Physics 3 Summer 1990 Lab 1 - Projectile Motion Theory Consider an object launched at time to = 0 at some angle θ from the horizontal with an initial velocity, v o . Neglecting air resistance, the only force acting on the projectile is the force of gravity and we have in essence a freely falling object (see figure 1). At first glance, the motion may seem more complicated than that of an object initially at rest dropped from a height, h . In that case, the object's motion is one dimensional while here we have motion in two dimensions.

Figure 1 Experiment has shown, however, that the vertical and horizontal motions are independent of each other. This simpifies the matter and allows us to treat the motion in the x-direction without reference to its motion in the y-direction and visa versa. Since there are no forces acting in the x-direction, the initial velocity in that direction should remain unchanged. Therefore, in a time t the projectile should move a distance x given by x = vxt = v o cosθ

.

(1)

Figure 2a shows the displacement of the object at equal time intervals for the x-direction only. The length of the arrow is proportional to the displacement during the interval, in this case a constant. The motion in the y-direction is subjected to the force of gravity and will experience an acceleration equal to -g . Its position as a function of time will be given by y = (-1/2) gt 2 + vyt + yo

(2)

where yo is the y position at t = 0 and vy is the y component of the initial velocity. Since yo = 0 and vy = vo sin θ , equation (2) can be rewritten as

y = (-1/2) gt 2 + tvo sinθ

(a)

.

(3)

(b)

(c)

Figure 2 The y component of the displacement at equal time intervals is shown in figure 2b. The trajectory (the path of the projectile) can be found by eliminating the time variable between equations (1) and (3). Solving for t from equation (1) yields t =

x vo cos θ

.

Substituting this expression for t into equation (3) gives

or

x y = -1g 2 vo cosθ y =

2

-g 2v2o cos 2 θ

+

x vo sinθ vo cosθ x2 + x tan θ

.

(4)

This equation has the general form y = ax2 + bx + c where c = 0 . It is a quadratic equation and if graphed would yield a parabola. Hence, the trajectory of this projectile is parabolic. The total motion is a combination of x and y motions as shown in figure 2c. In the preceding example, we began with the initial conditions (the initial position and the initial velocity) and from that computed the future path of the body. It is also possible to work backwards from the path of the body to arrive at the initial conditions and other pertinent data. This may be done graphically using a technique called graphical analysis or mathematically using equations (1),(3) and (4).

Graphical Analysis

Suppose that you have obtained the positions of a projectile (relative to the origin of some arbitrary reference frame) at equal time intervals (see figure 3a) and you wish to analyze the motion for such pertinent information as the object's acceleration. One way of doing this is to construct a vector diagram.

(a)

(b) Figure 3

This diagram is a graphical representation of the actual physical situation. The link between the diagram and the physical situation it represents is a scale which relates lengths on the diagram to actual distances in the real situation. The choice of scale is determined by the size of the diagram and the actual distances traversed by the projectile. Convenience is also a factor to be considered. If our projectile moved 25 meters and our diagram is 30 centimeters long, a convenient scale might be 1 cm on the diagram represents 1 meter in the physical situation. Once the scale has been chosen and the axes of the coordinate system have been drawn, the points representing the positions of the ball can be plotted. A position vector should then be drawn from the origin to each point on the graph (see figure 3b). Note that the position vectors uniquely specify the location of each data point, since the length gives the distance from the origin and the orientation gives the direction with respect to the coordinate axes. Now that the initial diagram is drawn, we can begin our analysis by determining the initial conditions. While any point, except the first or last, can be considered to define the initial conditions, we choose for convenience, the second from the left. The reason for this choice will become clear as we proceed. The initial position is, therefore, defined by the vector ro . The initial velocity is the instantaneous velocity at ro . This presents a problem since we can only compute average velocities from our data points. This is not a serious obstacle, however, for if we keep the time interval between the data points quite small the average velocity over an interval will closely approximate the instantaneous velocity at the mid-point of the interval. Therefore, the initial velocity of our projectile (its instantaneous velocity at ro ) can be approximated with good accuracy by its average velocity between positions r- 1 and r1 .

