Lab 5 - Projectile Motion

Transcription

1 Lab 5 Projectile Motion L5-1 Name Date Partners L05-1 L05-1 Name Date Partners Lab 5 - Projectile Motion Name Date Partners LAB 5 - PROJECTILE MOTION LAB 5 - PROJECTILE MOTION A famous illustration from Newton s Principia showing the A famous illustration relationship from Newton s between projectile Principiamotion showing the orbital relationship motion between projectile A famous illustration motion from Newton s orbital motion Principia showing the relationship between projectile motion orbital motion OVERVIEW OVERVIEW We learned in our study of kinematics that two-dimensional motion is a straightforward extension of one-dimensional motion. Projectile motion under the influence of gravity is a We subject learned with in which our study we are of of all kinematics familiar. that that We two-dimensional learn to shoot motion basketballs motion is a is straightforward in a an straightforward arc to swish ex- extension through of the of basket one-dimensional or to bounce motion. motion. off Projectile the Projectile backboard. motion motion under We learn the under influence how the influence to of lob gravity in volleyball of isgravity a subject is a subject tennis. with which with These we which are all examples we familiar. are all of We familiar. projectile learn to shoot We motion. basketballs learn The to shoot force in anbasketballs arc of to gravity swish through acts in an in arc theto basket vertical swish through direction, or to bounce the basket offair the resistance backboard. or to bounce acts Wein off learn both the how backboard. the tohorizontal lob in volleyball We learn vertical how tennis. to directions, lob These in volleyball are but examples we often tennis. neglect of projectile air These resistance motion. are examples The for small forceof of objects. projectile gravity acts motion. in the vertical The force direction, of gravity airacts resistance in the acts vertical in direction, both the horizontal air resistance vertical acts directions, in both the but we horizontal often neglect vertical air resistance directions, for small but objects. we often neglect In In this air experiment resistance wefor will small will explore objects. explore how the how motion the depends motion y depends upon theupon body sthe initial body s velocity initial elevation velocity angle. elevation Consider In this experiment we will explore how the motion angle. a bodyconsider with an initial a body speed with v an initial speed v 0 at angle depends upon the body s initial 0 at angle α with respect to the y velocity elevation V 0 α horizontal with respect axis. We to the analyze horizontal the body s axis. motion We in analyze two independentmotion coordinates x two (horizontal) independent y (vertical). coordinates [As we x 0y angle. Consider a body with an initial speed v the 0 at angle V body s α V 0 α (horizontal) arewith free respect to choose to y this the (vertical). origin horizontal of[as ouraxis. we coordinate We analyze are free system, the to choose we V x body s motion in two independent coordinates x 0y choose x = 0 y = 0 when t = 0 so as to simplify our calculations.] Figure 5.1 then shows that the components of the V 0x this origin of our coordinate system, we choose x = 0 α (horizontal) y (vertical). [As we are free to choose x y = 0 when t = 0 so as to simplify our calculations.] V Figure 0x 1. this velocity origin vector of our along coordinate the x system, y axes are we respectively: choose x = 0 Figure 1 then shows that the components of the velocity y = 0 when t = 0 so as to simplify our calculations.] Figure 1. vector along the v x y axes are respectively: Figure 1 then shows 0x = v 0 cosα v that the components 0y = v 0 sinα (5.1) Figure 5.1: of the velocity vector along the x y v0 axes x = v are 0 cosα respectively: v0 y = v 0 sinα (1) If we neglect air resistance, the only force affecting the motion of the object is gravity, which near the Earth s surface acts purely in the vertical direction If we neglect air resistance, vthe 0 x = only v 0 cosα force affecting v0 y = the v 0 sin motion α of object is gravity, (1) which near the Earth s surface acts purely in the vertical direction ( F ˆ g = mg = mgy). If (Fwe g = neglect mg = mgŷ). air resistance, There nothe force only at all force in the affecting horizontal the direction. motion of the object is gravity, which There is near no force the Earth s at all in surface the horizontal acts purely direction. in the vertical direction ( F ˆ g = mg = mgy). There is no force at all in the horizontal direction. Universityof of Virginia Physics Department PHYS 1429, Spring PHYS 1429, Spring 2011

