructor's Initials: ____________

Grading Checklist Points

LABORATORY JOURNAL:

PREDICTIONS (individual predictions completed in journal before each lab ses-sion)

LAB PROCEDURE (measurement plan recorded in journal, tables and graphs made in journal as data is collected, observations written in journal)

PROBLEM REPORT:*

ORGANIZATION (clear and readable; correct grammar and spelling; section head-ings provided; physics stated correctly)

DATA AND DATA TABLES (clear and readable; units and assigned uncertainties clearly stated)

RESULTS (results clearly indicated; correct, logical, and well-organized calcula-tions with uncertainties indicated; scales, labels and uncertainties on graphs; physics stated correctly)

CONCLUSIONS (comparison to prediction & theory discussed with physics stated correctly ; possible sources of uncertainties identified; attention called to experimental problems)

TOTAL

(incorrect or missing statement of physics will result in a maximum of 60% of the total points achieved; incorrect grammar or spelling will result in a maximum of 70% of the total points achieved)

Lab I - 36

BONUS POINTS FOR TEAMWORK (as specified by course policy)

* An "R" in the points column means to rewrite that section only and return it to your lab instructor within two days of the return of the report to you.

Documents

LABORATORY I: DESCRIPTION OF MOTION IN ONE …visual.physics.tamu.edu/LabManual/Lab_1.pdf · ... velocity, and acceleration) ... • Describe completely the motion of any object moving