Free Fall and Projectile Motion

[International Campus] Free Fall and Projectile Motion Objective Investigate the motions of a freely falling body and a projectile under the influence of gravity. Find the acceleration due to gravity. Theory ----------------------------- Reference -------------------------- Young & Freedman, University Physics (14 th ed.), Pearson, 2016 2.1 Displacement, Time, and Average Velocity (p.58~61) 2.2 Instantaneous Velocity (p.61~64) 2.3 Average and Instantaneous Acceleration (p.64~68) 2.4 Motion with Constant Acceleration (p.69~74) 2.5 Freely Falling Bodies (p.74~77) 3.3 Projectile Motion (p.99~106) ----------------------------------------------------------------------------- When a car moves from PP 1 to PP 2 in the +xx-direction as in figure 1, the xx-component of average velocity is the xxcomponent of displacement xx = xx 2 xx 1 divided by the time interval tt = tt 2 tt 1 during which the displacement occurs. Figure 2 is the xx-tt graph of the car s position as a function of time. The average velocity of the car equals the slope of the line pp 1 pp 2. But the average velocity during a time interval can t tell us how fast, or in what direction. To do this we need to know the instantaneous velocity, or the velocity at a specific instant of time or specific point along the path. The instantaneous velocity is the limit of the average velocity as the time interval approaches zero. On the xx-tt graph (Fig. 2), the instantaneous velocity at any point is equal to the slope of the tangent to the curve at that point. xx vv xx = lim tt 0 tt = dddd dddd (2) vv av-xx = xx 2 xx 1 = xx tt 2 tt 1 tt (1) Fig. 1 Positions of a car at two times during its run. Fig. 2 The position of a car as a function of time. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 1 / 15

Acceleration describes the rate of change of velocity with time. Suppose that at time tt 1 the object is at point PP 1 and has xx-component of velocity vv 1xx, and at a later time tt 2 it is at point PP 2 and has velocity vv 2xx. So the velocity changes by amount vv xx = vv 2xx vv 1xx during tt = tt 2 tt 1. We define the average acceleration of the object equals vv xx divided by tt. We can also derive an equation for the position xx as a function of time using Eqs. (1) and (5) when the xx-acceleration is constant. With the initial position xx 0 at time tt = 0 and the position xx at the later time tt, Eq. (1) becomes vv av-xx = xx xx 0 tt (6) aa av-xx = vv 2xx vv 1xx tt 2 tt 1 = vv xx tt We can now define instantaneous acceleration following the same procedure that we used to define instantaneous velocity. The instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero. (3) We can also get a second expression for vv av xx. In this case the average xx-velocity for the time interval from 0 to tt is simply the average of vv 0xx and vv xx. vv av xx = vv 0xx + vv xx 2 Substituting Eq. (5) into Eq. (7) yields (7) vv xx aa xx = lim tt 0 tt = ddvv xx dddd (4) vv av xx = 1 2 (vv 0xx + vv 0xx + aa xx tt) = vv 0xx + 1 2 aa xxtt (8) The simplest kind of accelerated motion is straight-line motion with constant acceleration. We can find the velocity vv xx of that motion using Eq. (3). We use vv 0xx for the xx-velocity at tt = 0; the xx-velocity at the later time tt is vv xx. Then Eq. (3) becomes aa xx = vv xx vv 0xx tt 0 or vv xx = vv 0xx + aa xx tt (5) We set Eq. (6) and Eq. (8) equal to each other and simplify xx = xx 0 + vv 0xx tt + 1 2 aa xxtt 2 (9) Figure 4 shows the graphs of Eq. (9) and Eq. (5). If there is zero xx-acceleration, the xx-tt graph is a straight line; if there is a constant xx-acceleration, the additional (1 2)aa xx tt 2 term curves the graph into a parabola (Fig. 4(a)). Also, if there is zero xx-acceleration, the vv xx -tt graph is a horizontal line; adding a constant xx-acceleration gives a slope to the vv xx -tt graph (Fig. 4(b)). Fig. 3 A vv xx -tt graph for an object moving with constant acceleration on xx-axis. Fig. 4 How a constant xx-acceleration affects a body s (a) xx-tt graph and (b) vv xx -tt graph. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 2 / 15