2 L5-2 Lab 5 Projectile Motion Since there is no horizontally applied force, there is no acceleration in the horizontal direction; hence the x component of the velocity will remain unchanged forever. The horizontal position of the body is then described by the expression for constant velocity: L05-2 Lab 5 Projectile Motion x(t) = v 0x t = (v 0 cosα)t (5.2) Since there is no horizontally applied force, there is no acceleration in the horizontal The force in the y (vertical) direction is gravitation (F y = mg). Since F = ma, a y = g. Integrating horizontal this with position respect of the to time body yields is then thedescribed vertical component by the expression of the velocity: for constant velocity: direction; hence the x component of the velocity will remain unchanged forever. The x( t) = v0 xt = ( v0 cosα )t (2) v y (t) = v 0y gt = v 0 sinα gt (5.3) The force in the y (vertical) direction is gravitation ( Fy = mg). Since F = ma, Theavertical component of the position can be obtained by integrating Eq. (5.3) with respect to y = g. Integrating this with respect to time yields the vertical component of the time t, yielding the result: velocity: v y ( t) = v0 y(t) = v 0y t 1 y gt = v0 sinα gt 2 gt2 = (v 0 sinα)t 1 (3) The vertical component of the position can be obtained 2 gt2 by integrating Eq. (3) with (5.4) respect to time t, yielding the result: We can use these equations to determine the 1 actual trajectory of the body in terms of the x y variables, with no explicit reference y( t) = v0toy tthe time gt = t. ( vthe 0 sinequation α ) t gt has a simple parabolic form (4) 2 2 We can use these equations to determine the ( actual x y = 4h 1 R)( x trajectory ) of the body in terms of the x y variables, with no explicit reference to the time t. The equation has a simple (5.5) parabolic form R where! x "! x " y = 4h# $# 1 $ (5) % R R = v2 0 sin(2α) &% R & where 2 (5.6) v 0 sin(2 g α) R = (6) g v2 2 0 sin2 2 v0 sin ((α) h = (7) 2 (5.7) 2g g R isr called is called the range the range is the is horizontal the horizontal distance distance that that the projectile the projectile travels travels on level on level ground. h is ground. the maximum h is the height maximum that the height projectile that the can projectile reach. Acan plot reach. of y/h A versus plot of x/r y/ h isversus shown in Figure x/ R 5.2 is (below). shown in Figure 2 (below) y /h x /R PHYS 1429, Spring 2011 Figure 2 Figure 5.2:

3 Lab 5 Projectile Motion L5-3 Although we won t measure it in this experiment, we can similarly show that the time of flight (T) is given by T = 2v 0 g INVESTIGATION 1: Measurement of Range You will need the following materials: projectile motion apparatus ling pad optical bench digital photogate timer meter stick masking tape level 3 m tape measure pressure-sensitive paper sin (α) (5.8) Before doing anything, listen while the TA goes through a description of the apparatus how to level it. It is crucial for the apparatus to be level or you will not obtain good results. The wooden table is heavy can easily slip during the measurements. After the brief TA discussion, familiarize yourself with the apparatus. The central feature of the setup is a spring-loaded gun mounted on a plate marked in degrees, as shown in Figure 5.3. The plate can be rotated around a horizontal axis to set the initial direction of motion: a desired angle can be set by loosening the knurled screw attached to the plate, aligning the required angle mark with the horizontal mark on the st then tightening the screw. Figure 5.3: The initial speed of the projectile (a steel ball) can be set to three different values by drawing back the spring-loaded firing pin until it is locked in one of the three preset positions. Notice the cable-release trigger make sure that its plunger is fully withdrawn (this is achieved by

4 L5-4 Lab 5 Projectile Motion pressing on the small ring near the end of the trigger cable), then try drawing back the firing pin listen to the three distinct clicks corresponding to the three positions. Unfortunately, sometimes the cable-release becomes disconnected, something you want to avoid, because it is difficult to reconnect it! Press the trigger s plunger the pin will fire. Now you can practice launching the projectile: try different angles different velocities to get a feeling of which angular range will l the projectile onto the table. When doing your measurements, record the ling spot of the ball on the table by means of a pressure sensitive paper taped to the board. Do not tape the paper in position until you are ready to start your measurements, because it is expensive, we can t afford to waste any. 1. Now you will want to verify that the apparatus is level that the ling table is at the same height as the pivot point (which is where we believe the ball also leaves the spring). First, place the level on the back of the gun apparatus, if needed, use the two screws holding up the apparatus to level the gun apparatus. 2. Use a meter stick the level to make sure that the table top surface is at the same height as the pivot point of the angle mark (the timing apparatus can easily be pulled out to see where the pivot point actually is) that the table is level. Ask your TA for assistance if this procedure is not clear. IT IS CRUCIAL THAT YOU HAVE DONE THIS CORRECTLY! Activity 1-1: Determining the Initial Speed As the final preparation, we want to measure the initial speed of the projectile. As it leaves the gun, the projectile crosses the light beam of two light bulbs (actually light emitting diodes or LEDs), shining onto two photocells, which are connected to a timer unit. When in pulse mode, the timer will start counting when the first light beam is interrupted, it will keep counting until the second beam is obscured. 1. Set the timer is to 0.1 ms resolution. NOTE: The timer will STILL record the time in seconds (but with 0.1 ms precision). 2. Set the launcher angle to 0 (horizontally). This will give you the best possible time resolution. 3. Withdraw the firing pin to one of the three positions place the ball into the gun. If the timer is running, press the RESET button. Launch the ball use the Styrofoam cup held slightly angled to catch the ball. The timer will record the time (in seconds!) it took the projectile to go from the first to the second beam. Knowing the distance d between the two beams (it is marked on the apparatus should be 20 mm) you can determine the average speed of the ball as it travels between the lights. Without more knowledge, we assume the uncertainty in d (σ d )is one half of the least significant digit: d = 20 ± 0.5 mm.