The average velocity is defined as the displacement (change in position) the projectile undergoes during a time interval divided by the time interval. The displacement is a vector which is determined by subtracting the position vector at the beginning of the time interval from the position vector at the end of the time interval. The velocity vector will be in the same direction as the displacement vector, but will differ in size by a factor of 1/∆t. Figure 4 shows the displacement vector used to determine the initial velocity vector. If our position vector diagram has been carefully drawn to scale, a ruler can be used to measure the length of the displacement vector; and then using the scale we can determine

Figure 4 magnitude of the actual displacement that the projectile underwent during that time interval.The average velocity may then be computed from this. For example, if our scale is cm on the diagram equals 1 meter in the actual situation and our displacement vector measures 3 cm in length, then the actual projectile moved 3 meters during that time interval. If the timeinterval was 1 second, the initial velocity was 3m/sec in the direction of the displacement. Once the initial position and initial velocity are known, we are ready to begin the calculation of the acceleration of the body. Since the acceleration is defined as the rate of change of the velocity, our first job is to calculate the instantaneous velocities at the other positions. This is done in the same way in which the initial velocity was done. When the lengths and orientations of the velocity vectors are known, we can construct a velocity vector diagram. The scale may be the same or it may be different from the position vector diagram. Convenience will be the principal factor. Since it is possible to have negative velocities in the y-direction, the y axis will have to be extended downward into the negative area. The lengths of the velocity vectors will represent the actual velocities and their orientation to the coordinate axes will be the same as the projectile's actual displacement. A protractor can be used to determine this orientation. Figure 5 shows a position vector diagram and its corresponding velocity vector diagram.

(a)

(b) Figure 5

Note that in figure 5a a new notation was introduced. For simplicity and to emphasize that position vectors and displacement vectors are two different things, the displacement vectors are denoted as s o , s 1 ... with s o = (r1 - r - 1 ), s 1 = (r2 - ro ) , etc. The next step in determining the acceleration is to determine the change in velocities. This is done in the same way as the determination of the displacement vectors. For example, the change in velocity during the time interval ∆t = t1 - to would be ∆v = v 1 - v o . Since the velocity vectors have been drawn to scale, we can compute the magnitude of the change in velocity vectors by using a ruler to measure their lengths and using the scale to convert the lengths to the dimensions of the actual physical situation. Once the actual changes in velocities are known, the acceleration of the projectile during the time intervals can be computed by dividing the respective changes in velocities by the time it took for those changes to occur. If care is used in the construction of the various diagrams, the average accelerations calculated in this way will be close approximations to the actual accelerations experienced by the projectile.

Mathematical Analysis The graphical method, although clearly illustrating the vector nature of the quantities involved, has serious limitations. 1. It is time consuming. 2. Since each measurement used in the calculation of a quantity increases the uncertainty of the final result and since the construction of the graphs involve numerous measurements, the uncertainty of the acceleration values can be considerable. They are at best a first

approximation. 3. Only simple motion can be analyzed. The vector diagrams are often too cumbersome for complex motions. A much more powerful method is to use only the initial experimental data (the positions of the projectile at the given times) and the mathematical expressions for the quantities we want. Once the position vectors are known, the velocities can be calculated from vi = ri+1 - ri-1 = s i 2 ∆t 2 ∆t

(5)

and the accelerations from

a i→i+1

s i+1 - s i v v i = 2∆t 2∆t = i+1 ∆t ∆t

.

(6a)

a i→i+1 = s i+1 - s2 i 2 (∆t)

.

(6b)

Note that equations (5) and (6) are vector equations and vector algebra must be used. For a complete discussion of vector algebra see Halliday and and Resnick's Fundamentals of Physics, chapter 3. The mathematical analysis can be made even simpler by resolving the initial position vectors into their components and analyzing the motion in the x-direction separately from the motion in the y-direction. In this way we reduce the mathematics to simple algebraic relationships. The velocities are calculated by vxi = xi+1 - xi-1 2 ∆t

(7a)

yi+1 - yi-1 2 ∆t

(7b)

vyi = and the accelerations by

a xi→i+1 =

a yi→i+1 =

vxi+1 - vxi ∆t vyi+1 - vyi ∆t

(8a)

(8b)

These calculations can be done by hand or with the use of computer. In this lab you will use the methods of graphical analysis and mathematical analysis described above to study the motion of a golf ball projectile. The primary object of the analysis will be to

obtain a value for the acceleration due to gravity, g .

