Two-Dimensional Projectile Motion

Transcription

1 Two-Dimensional Projectile Motion I. Introduction. This experiment involves the study of motion using a CCD video camera in which a sequence of video frames (a movie ) is recorded onto computer disk and subsequently analyzed with computer programs. In this experiment, an object in motion is illuminated by conventional or directed room lighting, and successive images of the motion are recorded every 1/30th of a second using a shuttered video camera. The object in motion is located relative to a reference coordinate system (imposed by the computer program) and scaled by reference ruler within the field of view of the camera. From this information, the position of the object as a function of time is determined. Analysis of these data yields the velocity and acceleration of the object as functions of time. In this experiment a projectile, a yellow ping-pong-sized ball, is shot from a springloaded launcher. The video movie of the ball's trajectory is analyzed to find the time dependence of the components of the velocity and acceleration of the ball in the horizontal and vertical directions. The results are compared with what is expected for projectile motion. y g spring launcher x Video camera CCD image II. Required Equipment. A. A spring-loaded launcher, which is used to shoot the projectile. The angle is adjustable. The projectile should land in a padded box which is provided. B. A reference ruler of 1 m length, with gradations every 10 cm, to provide a reference scale. 2-1

2 C. A mounted CCD video camera with shutter. The frame rate of the camera is 1/30th of a second which represents the time between successive video images. The camera should be located approximately 3m from the launcher setup. D. A neutral colored backdrop is mounted behind the grid to reduce unwanted reflections into the video camera. E. A computer system to control video recording. This computer will be running the AVID Video Shop software program. F. Convert-to-Movie application to reduce video size in computer memory to speed data transfer. G. A computer system to analyze the video movie. This computer will be running VIDEO POINT and Graphical Analysis GA programs. III. Procedure. A. Data Recording. (Refer to Section 4 of Computer Usage for additional information) Laboratory partners should work together. One controls the video recording; the other fires the projectile. Be careful not to disturb the alignment of the scale rulers or the CCD video camera. Specific steps are now indicated which should allow successful video data recording. 1. Double click on the Avid Videoshop icon on the desktop of the computer used to control video recording. Several dialog boxes will appear - the last being canvas size. Select 320 x 240 and click on OK. 2. Three screens will now be displayed: canvas: untitled1 ; recording folder; and sequencer. Click on the recording folder. 3. From the Menu Bar, pull down Windows and select recording. A recording window will be activated and a live video display will appear in this window. Move your hand in front of the video camera lens to verify that this is the case. 4. At this stage you are ready to test the projectile launcher. 5. Adjust the launcher for the trajectory you want and take several practice shots. When you are satisfied that the projectile follows a reasonable arc and lands in the padded box, you are ready to run the experiment and record your video. 6. The computer operator starts the movie recording (clicking on the [REC] button next to the live video display). When the blue thermometer under the video display starts to turn red, the computer operator tells the launcher to fire the projectile. The computer operator stops the movie (clicking on the [Stop] button) a few seconds after the mass hits the padded box. 7. You can now replay the movie to see if it has provided acceptable video data. To do so, click on the highest numbered untitled movie in the Recording Folder. (The highest number corresponds to the most recently recorded movie - which should be yours.) You may now run the movie, by selecting Play from the Menu Bar and selecting Play 2-2

