PHYS-101 LAB-02 One and Two Dimensional Motion 1. Objectives The objectives of this experiment are: to measure the acceleration due to gravity using one-dimensional motion, i.e. the motion of an object dropped from rest in free fall. to determine the independence on motion along two perpendicular axes in projectile motion. 2. Theory A. ) One-Dimensional Motion ( Free Fall ) Prior to the advent of high speed data collection techniques such as ultrasonic transducers and high speed digital cameras, a scientist would have to be clever to be able to measure the acceleration due to gravity. We take it for granted that the acceleration due to gravity of an object in free fall is approximately a y = g 9.8 m/ s 2 and is constant near the surface of the earth, but it is not as easy to measure as you would think. Take a baseball and drop it from shoulder height. Try to accurately measure the time it takes to reach the ground. It is pretty hard objective to accomplish. If only we could slow down the motion. Galileo was very interested in the motion of free falling objects. He thought of an ingenious method for slowing down the the motion of a body under the influence of gravity. He used an incline plane. Figure 2.1 The Inclined Plane Here the sum of the forces in the y direction ( taken perpendicular to the plane ) is zero, so the acceleration in that direction is also zero. The sum of the force in the x direction ( taken parallel to the plane and down the plane ) is equal to the mass times the acceleration in the x direction. N m a x = F x i=0 i m a x =mg sin a x =g sin 1
Galileo could slow down the motion enough to get accurate measurement of the time. Releasing the object from rest and measuring the distance traveled Galileo could find the acceleration, and use then value to compute the acceleration due to gravity. 1 x= x o v x t a x t 2 2 a x =g sin 1 2 x= x o v x t a x t a 2 g= x sin 2 x a x= 2 t Now using high speed digital cameras we can view the motion without the use of an inclined plane. You will drop a ball from about 1.5 2.0 meters and record the motion using the digital camera. Software will be used to calculate the y-position of the falling object as the frames are stepped through. You will use an object that has negligible, but not zero, drag forces and will not be effected by air resistance. If the acceleration due to gravity is indeed a constant, plotting y-position versus time should yield a parabola. Fitting a second order polynomial will provide you with the value of the acceleration due to gravity. You can also find the approximate y-component of velocity of the object. If the acceleration due to gravity is indeed a constant, plotting y-component of velocity versus time should yield a straight line. If you fit a straight line to this data, the slope will yield the acceleration due to gravity. o o Y Position versus Time Y Comp of Velocity versus Time 12 10 6 v(m/s) y(m) 8 4 2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0-2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-4 -6-8 -10-12 -14-16 t(s) t(s) Figure 2.2 Position Versus Time Figure 2.3 Position Versus Time v y =v o g t 1 2 y= y o v y t g t 2 B. ) Two-Dimensional Motion ( Projectile Motion ) If the same object is launched at an angle above the horizontal, the resulting two-dimensional motion can be resolved into two independent motions. You will take a movie of a projectile being launched at an angle above the horizontal. You will look at the motion along the x axis and along the y axis. You will complete the same analysis as you did for free fall except know you will consider the y direction and the x direction. You will use this analysis to verify the projectile motion equations presented in lecture. y 2 a y = g = 9.8 m/ s v y =v o g t 1 2 y= y o v o t g t 2 y y 2 a x =0 v x =v o =constant x= x o v o t x x v o =v o cos v o =v o sin x y
C. ) A word about Numerical Differentiation In this experiment you will estimate the instantaneous velocity by using numerical differentiation. In essence, you will take the average velocity between two points using as small a time step as possible. v= dx x x = lim as t gets as small as possible. dt t 0 t t Numerical differentiation is skill. If the velocity is constant, getting the numerical derivation is relatively easy and the time step can be rather large. If the object is accelerating, care must be taken. Consider the plot of position versus time for an object in Figure 2.4. Figure 2.4 Position versus Time for an Object Experiencing Acceleration The velocity at Point i should be the slope of the line tangent to point i. In order to compute the slope, you will need two points. Your first thought might be to use points i and i+1, but looking at Figure 2.5 shows that this might not be a good idea. You would fine the same problem using Point i and i+1, as shown in Figure 2.6. A better choice might be i-1 and i+1, see Figure 2.7. 3
Figure 2.5 Position versus Time Figure 2.6 Position versus Time Figure 2.6 Position versus Time It looks like the best method maybe the one in Figure 2.7. x i 1 x i 1 v i t i 1 t i 1 If time permits, take a look at all three methods. Your spreadsheet does all the work! 4