The Computer Up to now we have been discussing what to do with the experimental data for the motion of a projectile once we have it, but have said nothing about how the data is to be obtained in the first place. In this lab you will take the data in two ways. The first method involves the use of strobe photography. Using a spring loaded gun, a golf ball will be launched in front of a grid attached to a wall in the lab. A strobe light on the floor below the path of the ball will briefly (for approximately one millionth of a second) illuminate the ball at equal time intervals. Its position, with respect to the grid, at those times will be recorded on film using a Polaroid camera which will have its shutter manually held open for the duration of the ball's flight. The positions of the center of the ball will then be determined by hand from the resulting photograph using specially machined cylinders and pins. This is done by positioning the cylinder over an image on the photograph and marking its center by pushing a pin into the photograph through a hole drilled in the center of the cylinder. This method has obvious drawbacks. First, it is slow. The picture must be taken, the film developed, the photograph enlarged to allow for more accurate determination of the positions of the ball, the positions determined by hand using small, hard to handle cylinders and pins, and the data recorded. Secondly, there are many places for experimental errors to occur. For example, the cylinders must be machined to just the size of the image in the photograph. Because the ball is illuminated from below, the cylinders must be fitted to only partial images of the ball. The pins must exactly fit the hole in the center of the cylinder and the hole must be drilled exactly along the center axis of the cylinder or else the pins, when pushed into the photograph, will not mark the center of the ball. Lastly, it is not very accurate. In order for the grid lines to be clearly resolved in the photograph, they need to be at least 1 centimeter apart, thereby reducing the precision of the measurements. Given all of the possible sources for error, the positions of the ball can only be determined accurately to approximately plus or minus 0.5 centimeters. The second method involves the use of a computer to take the data. In this method a video camera and a specially modified Apple II+ computer replace the strobe light, the Polariod camera, and the grid. Through a combination of hardware and software, the video camera and the computer takes a digital "picture" of the golf ball's flight, recording the position of the center of the ball every 1/30th of a second. The output from the computer consists of a picture of the ball's flight similar to the one taken with the Polaroid camera, a graphical representation of the ball's flight with a cross marking the position of the center of the ball for each image, and a table listing the x and y coordinates of each image of the ball with respect to a coordinate system which is fed into the computer and calibrated by the user prior to taking the data. This method is fast and accurate (assuming the calibration procedure is accurately done).

References The following are the sections in Halliday and Resnick's Fundamentals of Physics 3rd edition which are pertinent to this lab. This material should be read before coming to class. 1. Chapter 2 2. Chapter 3 3. Chapter 4, sections 4-1 through 4-6

Experimental Purpose The general purpose of this lab is to apply the graphical and mathematical analysis techniques described in the introduction to the analysis of the motion of a golf ball projectile. The end result of that analysis will be to determine a value for the acceleration due to gravity, g. The data will be taken in two ways: (1) using a Polaroid photograph taken using strobe photography and specially machined cylinders and pins and (2) using a specially adapted Apple II+ personal computer.

Procedure Part I - Graphical Analysis 1. At the end of this write-up, you will find a reproduction of an enlarged Polaroid photograph of a golf ball launched in front of a large grid which was taken using the method described in the introduction. (Note: If you downloaded this writeup from the Public File Server the photograph can be obtained in the lab.) Separate that reproduction from the rest of the writeup. Your TA will demonstrate how the original picture was taken. The time interval between flashes of the strobe was 0.1 seconds. The photograph shows the position of the ball at several points in its trajectory. The position of the ball is determined by the location of its center. (Why?) From your TA, get a pin and one of the small aluminum cylinders which has been machined to be the same size as the images of the ball in the photograph. Insert the pin into the hole in the center of cylinder and carefully position the cylinder over the first image of the ball in the picture. Push the pin into the picture just far enough to make a small hole. Repeat this procedure for all of the other images in the picture. Return the cylinder and pin to the TA and get a ruler, a protractor and a triangle. 2. At the end of this write-up is a sample table which has been provided for you to help facilitate the habit of using tables to organize your data and to serve as a model for the construction of tables for future labs. (Note: For the remainder of the term, it will be expected that you will come prepared with tables and graphs of your own construction.) Remove that table from the writeup and record in it the x and y coordinates with respect to the grid of the position of each image of the ball. Each of the strings which make up the grid are one centimeter apart. (Note:

