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PASCO Specialty & Mfg. ME-6830, User's Manual

1. Total distance Alternate Method for Determining the Initial Speed of the Ball using photogates 1 Attach the photogate mounting bracket to the Launcher and attach two photogates to the bracket Check that the distance between the photogates is 0 10 m 10 cm 2 Plug the photogates into an interface or a timer 3 Adjust the angle of the Launcher to 90 degrees straight up 4 Puta plastic ball in the Projectile Launcher and use the ramrod to cock it at the medium range setting 5 Setup the data acquisition software or the timer to measure the time between the ball blocking the two photo gates 6 Shoot the ball three times and calculate the average of these times Record the data in the Table 5 2 7 Calculate the initial speed of the ball based on the 0 10 m distance between the photogates Record the value Table 5 2 Table 5 2 Initial Speed Using Photogates Trial Time 1 2 3 Average Time Initial Speed LEIHO 012 05375C 29 Projectile Launcher Exp 5 Conservation of Energy Part B Measure the Height of the Ball 1 Adjust the angle of the Launcher to 90 degrees straight up 2 Shoot the ball on the medium range setting several times and then measure the maximum height attained by the ball Record the maximum height in Table 5 3 3 Determine the mass of the ball and record it in Table 5 3 Analysis 1 Calculate the initial kinetic energy a
2. cos mR b cm where M is the mass of the pendulum and ball combined m is the mass of the ball g is the acceleration due to gravity Rm is the distance from the pivot point to the center of mass of the pendulum R is the distance from the pivot point to the ball O is the angle reached by the pendulum and is the moment of inertia of the pendulum with the ball in the catcher The value of J can be found by measuring the period of small oscillations of the pendulum ball combination and by using the equation Ba M8Rom 2 4n where T is the period Setup 1 Put the Projectile Launcher on the Ballistic Pendulum upright at the level of the ball catcher the low posi tion Make sure that the pendulum can hang vertically without touching the launcher 2 Clamp the Ballistic Pendulum base to a sturdy table Make sure that the clamp does not interfere with the pen dulum swing Procedure 1 Latch the pendulum at 90 so it is out of the way temporarily and then load the projectile launcher with the steel ball Allow the pendulum to hang freely and move the angle indicator to zero degrees 2 Fire the launcher and record the angle reached If you want to do the experiment with a lower or higher angle add to or remove mass from the pendulum Repeat these test measurements until you are satisfied with the mass of the pendulum 3 Once you have chosen the mass to use for your experiments remove the pendulum from the upri
3. Instruction Manual LEIH Oo 012 053750 Ballistic Pendulum Projectile Launcher ME 6830 ME 6831 Vertical upright Ballistic Pendulum Included Ramrod 2 D Collision Accessory Steel Balls 2 Plastic Balls 3 Not shown Safety Glasses 2 pair 800 772 8700 www pasco com The cover page shows the PASCO Ballistic Pendulum with the Short Range Projectile Launcher mounted on the vertical part of the base The Ballistic Pendulum is designed for traditional ballistic pendulum experiments and can also be used for projectile motion experiments and demonstrations When the Projectile Launcher is used for projectile motion experiments the launch angle at the upper launch position can vary from 0 to 90 degrees and the firing height is fixed for any launch angle The Launcher can also mounted in a horizontal position that is height adjustable The vertical base of the Ballistic Pendulum also has a dedicated position for the Launcher for Ballistic Pendulum experiments This manual contains copy ready experiments and demonstrations for the ballis tic pendulum and projectile launcher IZEIA ii Table of Contents EU sees rond A a a j oon 8682 4 oe ese SAGER ES US 1 A Aine Same sea deere a as ote de ow eee oe a 2 INMOGUCHION 44 oie cadine ceucdaeenaketicine E EEDE E EEA Aa ain ne we ERRE 2 General Operation of th
4. 6 Pick any x y data point from Table 4 1 Use the vertical distance y to calculate the time t Calculate the ini tial speed using this time and the horizontal distance x Record the results in Table 4 2 24 012 05375C LLEI O Model No ME 6830 Exp 4 Projectile Path 7 Calculate the percent difference between the two initial speeds that were found using the different methods Record the percent difference in Table 4 2 To calculate the percent difference let A be one of the initial speed values and let B be the other initial speed value A B 100 Er Data Table 4 2 Table 4 2 Compare Methods for Initial Speed Item Value Slope of graph Initial speed from slope Time of flight Initial speed from x y Percent difference Question 1 From the graph was the best fit line straight 2 What does the shape of the best fit line on the y versus x graph tell you about the relationship of y and x 3 Ifyou plotted a graph of y versus x how would the graph differ from the y versus x graph 4 What shape is the path of the projectile LEIHO 012 05375C 25 Projectile Launcher Exp 4 Projectile Path Notes 26 012 05375C IZEIA Model No ME 6830 Exp 5 Conservation of Energy Exp 5 Conservation of Energy Equipment Needed Item Item Projectile Launcher and plastic ball Plumb bob and string Meter stick or measuring tape Sticky tap