3. Procedure In this experiment you will take video of an object in free fall and an object in projectile motion. You will use these videos to analyze the motion of the object in free fall and the object exhibiting projectile Figure 3.1 The VideoPoint Icons motion. This will involve two pieces of software. As the name suggests, VideoPoint Capture is the software for controlling the video camera and capturing the video. VideoPoint Physics Fundamentals is the software that will be used to analyze the data. The analysis that the software provides is a list of x and y coordinates as a function of time. This position versus time data will be exported to a spreadsheet, where the velocity and acceleration as a function of time will be computed. Capturing the Video Clip: 1. Open the VideoPoint Capture Software by clicking on the icon on the desktop. The icons are shown in Figure 3.1. The opening screen should resemble Figure 3.2. Figure 3.2 VideoPoint Capture Opening Screen 5
2. Next, turn on the camera. Use the menu on the camera to select a shutter speed of 1/250 seconds. 3. Click on the Capture Button. This should bring up a screen which resembles Figure 3.3. You should the video input from the camera. If you do not see any video input but instead see a black screen, click on the settings button. A pop-up window will appear ( Figure 3.4 ) with a drop down menu will appear which lists the cameras attached to the computer along with other camera settings. Choose the proper camera and press the OK button. ( If you still do not see any video, make sure the camera is on and press record, then press stop, and then press the back arrow. ) Figure 3.3 VideoPoint Capture Normal Operation Figure 3.4 Selecting the camera and settings. 4. Data can now be collected by capturing a video. One person should hold ( or mount to a wall ) a meter stick. Recording the data is started by hitting the Record button. When recording, the Record button is grayed out. A ball is then dropped from a height of about two meters. Hit the Stop button after the ball hits the ground. The film may now be edited with the VideoPoint Capture Software. 6
Figure 3.5 VideoPoint Capture collecting data. Notice that the Record button is grayed out. 5. Once the Stop button is pressed, editing may begin. Save the video clip as a.mov movie file using the File drop down menu at the top of the screen. Editing the Video Clip: 6. Data analysis will be easier if the length of the video is cropped to include only the frames of interest. After the stop button, the editing screen will appear. The editing screen contains a set of controls much like the set of controls on any dvd player. There are buttons for play, fast forward, fast rewind, step forward, and step backward. When the play button is pressed, the video will begin to play, and the button is transformed into a stop button. These controls are shown in Figure 3.6. There is also a slidebar with a black diamond on the top. Sliding this black diamond along the slide bar advances/recedes the video clip. Use these controls to review the film clip to decide where to crop the video clip. There are also two white triangles. These triangles are to mark the start and end of the frames that are to be analyzed. 7
Black Diamond White Triangle - Start Figure 3.6 VideoPoint Capture Editing Screen White Triangle - Finish 7. After reviewing the video clip, use the white triangles to mark the start and finish of the frames that will be analyzed. Make sure that the start is marked after the ball has cleared the hand of the person dropping the ball. Deleting approximately five frames after the ball is released is recommended. Once you have chosen the start and finish, clip on the Confirm Edit button. This will crop the video clip to the chosen length. ( See Figure 3.7 ) 8. After checking that the film clip has the needed frames, save the film clip. Figure 3.7 VideoPoint Capture Editing Screen 8
Analyzing the Video Clip: 9. Open the VideoPoint Physics Fundamentals Software. Using the File drop down menu, load the video clip movie file saved in step 8. Figure 3.8 VideoPoint Physics Fundamentals Preview Screen 10. The first screen can be used to view the video clip. This screen is labeled as the Preview Screen in the tabs which run along the top of the screen. These tabs may be used to navigate through the program. As each screen progresses, a hint will appear in a pop up window. These hints can be closed using the X button in the upper, right hand corner of the pop up window. Once the clip has been viewed, advance to the next screen using the forward arrow button. 11. This screen is the calibrate screen. Here a yellow calibration tool can be found. Move one end of the calibration tool to meet with one end of the meter street ( or some other known standard ) and move the other end of the yellow calibration tool to meet with the other end of the meter stick. Enter the length of the object between the jaws of the calibration tool, one meter if the meter stick is used. Advance to the next screen. 9