The decimeter lines in the grid are made of dark green string which does not show up in the photograph. When determining the coordinates of an image, be careful not to omit counting the invisible decimeter lines.) 3. In the lab there is special graph paper which looks like the grid in the photograph. From the position data recorded in step 2, construct on a sheet of that graph paper a position vector diagram similar to figure 4. 4. On the same graph, draw in the displacement vectors. If you don't know how to do this, review the section on graphical analysis in the introduction. Compute the magnitudes of each of the displacement vectors and record them in the margin of the graph paper. (Note: Remember that your graph is a scaled down representation of the flight of the ball in the real world and that when you compute these magnitudes you want to do so in real world units, not in the arbitrary units of your graph paper. One way of doing this is to note that since the graph paper is a replica of the grid, each small division on the graph paper represents 1 centimeter in the real world. With this information, you can determine the scale for the diagram. In other words, you can compute what a distance of 1 centimeter as measured on the graph represents in centimeters in the real world. That done you can then measure the lengths of the displacement vectors on the graph in centimeters and use the scale to convert to real world units. As it turns out, however, the scale thus computed is an inconvenient value of 1 centimeter on the graph = 5.32 centimeters in the real world. An easier method is to use a sheet of the special graph paper as a ruler. If that graph paper is carefully folded along a decimeter line, it can act as ruler which will automatically convert from the graph units to centimeters in the real world. Use whichever method you feel most comfortable with.) 5. From the magnitudes of the displacement vectors, compute the magnitudes of the instantaneous velocity vectors at the various positions and record them in a separate margin of the graph. (What is the direction of the velocity vectors?) 6. From the instantaneous velocity data, construct on a separate sheet of graph paper a velocity vector diagram similar to the one shown in figure 5b. 7. On the same graph, draw in the change in velocity vectors. Compute their magnitudes and record them in the margin of the graph paper. From the magnitudes of the change in velocity vectors, compute the magnitude of the average accelerations during the intervals between positions and record them in a separate margin of the graph paper.

Part II - Mathematical Analysis 1. Using equations (7a) through (8b), fill in the sample table.

Part III - The Computer The purpose of this part of the lab is to show you two uses for the computer in scientific work. The first use is as a data acquisition device and the second is as a data analysis device. A secondary purpose is to familiarize you with the use of the Kiewit system as it will be used for data analysis in future labs. Procedural step 1 deals with the data acquisition aspect and procedural step 2 deals with the data analysis. The steps can be done in any order. If the data acquisition computer is in use, go on to step 2. 1. Data Acquisition. The experimental system consists of an Apple II+ computer, two monitors, a video camera, a printer, a floppy disk containing the software and a black box on wheels to which a large brass cylinder is attached. The right hand monitor is used to display the experimental data and the left hand monitor displays what the camera is actually looking at. All the electrical equipment is plugged into a plug strip on the base of the cart and can be turned on with one switch located to the left of the computer. The black box provides the necessary black background against which the camera can "see" the golf ball and the brass cylinder is used to launch the ball. The ball is launched by allowing it to roll through the brass cylinder and bounce off the floor. The motion of the ball after it bounces off the floor is the motion you will be analyzing. The following instructions are for beginning from scratch. If the computer has already been used, then chances are that all you will have to do is press any key on the keyboard to set up the system for data acquisition. a. Somewhere near the computer on the floor, you will find a piece of tape. Position the front edge (the edge with the camera on it) of the black base of the cart over that tape. Somewhere near the black box, you will also find a piece of tape. Position the front edge of the black box over the tape. These positions were used to calibrate the software so that it will give the x and y coordinates in centimeters. The two pieces of apparatus must be in these positions to give accurate results. b. Insert the floppy disk labeled "APPLE STROBE" into disk drive 1 and turn on the apparatus. If you have never used a floppy disk before, ask your TA to show you the correct way to insert the disk. The needed software will automatically load itself. c. When the monitors have had time to warm up, aim the camera so that the black box is centered in the display on the left hand monitor. Imagine a straight line drawn from the camera lens to the back panel of the black box. Check to see that this line meets the back panel of the black box at right angles in both the x and y directions. In other words, check to see that the line of sight of the camera is perpendicular to the plane in which the ball is to travel. If it is not, adjust the orientation of either the camera cart or of the black box until this condition is met. d. By now the software should have loaded and some instructions should have appeared on