5. it is possible to demonstrate this effect Setup 1 Clamp the Ballistic Pendulum Projectile Launcher to a sturdy table with the launcher mounted in the lower position on the upright Procedure 1 Shoot the ball into the pendulum and record the angle reached 2 Remove the pendulum and reinstall it in the reversed position so that the ball opening is away from the launcher 3 Shoot the same ball again at the same launcher setting and note the angle reached Analysis e The collision between the ball and the pendulum is not perfectly elastic so kinetic energy is still not con served However the collision is more nearly elastic than the completely inelastic collision in step 1 There is a greater transfer of kinetic energy LEIHO 012 05375C 49 Projectile Launcher Exp 13 Demo Elastic and Inelastic Collisions 50 012 05375C 7 Wefe Model No ME 6830 Teacher s Guide Teacher s Guide Exp 1 Projectile Motion NOTE For best results make sure that the Launcher is securely clamped to a sturdy table Any movement of the Launcher will result in inconsistent data The muzzle speed of the Launcher tested was 6 5 m s on the long range setting To find the range at the chosen angle it is necessary to solve the quadratic equation given in the Theory section The solution is vo sin vysin t2g yg y 8g Analysis 1 The difference depended on the angle at which the Launcher was fir
6. In the system consisting of just the balls both balls are falling under the influence of gravity so Momentum is not conserved in the vertical direction However there is no net force in the horizontal plane if air resistance is ignored so momentum is conserved in the horizontal plane y th oe af b Before collision since all the momentum is in the direc a before collision b after collision tion of Ball 1 m it is convenient to define the x axis in this direction Momentum before the collision is Figure 6 1 Conservation of Momentum A Phefore MVox where vg is the initial speed of Ball 1 and x is the unit vector in the x direction The momenta of the two balls after the collision consists of both horizontal and vertical components so the momentum after the collision is A A Pafter M Viy MV3 X mv MyVoy where v v cos 8 v y v sin 01 vo vo cos 8 and Voy Vo sin 0 Since there is no momentum in the y direction before the collision there is zero net momentum in the y direction after the collision Therefore t MV MV Equating the momentum in the x direction before the collision to the momentum in the x direction after the colli sion gives m vg MV tm In a perfectly elastic collision kinetic energy is conserved as well as momentum Le ll MVG MV 5m7 2 Also when energy is conserved the paths of two balls of equal mass will be at right angles to each
7. between the muzzle of a Launcher and the place where the projectile hits given by X Volt where vg is the initial speed of the projectile as it leaves the muzzle and t is the time of flight The vertical position y of the projectile at time t is given by 2 Yy Yo 5 8 where yg is the initial height of the projectile and g is the acceleration due to gravity Solving the x equation for and substituting the expression in the y equation gives 1 x 2 2 y yoe ax b where a and b are constants The y equation y ax b describes a parabola Pre Lab Before the demonstration begins find the initial speed of the bal Use two photogates and a photogate mounting bracket with a PASCO Interface or Timer or shoot the ball horizontally and measure x and y Use y to calculate the time of flight See experiments 1 and 2 Setup 1 Clamp the Projectile Launcher to a sturdy table near one end of the table with the Launchers aimed away from the table so the balls will land on the floor 2 Adjust the angle of the Launcher to 0 so the balls will fire horizontally Procedure 1 Measure and record the initial height y of the ball at muzzle level 7 Wrefe 012 05375C a7 Projectile Launcher Exp 12 Demo Shooting Through Hoops 2 Calculate and record the horizontal and vertical positions of the ball each 1 10 second until the vertical posi tion is zero Table 10 1 X and y positions t