5 Lab 5 Projectile Motion L5-5 Question 1-1: Discuss among your group why we use the procedure suggested (with the launcher horizontal) to determine the initial speed. [Hint: Consider the forces on the ball.] Write your conclusions here. 4. Open Excel, find the TeachingLabs directory, open 1429, open the file L05.Projectile Motion. Note at the bottom there are three tabs (called sheets): Table 1-1, Table 1-2, Table 1-3 for the three different measurements you will be making. Make sure you are looking at Table 1-1 (if not, click on the proper tab near the bottom of the screen). You will be entering your data into the yellow background areas. 5. Measure the time (t) for at least four trials for each click setting. It is important to work together. One student should enter the data into the Excel file while the other students take the data. Take turns during the lab doing each. Excel will perform the calculations for you, but it is important that you underst what is being calculated. You can click on a given cell in Excel look above the column headings to see the function being calculated. For example in cell F5, you will see the following: =AVERAGE(B5:E5)*1000. This means that we are finding the average time in seconds between cells B5 E5 then converting the time from seconds to milliseconds (ms) by multiplying by You should click on each cell in a row to underst what is being calculated. For each setting the Excel file calculates the average time (t avg = t, Column F), the stard deviation σ t ( Column G), the stard error in the mean (σ t) (Column H), one-half of the least significant digit in the timer (Column I). Your uncertainty in t (σ t), will be the larger of the stard error in the mean or one-half of the least significant digit (0.05 ms) [ that is in the Column J headed]. In Column M we calculate the initial speed, v 0 = d/ t. In order to calculate the uncertainty in the velocity we need to use the propagation of errors methodology (see Appendix D), because we have uncertainties in both distance d time t. Therefore in Columns K L we find the fractional errors in the time distance, respectively. These values are used to determine the uncertainty in velocity given in Column N. Finally in Column O we find the fractional uncertainty in the velocity, which we will use later to determine the uncertainty in a calculation of the range. 6. Print out one copy of Table 1-1 for your group report. Question 1-2: Look at the function being calculated in cell N6 of the Excel file. What is being calculated? Explain the equation being used.

6 L5-6 Lab 5 Projectile Motion L05-6 Activity 1-2: Range as a Function of Initial Angle Lab 5 Projectile Motion Question For this activity 1-3: Write you will down execute two reasons a set of launches why it might for fivebe different best to values use the ofhighest the launch speed angle: to 15 take, 30 these, 45 data., 60, 75. Take these range data with the highest initial speed. Question 1-3: Write down two reasons why it might be best to use the highest speed to take these data. Prediction 1-1: Consider a coordinate system where the x y coordinates are both Prediction zero when 1-1: the Consider ball passes a coordinate through system the pivot where the ls x on ythe coordinates top of the areling both board zero when at that thesame ball passes level through ( y = 0). the The pivotball ls then on starts the top at of y the = 0ling reaches board at y that = 0 again same when level ling (y = 0). on The the ball board. then starts The at range y = 0 will simply reaches be y = the 0 again x coordinate when ling where on it hits. the board. Draw The your range quantitative will simply prediction be the x coordinate for the trajectory where it hits. on Draw the following your quantitative graph for prediction for the trajectory on the following graph for an initial angle of 45. an initial angle of 45. Vertical displacement ( y ) Lab 5 Projectile Motion L Horizontal displacement (x ) Prediction Prediction 1-2: 1-2: Mark your Mark quantitative your quantitative predictions for predictions the range R for (see the Equation range 6) for R (see each Equation of the launch 6) for angles each of on the the following launch angles graph. on [Note: the following These will be graph. points.] [Note: These will be points.] Range R (m) Initial Angle a (degrees) 1. Remove the photogate so that you can more easily measure the distances. Turn off the photogate s power. PHYS 1429, Spring For each launch, the range of the projectile will be recorded by a dot on the pressuresensitive paper. You will notice that even if you keep the initial speed inclination fixed, the ling positions will be spread over several millimeters. The actual range