Figure 3.9 VideoPoint Physics Fundamentals Calibration Screen 12. This is the Set Up Analysis frame. Set the origin using Move Origin x and y position boxes. The axis may also be rotated and the t = 0.0 seconds frame may also be set. Go to the next frame. Figure 3.10 VideoPoint Physics Fundamentals Set Up Analysis Screen 13. This is the Collect Data screen. Here the mouse can be used to select data points on the screen while in the Data Collection Mode. As the data point is selected, the video clip advances to the next frame. The set of data points may be completely deleted by selecting the data set name ( ex. Points S1 ) and pressing the Delete button. A new set may be started with the Add New Points drop down menu and hitting the OK button. Point locations may be moved while in the Point Editing Mode. When in the Point Editing Mode, the mouse can grab the point and drag it to a new position. Once the points have been selected move to the next screen. 10
Point Editing Mode Data Collection Mode Figure 3.10 VideoPoint Physics Fundamentals Collect Data Screen 14. Export the data set into a spreadsheet using the file drop down menu. 11
Analyzing the Data Collected: 15. Open the spreadsheet containing the data collected. 16. Plot the y position versus time data. Add a trend line using a second order polynomial fit. Show the equation of the line and the R-squared valued. 17. Calculate the y component of the velocity using the equation: vy= i y i 1 y i 1 y i 1 y i 1 = t i 1 t i 1 2 t 18. Plot the y component of the velocity versus time. Add a trend line with a linear fit. Show the equation of the line and the R-squared value. 19. Calculate the y component of the acceleration using the equation: ay= i v y v y i 1 i 1 t i 1 t i 1 = v y v y i 1 i 1 2 t 20. Plot the y component of the acceleration versus time. Add a trend line with a linear fit. Show the equation of the line and the R-squared value. Projectile Motion: 21. Repeat the above procedure for projectile motion. Use the Pasco projectile launcher. Use an angle of 30 degrees and use the second step in the launching mechanism. 22. Open the spreadsheet containing the data collected. 23. Plot the y position versus time data. Add a trend line using a second order polynomial fit. Show the equation of the line and the R-squared valued. 24. Calculate the y component of the velocity using the equation: vy= i y i 1 y i 1 y i 1 y i 1 = t i 1 t i 1 2 t 25. Plot the y component of the velocity versus time. Add a trend line with a linear fit. Show the equation of the line and the R-squared value. 26. Calculate the y component of the acceleration using the equation: ay= i v y v y i 1 i 1 t i 1 t i 1 = v y v y i 1 i 1 2 t 27. Plot the y component of the acceleration versus time. Add a trend line with a linear fit. Show the equation of the line and the R-squared value. 28. Plot the x position versus time data. Add a trend line using a linear fit. Show the equation of the line and the R-squared valued. 12
29. Calculate the x component of the velocity using the equation: vx= i x i 1 xi 1 x i 1 xi 1 = t i 1 t i 1 2 t 30. Plot the x component of the velocity versus time. Add a trend line with a linear fit. Show the equation of the line and the R-squared value. 13
Data Analysis: Free Fall: 1. Describe the relationship between the acceleration and time, the velocity and time, and the yposition and time. 2. What is the value for the acceleration due to gravity found using the y position versus time plot? 3. What is the error between the acceleration due to gravity found using the y position versus time plot and the accepted value of 9.8 m/s2? 4. What is the value for the acceleration due to gravity found using the y component of velocity versus time plot? 5. What is the error between the value for the acceleration due to gravity found using the y component of velocity versus time plot and the accepted value of 9.8 m/s2? 6. Which method gave you a better approximation for the acceleration due to gravity? Explain why that method gave the better approximation. 14
Projectile Motion: 7. Describe the relationship between the acceleration and time, the velocity and time, and the position and time for both the horizontal and vertical component? 8. What is the value for the acceleration due to gravity found using the y position versus time plot? 9. What is the error between the acceleration due to gravity found using the y position versus time plot and the accepted value of 9.8 m/s2? 10. What is the value for the acceleration due to gravity found using the y component of velocity versus time plot? 11. What is the error between the value for the acceleration due to gravity found using the y component of velocity versus time plot and the accepted value of 9.8 m/s2? 12. Which method gave you a better approximation for the acceleration due to gravity? Explain why that method gave the better approximation. 15
4. Pre-Lab LAB-02 One and Two Dimensional Motion Name: Sec/Group: Date: 1. Determine the acceleration of a particle down a smooth plane, inclined at an angle respect to the horizontal. Draw the acceleration vectors on the diagram below. with 2. A particle is launched in a gravitational field ( g ) with an initial velocity v o at an angle with respect to the horizontal. a.) Write down the equations for displacement along the x and along the y directions, as a function of time. b.) Combine the equations in part 2a.) to eliminate time ( t ) and explicitly show that the resulting equation of the form y = f ( x ) shows that the trajectory is parabolic. 16