the right hand monitor. Press any key. A rectangular box should appear on the right hand monitor. Using the two game paddles to the right of the computer, position the rectangle so that the ball will pass through its middle near the top of the ball's upward trajectory after it bounces off the floor. Press the space bar. The right monitor screen should become whitish. Drop the ball through the brass cylinder. If the alignment of the triggering rectangle was correct, you should hear the computer start to click rapidly. If this does not happen, reposition the trigger rectangle and try again. (Note: to reposition the trigger box, it will be neccessary to turn the apparatus off and then on again.) If the apparatus continues not to trigger, see your TA. e. When the data acquisition is complete, a picture of the path of the ball will appear on the left hand monitor and will be sent to the printer. Next, the positions of the ball will be computed and displayed as crosses on the left hand monitor. Once the display is complete, this graphic representation of the trajectory will also be sent to the printer. Finally, the x and y coordinates of each image of the ball will be printed out in tabular form on the printer. When all that is complete, tear off your data and let the next group take some data. 2. Data Analysis. For those of you with little or no knowledge of the use of the Kiewit system, we will begin with instructions on how to sign on, call up and run the appropriate program. These instructions do assume (1) a basic knowledge of how to use a Macintosh and (2) that the Macintosh that you are using is connected to the Appletalk network. If one or both of these assumptions are false, see your TA for help. In all that follows, keys that you need to press on the keyboard or commands that you need to enter are given in capital letters. After each Kiewit command or response that you type in, press RETURN. a. Turn on the Macintosh and insert a disk containing the DarTerminal communications application. Start the DarTerminal application by double clicking on the DarTerminal icon or by using the OPEN option from the file menu. When the sign-on dialog box appears, type "D1". Kiewit will respond with a series of messages and then it will ask for your user number. b. Enter your user number followed by a comma and your password. Your user number is a six character string consisting of five numbers and a letter. It is given on your student ID card. Generally your password will be your birthdate. For example, a student with an ID number of D98700 and a birthdate of June 7, 1956, would type "D98700,06-07-56". If this is entered properly, Kiewit will respond with the word "Ready". c. All of the data analysis programs that you will use in P3 are stored in a subcatalog called P3 in the public library called PHYSLIB***. The general command format for calling up a program from this subcatalog is OLD PHYSLIB***:P3:. For this lab the program's filename is PROJ. Therefore, for this lab the data analysis program is called up by typing OLD PHYSLIB***:P3:PROJ . (Note: The dot after PROJ is a period and

should not be included when typing the OLD command into the computer.) Call up PROJ. d. To run the program, type RUN. The program will prompt you when it needs information or wants you to select an output option. This program is compiled. The documented source code can be examined by calling up and listing the file PROJ.DOC in PHYSLIB***:P3. Using the instructions above, call up and run PROJ. Enter the data taken from either the Polaroid camera picture or from the APPLE STROBE apparatus. Obtain a hard copy of all of the tables, all of the graphs and the equations obtained through the use of the curve fitting routine. (Note: all graphics routines must be done on the Macintosh.) Your TA will show you how to get hard copy from a Macintosh located in the lab.

Part IV - Optional Replace the golf ball with a ping-pong ball or a styrofoam ball and take the data with and/or without a resistive air flow. Your TA will show you how to set up either data acquisition system to do this. Repeat the analysis done above. How do the x and y components of the velocity and acceleration compare to those obtained for the golf ball? (You may analyze this system using either the graphical or mathematical approach, whichever you believe to be easier and more precise.)

Lab Report You should use the lab notebook format described by your TA in his/her comments at the beginning of the lab session. Your lab report should include the answers to all of the questions asked in the introduction or procedure, all raw and derived data, and an estimate of the magnitude and sources of error in any data recorded. When answering any question or when giving any comparison or explanation, always refer to specific data to support your statements. For this lab, also include the following: 1. a copy of the position and velocity vector diagrams; 2. the computations of the magnitudes of the instantaneous velocities and average accelerations obtained from the appropriate vector diagrams; 3. the table containing the magnitudes of the x and y components of the position vectors, the magnitudes of the x and y components of the instantaneous velocity vectors and magnitudes of the x and y components of the average accelerations; show an example of how these components were calculated; 4. all of the hard copy - tables, graphs and curve fitting equations - obtained in step 2 part III;

5. a comparision of the results of the experiment using the Polaroid camera-strobe system and using the computer system; 6. any analysis you did on the optional part; and 7. the answers to these questions: a. Is the x-component of the velocity constant within experimental uncertainty? If not, suggest a reason for this discrepancy from theory. b. Is the magnitude of the acceleration constant within experimental uncertainty? If not, suggest a reason for this discrepancy from theory. c. Is the direction of the acceleration vector constant? What is its direction? Why does the acceleration have the direction it does? d. What is the average numerical value of the acceleration of the golf ball in the y direction? How does it compare with the accepted value? What is the accepted value? e. How do the results of the graphical and mathematical approaches compare? f. Explain the physical significance of each of the coefficients in each of the equations obtained using the curve fitting routine. Are they the values you would expect them to be? If not, what values would you expect and how would explain the discrepancy? g. Do the graphs obtained with PROJ have the shapes you would expect them to have? If not, explain any discrepancies. For each graph tell whether or not there is any physical significance to the slope of the curve or the area under the curve. If there is a physical significance, compute the slope or the area under the curve and compare the value you get with what you would expect it to be. h. Which data acquisition set do you think is the more the more accurate? Why?