PROJECTILE MOTION PURPOSE

Transcription

1 PURPOSE The purpose of this experiment is to study the motion of an object in two dimensions. The motion of the projectile is analyzed using Newton's laws of motion. During the motion of the projectile, the earth's gravitational force acts vertically downward while a constant velocity is maintained in the horizontal direction. THEORY Newton's second law expresses the relationship between the acceleration that a body can experience at any given instant during its motion and the resultant force acting on the body at that instant. This relation is written as: F = m a (1) where m is the mass of the object. Thus, the acceleration is a = F / m (2) This is a vector equation. The vector is resolved into components which lie along the axes of a suitable coordinate system allowing Equation 2 to be represented by a set of component equations, each one of which can be solved to describe the object's motion along the respective coordinate axis. In order to study the motion of an object, we establish a suitable coordinate system (reference frame) to which the object's motion can be referenced. This coordinate system is shown in Figure 1. Figure 1 III-1

2 The motion of the object shown in Figure 1 can be described completely by the two component equations obtained from eqn. (2). The motion of the object in the horizontal (x) direction must satisfy the equation: a x = F x / m (3a) while motion in the vertical (y) direction must satisfy the equation: a y = F y / m (3b) Equations 3a and 3b determine the components of the acceleration. From these two components, one can integrate to obtain the equations x = x(t) and y = y(t), which describe the motion of the object along the x and y axes as a function of time. For the motion of a projectile in the earth's gravitational field, the components of the acceleration (Equations 3a and 3b) are: a x = 0 a y = - g (4a) (4b) where g = gravitational acceleration. The gravitational force is the only force that acts if we neglect air resistance. Since the projectile is a comparatively massive steel ball that has small cross-sectional area, this is a reasonable approximation. Equation (4a) governs the motion in the x direction. Since no forces act in the x direction, the x component of acceleration is zero. This implies that the projectile maintains a constant velocity in the x direction. Equation 4b governs the motion in the y direction. Motion in this direction corresponds to that of a freely falling body as studied in the Freefall experiment. The coordinates of the projectile at any time, t, are given by: x(t) = x o + v ox t y(t) = y o + v oy t - (g/2) t 2 (5a) (5b) Here (x o, y o ) and (v ox, v oy ) are the initial values (t=0) of the x and y components of the position and velocity, respectively. At any given point in the path of the projectile, the velocity vector can be obtained from: v = v x i + v y j (6) where v x = v ox, v y = v oy - g t, and III-2

3 v = ( v x 2 + v y 2 ) 1/2 (7) The trajectory of the projectile is obtained by eliminating the time, t, from eqns. (5a) and (5b) giving: y = y o + (v oy /v ox ) (x-x o ) - (1/2) (g/v ox 2 ) (x-x o ) 2 (8) This is a parabolic equation that is valid only if air resistance is ignored. In this experiment the range of travel, R, is the horizontal distance covered by the object at the point where it crosses the horizontal axis in Figure 1. EXPERIMENTAL PROCEDURE Strobe photography is used to observe the motion of an object in flight and to provide a permanent record of its motion in the form of a photograph. The principle of operation is quite simple and basic to a number of devices that use the principle of remote sensing. A strobe unit produces a very bright flash of light of short duration (the pulse duration is several millionths of a second) with the bursts of light occurring at regular time intervals. The flash of light is reflected by the moving object into a camera, the shutter of which is held open during the flight, producing a permanent record on film of the instantaneous location of the object at successive time intervals. In order to locate the position of the object, photographs are made against a ruled background or "grid". The grid is constructed of specially reflecting nylon cord stretched across a frame at 2 cm intervals. The strobe unit consists of a flash tube and a timing source which produces high intensity flashes of light at a rate of 60 Hz which we set by synchronizing to the AC line frequency. A video camera attached to a frame grabber is used to acquire a multiple-exposure image. This image is analyzed with the computer to find position as a function of time. The projectile is a steel ball launched horizontally by a spring gun so that the initial velocity, v oy, along the y-axis is zero. Each group will prepare a digitized image of the motion of the projectile. Record the mass of the steel ball and the strobe frequency in Data Table 1. Analysis of the image will proceed as described in the following section. VIDEO GRABBER SOFTWARE (Step-by-Step User's Guide) DESCRIPTION: The Video Grabber software allows you to A) capture a video image from attached hardware, and B) define a coordinate system on the image (which consists of setting up the coordinate axes and a measuring scale) with which you can select data points to be saved to an Excel-compatible file. STEP 0: To ensure the image is clear enough for you to identify points, you must set the Monitors Control Panel (available under the Apple menu) to 256 Grays. The black and white setting is unacceptable. III-3

4 STEP 1: After fixing the desired image on the video monitor, choose Transfer (Command-T) from the Grabber menu to capture the image. STEP 2: It is sometimes easier to pick out data points by inverting the image. To do this, choose Invert Image from the Grabber menu.. If it is still difficult to select the desired points, see the lab instructor about reconfiguring the video hardware, or adjusting the position of the strobe lamps. (Do not attempt this yourself.) STEP 3: Next select the coordinate axes. The points (0,0) and (0,100) on the illuminated grid have been marked with white dots to enable you to distinguish them in the captured image. Choose Select Axes (Command-2) from the Grabber menu and drag the pointer from the (0,0) dot to the (0,100) dot, taking care to be as precise as possible in positioning the pointer. STEP 4: Next, select the scale. To do this, you need to identify 2 points in the image and tell the program how far apart they are. The two points (0,0) and (0,100) are 100cm apart, so use these. Choose Select Scale (Command-1) from the Grabber menu and drag the pointer from the (0,0) dot to the (0,100) dot in the image. Again, try to position the cursor accurately. STEP 5: Select Choose Points from the Grabber menu and use the pointer to click on data points. These points will be stored as ordered pairs of numbers, based on our choice of axes and scale. STEP 6: Save the points by choosing Save Points As... from the File menu. Rename the file and open it with MS Excel upon quitting Video Grabber. ***OPTIONAL NOTE: Since selecting axes, scale, and points can be done without the hardware, and since other students are waiting in line to use the setup, it's best to save the image to disk and transfer to one of the machines outside. To do this, select Save Image As... in the File menu, give it a name, and reopen the file using Video Grabber on one of the laboratory microcomputers. DATA ANALYSIS Analysis of horizontal and vertical equations of motion and range Perform a linear fit for x(t) using the data recorded in Table 1. Perform parabolic (2 nd order polynomial) fits for y(t) and y(x) using the data recorded in Table 1. Record the appropriate coefficients in Table 2 with units. Calculate the Range. III-4

5 NAME Sec/Group Date DATA TABLE 1 Mass of steel ball kg Strobe frequency Hz Time interval between successive images s Strobe image # x-coordinate (m) y-coordinate (m) elapsed time (s) III-5

6 DATA TABLE 2: The Equations of Motion Horizontal Motion (x vs. t) x = C 1 + C 2 t C 1 = ( ) C 2 = ( ) v ox = m/s a x = m/s 2 Vertical Motion (y vs. t) y = D 1 + D 2 t+ D 3 t 2 D 1 = ( ) D 2 = ( ) D 3 = ( ) v oy = m/s a y = m/s2 Trajectory (y vs. x) y = F 1 + F 2 x + F 3 x 2 F 1 = ( ) F 2 = ( ) F 3 = ( ) Range Calculated Measured (from Data Table 1) III-6