3 Forward. This will play the movie from start to finish. Watch your movie of projectile motion. 8. If the images look good and the projectile arc is good, go to step 9 to save your movie. If the images are of poor quality, or the motion of the object is not well recorded for whatever reason, return to step 5 and repeat steps Go to the File Menu and select Save as Movie. Save the movie to the Movie Folder on the local disk. 10. Once the movie has been saved, then go to the Apple icon on the Menu bar. Under Recent Applications, select the program Convert-to-Movie (or Select Convert-to-Movie from the desktop). When Convert-to-Movie opens, from the File Menu, open the movie that you created. Okay all the default options that you encounter in the two dialog windows that appear. Then save the converted movie to the local disk, then copy it to your lab bench computer s movie folder. Save the movie under the conventional name for the laboratory: [bench.room.movie], viz movie. Once you have completed this step, you are ready to begin analysis of your video on the lab bench computer. B. Video Analysis. The CCD camera records video images every 1/30th (0.033) of a second, and successive images of the movie show the position of the projectile (ball) 1/30th of a second apart along its trajectory. By measuring the coordinates of these positions as determined from the reference coordinate system and properly scaled to physical dimensions using a meter scale in the video field, it is straightforward to find the (x,y) position of the ball and the x,y components of the average velocity v of the ball (v x,v y ) as well as the average acceleration a and its components (a x, a y ). Here bold-face letters are used as symbols for vector quantities just as in the text books. Let us assume that the motion takes place in the x,y plane with x positive to the right and y positive upwards, respectively, and with the origin of the coordinate system chosen at any convenient point using the computer program. Newton's laws are vector laws and the acceleration due to gravity is downward near the surface of the earth. If the rulers and video camera are properly aligned with the vertical direction, the x component of projectile velocity should be constant (if air resistance can be neglected). The y component of velocity should vary as a function of time due to the gravitational acceleration. 1. To record the successive x,y positions of the ball from the video image, start up the Video Point program at your lab bench, by double-clicking on the VideoPoint Icon on the desktop. From the File menu, select open movie and double-click on your movie in the movie folder on the hard drive. 2. A dialog box asking for the number of points will appear on the screen. This is to determine how many objects will be in motion, and whose coordinates must be recorded. Since in this experiment only the ball is in motion, the number should be 1. Then click OK. 3. Three screens should now appear, as well as a vertical tool bar along the left-hand side of the screen. Your movie will appear on the upper left, a coordinate systems box on the upper right, and a data table in the lower left. You can play your movie backwards and forwards by moving the slider just beneath the movie screen. 4. It is convenient to expand the movie screen to a larger size to make measurement easier. Click on the expander box in the upper right corner of the window title bar, or pull down the 2-3

4 Movie menu and select Double Size. The movie screen should now nearly fill your screen. 5. Next actual physical dimensions need to be established for the movie. The movie is recorded in CCD camera coordinates, and a scaling factor must be provided to relate these coordinates to actual laboratory dimensions. This is accomplished by scaling the movie - by locating and measuring the length of a meter stick in the field of view. This allows CCD camera coordinates to be directly calibrated in terms of meters. To accomplish this task, pull down the movie menu and select Scale Movie. A dialog box will also appear in the middle of the screen. Set the known length to 1.00 and units of (m) for meters, and click on continue. The dialog box will disappear and you are now ready to measure a standard 1 meter length in the field of view. 6. Click on the left end of the meter stick and then click on the right end of the meter stick. These measurements will then tell the computer program that the separation between these two positions is 1 meter and to scale the movie dimensions accordingly. This also establishes the error with which we can measure x and y coordinates using the video recording system. For example, the meter stick covers approximately 150 video pixels. So each pixel is effectively 1/150 of a meter wide or.0066 m. Thus the error in locating an object can be no better than ± half of this value or δx = ±.0033 m. For the y direction, the error will be the same δy = ±.0033 m. We will need these error values later in the analysis of the experiment. 7. You are now ready to measure (or digitize ) the coordinates of successive images of the projected ball. Leave the movie screen at the current double size. Move the slider at the bottom of the screen until you see the ball begin to enter the field of view at the left. Move the cursor and center it on the ball and click the mouse. The location of the ball will be recorded and its (x,y) position and the time of the image will be digitized into the data table. After clicking, the computer will automatically advance to the next frame of the movie. Move the cursor so that it is centered on the next image of the ball and click. This image will be digitized, and, again, the movie frame automatically advanced. Repeat this exercise for as many images as are clearly visible. You should be able to digitize ball images in your video. 8. Then return to the Movie menu and click on normal size. You should now see that the data table has been filled. Move the cursor to the data table and select all the rows with (t,x,y) information. Then pull down the Edit menu and select copy. This will write the data table to the computer clipboard. From the clipboard, data can be read into the Graphics Analysis program. C. Graphical Analysis. You are now in a position to graph and analyze the tabulated data, but first the Graphical Analysis program must be started. To do this, go to the Apple icon on the Menu Bar, and select successively: Recent Applications, and Graphical Analysis 3.1, or select Graphical Analysis 3.1 on the desktop. 1. Click on the Data Table. Currently it will show 2 columns. We need to create a third column to paste in t, x, and y information from the clipboard. From the Data Menu, select New Column. Now your table will show three columns. 2. Click on the upper left-hand element of the table (currently blank). Pull down the Edit Menu and select Paste. Your data table from Video Point should now be pasted into the table. 2-4