The most familiar example of motion with constant acceleration is a body falling under the influence of the earth s gravitational attraction. If the distance of the fall is small compared with the radius of the earth, and if the effects of the air can be neglected, all bodies fall with the same downward acceleration. This is called free fall. Fig. 5 shows successive images of falling bodies separated by equal time intervals. The red ball is dropped from rest. There are equal time intervals between images, so the average velocity of the ball between successive images is proportional to the distance between them. The increasing distances between images show that the velocity is continuously changing. Careful measurement shows that the acceleration of the freely falling ball is constant. This acceleration is called the acceleration due to gravity. We denote its magnitude with gg. The approximate value of gg near the earth s surface is 9.8 m s 2. A projectile, such as a thrown baseball, is any body that is given an initial velocity and then follows a path determined entirely by the effects of gravitational acceleration. (We neglect the effects of air resistance.) The motion of the yellow ball in Fig. 5 is two-dimensional. We will call the plane of motion the xxxx-coordinate plane, with the xx-axis horizontal and the yy-axis vertically upward. The xx-component of acceleration is zero, and the yy-component is constant and equal to gg. So we can analyze projectile motion as a combination of horizontal motion with constant velocity and vertical motion with constant acceleration. We can then express all the vector relationships for the position, velocity, and acceleration by separate equations aa xx = 0 aa yy = gg (10) Since both are constant, we can use Eqs. (5) and (9) directly. For example, as in Fig. 6, suppose that at time tt = 0 our projectile is at the point (xx 0, yy 0 ) = (0, 0) and that at this time its velocity components have the initial values vv 0xx = vv 0 cos αα 0 and vv 0yy = vv 0 sin αα 0. From Eqs. (5), (9) and (10), we find vv xx = vv 0xx + aa xx tt = vv 0 cos αα 0 (11) vv yy = vv 0yy + aa yy tt = vv 0 sin αα 0 gggg (12) xx = xx 0 + vv 0xx tt + 1 2 aa xxtt 2 = (vv 0 cos αα 0 )tt (13) yy = yy 0 + vv 0yy tt + 1 2 aa yytt 2 = (vv 0 sin αα 0 )tt 1 2 ggtt2 (14) The time tt 1 when the projectile hits the ground is 0 = (vv 0 sin αα 0 )tt 1 1 2 ggtt 1 2 or tt 1 = 2vv 0 sin αα 0 gg (15) The horizontal range RR is the value of xx at this time. Substituting equation (15) into equation (13) yields RR = (vv 0 cos αα 0 )tt 1 = 2vv 0 2 sin αα 0 cos αα 0 gg = vv 0 2 gg sin 2αα 0 (16) In Eq. (16), the maximum value of sin 2αα 0 is 1. This occurs when 2αα 0 = 90 or αα 0 = 45. This angle gives the maximum RR for a given initial speed if air resistance can be neglected. Fig. 5 The red ball is dropped from rest, and the yellow ball is simultaneously projected horizontally; successive images in this stroboscopic photograph are separated by equal time intervals. At any given time, both balls have the same yy-position, yy-velocity, and yy-acceleration, despite having different xx-positions and xx-velocities. Fig. 6 The trajectory of a projectile is a combination of horizontal motion with constant velocity and vertical motion with constant acceleration. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 3 / 15

Equipment 1. List Item(s) Qty. Description PC / Software Data Analysis: Capstone 1 Records, displays and analyzes data measured by various sensors. Interface 1 Data acquisition interface designed for use with various sensors, including power supplies which provide up to 15 watts of power. Photogate (Rod, Cable, and Screw included) 1 set Measures high-speed or short-duration events. A-shaped Base Multi-clamp 1 1 Provide stable support for experiment set-ups. Support Rod (600mm) 1 Provides stable support for experiment set-ups. Cushioned Baskets 1 Absorb shock on impact. Picket Fence set 1 set PF#1: The edges of the bands are 50mm apart. (Opaque: 20mm / Transparent: 30mm) PF#2: The edges of the bands are 40mm apart. (Opaque: 20mm / Transparent: 20mm) Projectile Launcher 1 Launches a ball at any angle from zero to ninety degrees with three range settings. Photogate Bracket 1 Mounts the Photogate on the Projectile Launcher. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 4 / 15