7 Lab 5 Projectile Motion L Remove the photogate so that you can more easily measure the distances. Turn off the photogate s power. 2. For each launch, the range of the projectile will be recorded by a dot on the pressuresensitive paper. You will notice that even if you keep the initial speed inclination fixed, the ling positions will be spread over several millimeters. The actual range you will use for plotting your data will be the average of the measured points. 3. To avoid confusions due to spurious marks on the paper caused by bounces of the projectile, it is advisable, first, to label each mark as the ball makes it, second, to perform the measurements with the longest ranges first. You might, for example, circle each mark to distinguish it from the newer ones. Do not take data until instructed to do so! Ask your TA for permission to proceed. Question 1-4: Which angle corresponds to the longest range? Explain. That should be your first angle. 4. Click on the tab for the Table 1-2 sheet in Excel. Record your initial speed (also enter it into cell I15): v 0 : 5. Now take your data for this v 0 insert your values into Table 1-2 of the Excel file. Your TA may have some suggestions as to how to most easily measure the individual ranges. Make your measurements in centimeters (we will have Excel make the conversion to meters in rows 11 12). Remember to work together. One student can input the data into Excel, while the other students perform the measurements. Move quickly along to take these data. Question 1-5: Think about your measurement of the range. How accurate do you think that measurement is? At the very most it can be no better than you can read the measurement device (meter stick, tape measure, whatever...) Write below what you think this limiting measurement value is enter it into Table 1-2 into row 14 of the Excel file. Measurement limit: Discuss how you determined this value: 6. For each angle, Excel calculates the average range, R, the stard deviation,σ R, the stard error in the mean σ R. Your uncertainty in R will be the larger of the

8 L5-8 Lab 5 Projectile Motion stard error in the mean or the value of your measurement limit from Question1-5. The uncertainty in R is given in Row 15. Question 1-6: Did you expect any angles to have the same range? Does it appear from your data that the ranges for any of the angles are about the same? Discuss. 7. Look carefully at Table 1-2 in your Excel file to see what is being calculated. For example, the theoretical range is calculated in Column J depending on the angles shown in Column H the velocity given in Cell I15. The average experimental range ( R) is determined in Column K. The graph shows the theoretical range R(α) for an initial speed v 0 is plotted versussin(2α). Your experimental measurements. [The theory calculation depends on the velocity measurement you made in the previous activity.] Question 1-7: Explain what is being calculated in cell D13 of Table 1-2 in Excel. Write down the equation used in this cell below. What equation on the summary page of Appendix D are we determining? 8. Print out include the graph with your report. 9. Try to add error bars by h to the measured R data on your graph. These should be short lines of your maximum uncertainty in R that extend above below the data point. Often these lines have T like ends. See the next question if you have a problem. Question 1-8: Are your uncertainties in R smaller than the size of your data point markers? If so, is that good? Would taking more statistics help? Question 1-9: How well do your measurements agree with your predictions? Are the deviations consistent with the accuracy you expect of your measurements? Explain why or why not. Discuss possible sources of systematic errors that may be present.

9 Lab 5 Projectile Motion L5-9 Activity 1-3: Range as a Function of Initial Speed 1. Click on the tab for the Table 1-3 sheet in Excel. 2. Now you want to choose one suitable angle measure the range R for each of the three initial speeds that you determined earlier. Write down your chosen angle enter it into cell E2: Launch angle: Question 1-10: What criteria did you use for choosing your launch angle? Would it better to use a shallow or high angle? Discuss. 3. Take five readings for each of the three initial speeds. Insert your results into Table 1-3 of the Excel file. Enter the data in centimeters (Excel will once again do the conversions for us). 4. The Excel file will calculate the average range data uncertainties as before. This information is here so you will know what we are doing in the lab. 5. Excel has plotted your range data versus v 2. The plot also indicates the theoretical range calculated using the velocities you determined in Activity 1-1 show in Table 1-1 of the Excel file. Because there are uncertainties in the velocity measurements (see Columns N O of Table 1-1), there are also uncertainties in the calculation of our theoretical ranges. The error bars on the theory calculations in Table 1-3 are reflective of the velocity uncertainties. We do the range theoretical calculations in Row 13 determine the uncertainty in Row 14. Question 1-11: Examine Cell D14 in Table 1-3 of the Excel file. Write down the equation in the cell below. What is being calculated? Explain each term in the equation. 6. Print out your graph, add error bars to your range data (if larger than the data points), attach the graph to your report. Question 1-12: Do your experimental data agree reasonably well with your theoretical predictions? How many of your data points overlap your theoretical calculation? Is this what you expect? Do you believe your errors are statistical or systematic? Explain.

10 L5-10 Lab 5 Projectile Motion Please clean up your lab area