5 3. The table headings are arbitrarily called X, Y, Z by the analysis program. They are really t, x, y as recorded in the Video Point, and so the headings (and associated units) need to be changed. Click on X. An X should appear in the text entry box at the top of the table. Move your cursor there and delete the X and replace it with t, and click on the checkmark. 4. Now click on the units box below your newly labeled t column. Then click on the text entry box and type in s for seconds and click on the checkmark. 5. Repeat the above exercise for the second and third columns in your table changing them to x and y in units m for meters. 6. Now from the File menu select Save as and save your data table to the Desktop under a typical name: bench.room.table, e.g., table. 7. You are now ready to analyze projectile motion. Study of the coordinate motion of the ball: x vs t 1. You are now in a position to study the coordinates of the ball as a function of time and of each other. From the Graph menu, select New Graph. A graph will appear on the right hand side of the screen. Set the vertical axis to x and the horizontal axis to t. A fairly linear relationship should appear for the data in the graph, which displays the x-coordinate of the ball as a function of time. 2. The error bars displayed by the program will not be correct as is. To assign the proper values, first double click on the x label in the data table. In the pop up menu, select options and Error Bar Calculations. Enter a value.0033 m in Error Constant box. This is the numerical value for the error in x (δx =.0033 m) which we calculate in the above. Second, select the graph and the options on the menu bar. Select graph options on the drop menu. Select y error bars. We will assume that time is measured so well that we can ignore its error. Enter.000 for the error in the horizontal axis (i.e. δt =.000 s). Click on OK. 3. Select Linear fit from the tool bar. Double click on the fit object. On the pop menu, select standard deviations for the slope and Y-intercept. The fit object will display the slope and Y intercept of the fitted line and their standard deviations. A regression analysis is a standard procedure for applying fitting functions to data. We are not going to dig into the details of such numerical methods here, but they are basically a more sophisticated application of error analysis following the strategies outlined in Measurement and Error. The computer program allows us to implement these procedures rapidly, and we will use the results here. 4. From the File Menu, select Page Setup and select sideways printing (Landscape Mode) and click OK. 5. From the File Menu, select Print Graph and print a copy of the x Vs t figure for each member of your group. y Vs t 1. Now, using the cursor, change the vertical axis to y. The graph should now display data points corresponding to the y-coordinate of the ball as a function of time. The graph should 2-5

6 also look distinctly non-linear. 2. Assign errors to the y values of the ball s trajectory in the same way that you assigned errors to the x values of the ball s trajectory. 3. A linear fit will not be appropriate to your y vs. t data. Select Curve Fit from the tool bar. Our knowledge of the expected behavior of an object moving only under the influence of gravity, suggests that the data should fit a parabola. On the pop up menu, select Quadratic from General Equation. Select Try Fit and OK. If you have not selected all the data points, you may do this in the small plot in the Curve Fit pop up menu. If errors are not shown for the fit parameters, double click on the fit object and select Show Uncertainty. Do a print graph for all members of your group. y Vs x 1. Now change the horizontal axis to x. Click on the label of the x axis, and then select x on the pop up dialog menu that appears. Click Options on the menu bar and Graph Options on the drop down menu. Select Error bars for both x and y. You have already calculated the values when you were on the Options page of the Column Options pop up menu. A linear fit will not be appropriate to your y vs. x data just as it was not suitable for your y vs. t data. 2. Select the graph and select curve fit from the menu bar. Motivated by our study of mechanics, we can guess at a useful form for a fit. On the pop up menu, select Quadratic from General Equation. Select Try Fit and OK. If you have not selected all the data points, you may do this in the small plot in the Curve Fit pop up menu. If errors are not shown for the fit parameters, double click on the fit object and select Show Uncertainty. Do a print graph for all members of your group. Study of the velocity and acceleration of the ball. To analyze the components of the velocity and acceleration of the ball requires the determination of differences in the ball positions and times. As we did with position, we will examine the velocity components versus time: v x = x/ t, v y = y/ t, and slope = (v y /v x ). x is the difference in x component of position and t is the corresponding change in elapsed time between each of the data points. In part B, t was given as 1/30 th sec. To form v x, v y, and slope, you will need to create 3 new columns in the data table. To calculate the x component of the ball s velocity, select Data from the menu bar. Select New Calculated Column. On the pop up menu, select a name, e.g. vx, and enter appropriate units. In the equation box, select delta() from the Function drop down menu. Select x from the Variables drop down menu. Then enter a slash, /, for division. Then select delta() and t for the denominator. Click OK. Repeat this process for the y component of the ball s velocity. In the third calculated column, create a quantity that we have called slope. This quantity is dimensionless. It will be equal to the y component of the velocity divided by the x component of velocity. 2-6