Item(s) Qty. Description Projectile (Green Plastic Ball) 1 Green plastic ball which can be loaded into the Projectile Launcher. Table Clamp 1 Clamps the Projectile Launcher to a lab table. Carbon Paper White Paper 1 1 Leaves a mark when a projectile ball hits it. (White Paper is not provided.) Box 1 Provides a horizontal surface so the projectile ball can reaches the same level as the muzzle of the Projectile Launcher. Measuring Tape 1 Measures distance. Vernier Caliper 1 Measures external, internal diameter or depth of an object with a precision to 0.05mm. 2. Details (1) Interface (2) Capstone: Data Acquisition and Analysis Software The 850 Universal Interface is a data acquisition interface designed for use with various sensors to measure physical quantities; position, velocity, acceleration, force, pressure, magnetic field, voltage, current, light intensity, temperature, etc. It also has built-in signal generator/power outputs which provide up to 15 watts DC or AC in a variety of waveforms such as sine, square, sawtooth, etc. The Capstone software records, displays and analyzes data measured by the sensor connected to the 850 interface. It also controls the built-in signal generator of the interface. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 5 / 15

(3) Photogate The Photogate sensor is an optical timing device used for very precise measurements of high-speed or short-duration events. It consists of a light source (infrared LED) and a light detector (photodiode). When an object moves through and blocks the infrared beam between the source and the detector, a signal is produced which can be detected by the interface. When the infrared beam is blocked, the output signal of the photogate becomes 0 and the LED lamp on the photogate goes on. When the beam is not blocked, the output signal becomes 1 and the LED goes off. This transition of signal can be used to calculate quantities such as the period of a pendulum, the velocity of an object, etc. The Projectile Launcher has three range settings so that balls can be launched with three different initial speeds. One or two Photogates can be attached to the Projectile Launcher using the Photogate Bracket so that the photogates can measure the initial speed of the ball. (5) Vernier Caliper (4) Projectile Launcher The Projectile Launcher is designed for projectile motion experiments. Balls can be launched from any angle from zero to ninety degrees measured from horizontal. The angle is easily adjusted using thumbscrews, and the built-in protractor and plumb-bob give and accurate way to measure the angle of inclination. 1 22 mm is to the immediate left of the zero on the vernier scale. Hence, the main scale reading is 22 mm. 2 Look closely for and alignment of the scale lines of the main scale and vernier scale. In the figure, the aligned (13 th ) line corresponds to 0.65 mm (= 0.05 13). 3 The final measurement is given by the sum of the two readings. This gives 22.65 mm (= 22 + 0.65). 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 6 / 15

Procedure Experiment 1. Free Fall (1) Set up equipment as below. The interface is automatically detected by Capstone. Click [Hardware Setup] in the [Tools] palette to configure the interface. (3) Set up Capstone software (3-1) Add a Photogate. Click the input port which you plugged the Photogate into and select [Photogate] from the list. (2) Turn on the interface and run Capstone software. The Photogate s icon will be added to the panel and [Timer Setup] icon will appear in the [Tools] palette. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 7 / 15

(3-2) Create a timer. 4 Make sure [Position] is checked. Click [Timer Setup] in the [Tools] palette and follow the steps below. 1 Select [Pre-Configured Timer] and click [Next]. 5 Enter a suitable value in the [Flag Spacing]. 2 Check [Photogate, Ch1] and click [Next]. You have two kinds of Picket Fences as follows. Spacing Description PF#1 0.05m Opaque 0.02m + Transp. 0.03m PF#2 0.04m Opaque 0.02m + Transp. 0.02m 3 Select [Picket Fence]. 6 Enter any name and finish the timer setup. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 8 / 15