7 It is necessary to estimate the errors for vx and vy before making plots. Errors will be ignored for the calculated quantity slope. Specifically, consider vx: vx = x/ t = delta( x )/delta( t ) Referring to the Measurement and Error section of your manual, because vx is obtained by division: δ(vx)/vx = [(δ( x)/ x) 2 + (δ( t)/ t) 2 ] However, we are assuming that the error in t =0. Therefore: δ(vx)/vx = [(δ( x)/ x) 2 ] = δ( x)/ x x = x n x n-1 Where x n is the nth value of x and x n-1 is the previous value of x. Again referring to the Measurement and Error section of your manual for the error in the difference between two measured values: δ( x) = [(δx n ) 2 + (δx n-1 ) 2 ] Since the error in each value of x is assumed to be the same: δ( x) = [2] δ(x n ) Therefore; δ(vx)/vx = δ( x)/ x = [2]δ(x n )/ x δ(vx) = vx [2]δ(x n )/ x = ( x/ t) [2]δ(x n )/ x δ(vx) = [2]δ(x n )/ t = [2]( )(30) = 0.14 m/s Return to the Column Options pop up menu by double clicking on the column label in the data table. Select the options page. Check Error Bar Calculations and enter the value of the calculated error. Repeat the process for vy. Ignore the errors for slope. Do a Print Data Table for all group members. You can now make graphs of these quantities and obtain fit values to functional forms applied to the data. 2-7

8 v x vs t 1. Click on the axis label of your plot. Select vx on the y axis and t on the x axis. Since you have calculated errors for vx and entered them in Column Options menu, the appropriate error bars should appear. If not, go to Options and Graph Options and select y error bars. 2. Select Linear Fit from the toolbar. If errors are not provided for the slope and intercept in the fit object, double click on the fit object and select standard deviations on the on the pop up menu. 3. You may notice that since the values of vx are quite similar (vx appears to be essentially constant.), you may need to manually scale the vx axis to put it on larger scale, comparable to scale that will necessary for vy. You may do this in the Graph Options menu. You may want to revisit this plot after completing your vy plot. 4. Print a graph for all members of your group. v y vs t 1. Click on the axis label of your plot. Select vy on the y axis and t on the x axis. Since you have calculated errors for vy and entered them in Column Options menu, the appropriate error bars should appear. If not, go to Options and Graph Options and select y error bars. If you have manually scaled your vx, plot you will need to return to the Graph Options and select autoscale. 2. Select Linear Fit from the toolbar. If errors are not provided for the slope and intercept in the fit object, double click on the fit object and select standard deviations on the on the pop up menu. 3. Print a graph for all members of your group. slope vs y 1. Now change the vertical axis to slope. Set the horizontal axis to y. Deactivate the error bars in the Graph Options menu. No error bars will be used for this plot. They are not zero, but we will ignore them for this qualitative study. You should produce an unusual looking plot. Do a print graph for all group members. Congratulations! You have now completed all graphing. D. Before you leave the laboratory: You should now have for all group members: 2-8

9 1. A copy of the full data table with six separate columns for t,x,y,vx,vy, slopes. 2. A plot of x vs t. 3. A plot of y vs t. 4. A plot of y vs x. 5. A plot of vx vs t. 6. A plot of vy vs t. 7. A plot of slope vs y. You should get your graphs initialed. You may complete the rest of the analysis outside the laboratory. D. Physics Analysis (PA). To be performed outside the laboratory. PA1. x vs t Is your plot linear? What physical quantities do the coefficients A and B represent? PA2. y vs t Is your function consistent with a parabola? What physical quantities do the coefficients A, B and C represent? PA3. y vs x Is your function consistent with a parabola? What does this suggest about the nature of projectile motion near the surface of the earth? PA4. vx vs t What do you expect for the functional relationship between vx and t? What do the fit parameters A and B represent here? If there is a finite acceleration term here, how might such a value occur? PA5. vy vs t What do you expect for the functional relationship between vy and t? What do the fit parameters slope and intercept represent here? How does slope compare with the quantity C in PA2 above? From the error reported in your graph, are you consistent with the textbook value for gravitational acceleration g = 9.8 m/s 2. PA 6. slope vs y What is the significance of the value of y where slope = 0? Hint: refer to your plot of y vs x. PA 7. Consider your data for projectile motion. Are they consistent with acceleration in the y direction only; constant velocity in the x direction? What were potential sources of error in this experiment? How might such error sources be reduced and the experimental results improved? PA 8. How can this experiment be improved? Write down 3 suggestions. 2-9