(3-3) Create a graph display and a data table. Click and drag the [Graph] icon from the [Displays] palette into the workbook page. Click and drag the [Table] icon from the [Displays] palette into the workbook page. Select [Time(s)] for the first column and [Position(m)] for the second column. A graph display will appear. You now have two displays in the workbook page. (4) Begin recording data. You can select the measurement of each axis by clicking <Select Measurement>. Select [Time(s)] for the xx-axis and [Position(m)] of the Picket Fence for the yy-axis. Click [Record] in the [Controls] palette. Capstone begins recording all available data. [Record] button will toggle to [Stop]. You can stop data collection by clicking [Stop]. Collected data are stored in memory and appear in all displays. The data run is listed in the legend for each display. (You can delete any run by clicking [Delete Last Run] or dropdown menu in the [Controls] panel.) 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 9 / 15

(5) Drop a Picket Fence. Make sure the opaque bands of the Picket Fence block the infrared beam of the Photogate during they pass through the Photogate. Calculate the average speed at each interval, and plot vv-tt graph. 1~2 tt tt 1 + tt 2 2 vv xx 2 xx 1 tt 2 tt 1 2~3 3~4 tt 2 + tt 3 2 tt 3 + tt 4 2 xx 3 xx 2 tt 3 tt 2 xx 4 xx 3 tt 4 tt 3 (6) Stop data collection. Click the [Stop] button to stop data collection. (7) Analyze the data. Find the slope of vv-tt graph using the method of least squares (refer to the appendix). The acceleration of the Picket Fence is equal to the slope of the vv-tt graph. Time(s) Position(m) (8) Repeat experiments. 1 tt 1 xx 1 2 tt 2 xx 2 3 tt 3 xx 3 1 Repeat measurement with the same Picket Fence (more than three times). 1 st 2 nd 3 rd aa result aa AVG gg 9.8 m/s 2 Repeat measurement using the other Picket Fence. Change [Flag Spacing] parameter. See step (3-2)-5. Save the data file with a different name for each Picket Fence. If you change [Flag Spacing], all premeasured data will be recalculated. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 10 / 15

Experiment 2. Projectile Motion CAUTION NEVER LOOK INTO THE MUZZLE of the Projectile Launcher when it is loaded. Accidental shooting could cause blindness or serious loss of vision. (1) Set up equipment. Attach the Photogate Bracket to the Launcher and attach the Photogate to the Bracket. When the ball passes the Photogate, the initial speed of the ball is calculated as Speed = Moving distance (= diameter)of the ball Time interval the ball blocks the Photogate Thus, make sure the Photogate beam is exactly on the path of the center of the ball. Tighten the Bracket thumbscrew to secure the Photogate in place. (The Photogate could be slightly shifted on impact. Whenever you shoot the ball, make sure the Photogate is in position.) Using the table clamps, clamp the Launcher to the left end of the table. Connect the Photogate to the interface. Align the square nuts of the Bracket with the T-shaped slot on the bottom of the Launcher barrel and slide the nuts into the slot until the Photogate is as close to the muzzle as possible without blocking the Photogate beam. (2) Measure the diameter of the ball. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 11 / 15

(3) Add a Photogate. (5) Create and configure a digital meter. Click the port which you plugged the Photogate into and select [Photogate] from the list. Click and drag the [Digits] icon from the [Displays] palette into the workbook page. Select [Speed(m/s)] for <Select Measurement>. (4) Create and configure a timer. Click [Timer Setup] in the [Tools] palette and follow the steps below to create a timer. 3 Select [One Photogate (Single Flag)] for the type of timer. 4 Make sure [Speed] is checked. 5 Enter the measured diameter of the ball for [Flag Width]. (6) Load the launcher. Place the GREEN ball in the muzzle of the launcher. And then push the ball down the barrel with the ramrod until the trigger catches the MEDIUM RANGE setting of the piston. (The trigger will click into place.) You can use a different range setting, if required. If you cannot coke the piston due to a structural problem, pull and return the trigger while you are pushing the ramrod. NOTE The Launcher has three range settings. The reference lines of the ramrod show the positions of each setting. Remember, if you cock the piston with a ball in the piston, the piston is in the MEDIUM RANGE position when the first (left) line (not the middle line) of the ramrod reaches near the entrance of the muzzle, CAUTION NEVER LOOK INTO THE MUZZLE of the launcher when it is loaded. Accidental shooting can cause blindness or serious loss of vision. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 12 / 15

(7) Measure the initial speed of the ball. (9) Place white paper and carbon paper on the box. 1 Click [Record] in the [Controls] palette. 2 Shoot the ball by pulling straight up the trigger. 1 Fire a test shot to locate where the ball hits. 2 Put a piece of white paper on the box at this location. 3 Put a piece of carbon paper (carbon-side down) on top of the white paper. When the ball hits the carbon paper, it will leave a mark on the white paper underneath. 3 Click [Stop] and record the speed of the ball. 4 Repeat more than 3 times with the same range setting. 5 Find the average speed of the ball. The average represents the initial speed vv 0 of the projectile. 1 st 2 nd 3 rd vv result vv AVG vv 0 = (m/s) (10) Begin experiment. (8) Adjust the height and the angle of the Launcher. The height and angle of inclination above the horizontal is adjusted by loosening the two thumbscrews and rotating the Launcher barrel to the desired angle. Use the plumb bob and the protractor on the label to select the angle. Tighten both thumbscrews when the angle is set. 1 Fire three shots with the same range setting of step (7). 2 Carefully remove the carbon paper. 3 Use a measuring tape to measure the horizontal distance RR from the muzzle to the dots. 4 Repeat measuring RR for angles below. αα 0 25 35 45 55 65 RR 1 st 2 nd 3 rd AVG 5 Calculate the error between the theoretical distance and the actual average distance for all angles. RR = (vv 0 cos αα 0 )tt 1 = 2vv 0 2 sin αα 0 cos αα 0 gg = vv 0 2 gg sin 2αα 0 (16) Q What angle give the maximum range of RR? What pairs of angles have a common range? A 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 13 / 15

Appendix 1. Method of Least Squares Whenever we perform an experiment, we need to extract useful information from the collected data. We usually measure one variable under a variety of conditions with regard to a second variable. The method of least squares is a useful statistical technique to estimate a mathematical expression for the relationship between the two variables. The sum of all of the squares of the deviations is called the residual χχ 2, given by χχ 2 = yy ii ff(xx ii ) 2 (1) Suppose the collected data have a linear relationship, then the model ff(xx) can be expressed in the general form Through series of observation, we get a series of nn measurements of the pair (xx ii, yy ii ), where ii is an index that runs from 1 to nn. Suppose a certain mathematical model yy = ff(xx) best describes the relationship between xx ii and yy ii. Here, we have two value sets; yy ii is experimental value obtained through series of observation, and ff(xx ii ) is the function value calculated by the model yy = ff(xx). Consider the distances (or deviations) between yy ii and ff(xx ii ) as shown below. If those deviations are as small as possible, we can say the model yy = ff(xx) is a really good model for the data. The method of least squares attempts to minimize the square of the deviations. ff(xx) = aa + bbbb (2) Substituting Eq. (2) into Eq, (1) yields χχ 2 = yy ii (aa + bbxx ii ) 2 = (yy ii aa bbxx ii ) 2 (3) To find ff(xx) that best fits the data, the residual should be as small as possible, i.e. parameters aa and bb should be chosen so that they minimize χχ 2. Differentiating equation (3) with respect to aa and bb and setting these differentials equal to zero produces the following equations for the optimum values of the parameters. χχ 2 = 2 yy ii + 2bb xx ii + 2aaaa = 0 χχ 2 = 2 xx iiyy ii + 2aa xx ii + 2bb xx 2 ii = 0 (4) Finally we obtain aa = xx ii 2 ( yy ii ) ( xx ii )( xx ii yy ii ) nn xx ii 2 ( xx ii ) 2 bb = nn( xx iiyy ii ) ( xx ii )( yy ii ) nn xx ii 2 ( xx ii ) 2 (5) 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 14 / 15

Result & Discussion Your TA will inform you of the guidelines for writing the laboratory report during the lecture. End of LAB Checklist Please put your equipment in order as shown below. Delete your data files from your lab computer. Turn off the Computer and the Interface. Clamp the Projectile Launcher to the left end of the table. Keep the Photogate Bracket assembled to the Photogate. Keep the Carbon Paper in the plastic bag. 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA ( +82 32 749 3430) Page 15 / 15