8. each dot of the inelastic collision shot Use the angle and the length of the lines for the shot to calculate the x component and the y component for each ball in the inelastic collision shot Record the values After the collision add the x momentum for Ball 1 and the x momentum for Ball 2 and record the result in Table 6 2 as Final x momentum Calculate the initial kinetic energy of Ball 1 and the sum of the kinetic energy of Ball 1 and Ball 2 after the collision Calculate the percent differences Table 6 2 Data for the Inelastic Collisions Item Value Item Value Percent difference Initial x momentum Final x momentum Ball 1 Ball 1 Ball 2 Final y momentum Final y momentum Ball 1 Ball 2 Initial kinetic energy Final kinetic energy Ball 1 Ball 1 Ball 2 Questions Was momentum conserved in the x direction for each type of collision Was momentum conserved in the y direction for each type of collision Was kinetic energy conserved for the elastic collision Was kinetic energy conserved for the inelastic collision For the elastic collision was the angle between the paths of the balls after the collision equal to 90 degrees as expected For the inelastic collision what was the angle between the paths of the balls after the collision Why is it less than 90 degrees 012 05375C LLEI O Model No ME 6830 Exp 7 Vary the Angle to Maximize the Height Exp 7
9. energy and substitute into it the equation for angular momentum m 1 2 KE al 0 ae gt V Figure 1 Ly Io ee KE 21 where is the moment of inertia of the pendulum ball combination and is the angular velocity immediately after the collision As you did previously solve this last equation for angular momentum L J2I KE This angular momentum is equal to the angular momentum of the ball before the collision as measured from the pendulum pivot point 2 Ly mR oO mR v where R is the distance from the pendulum pivot to the ball NOTE This radius is not in general equal to R which is the distance from the pivot point to the center of mass for the pendulum ball system These two angular momenta are equal to each other so mR 2IMgR 1 cos cm Solve for v v 2IMgR 1 cos Figure 2 oak ae mR Now you need to find J the moment of inertia of the pendulum and ball To do this start with the rotational equiv alent of Newton s Second Law t a where Tt is torque Z is moment of inertia and a is angular acceleration The force on the center of mass of the pen dulum is Mg and the component of force directed towards the center of the pendulum swing is F Mg sin 0 See Figure 2 LEIHO 012 05375C 7 Ballistic Pendulum Projectile Launcher Ballistic Pendulum Theory The torque on the pendu
10. has a stronger spring and is useful for large classroom demonstrations e Fixed Elevation Independent of Launch Angle The Projectile Launcher can pivot at the muzzle end so the elevation of the ball as it leaves the barrel does not change as the angle is varied The upright part of the Bal listic Pendulum base has three positions for mounting the Launcher At the top is a hole and curved slot for use when you want to change the launch angle The vertical slots let you mount the Launcher horizontally at different heights so you can fire a ball into targets such as a ball catcher on a PASCO Cart on a track At the bottom are two holes for use when you want to fire a ball horizontally into the Ballistic Pendulum e Repeatable Results The piston keeps the ball from rubbing on the inside of the barrel as it travels so there is no spin on the ball as it launches When the base is secured to a table with a C clamp there is very little recoil The trigger is pulled with a string to minimize jerking e Barrel Sights and Safety Precautions There are sites built in to the barrel for aiming the Projectile Launcher View the sites by looking through the back end of the barrel WARNING Never look down the front of the barrel because it may be loaded Safety glasses are provided so use them Look for the yellow indicator through any of the five slots on the top of the barrel because the yel A low indicator shows the position of the piston If the indicator
11. is between the first and second slots relative to the muzzle end the piston is not cocked e Computer Compatible One or two photogates can be attached to the Projec tile Launcher using the ME 6821A Photogate Mounting Bracket When used Wear Safety Glasses with a PASCO interface and data acquisition software the photogates can mea sure the muzzle speed of the ball Use a photogate and the ME 6810 Time of Flight Accessory to measure the time of flight of the ball e Compact Storage When the barrel of the Launcher is aligned vertically with the base the Launcher takes up minimal space The included ramrod and the Ballistic Pendulum base have hook and pile material that allows the ramrod to be stored on the base LEIHO 012 05375C 3 Ballistic Pendulum Projectile Launcher General Operation of the Projectile Launcher General Operation of the Projectile Launcher Trigger MEDIUM SHORT Yellow Band in Window RANGE RANGE Indicates Range ET Use 25 mm DO NOT LOOK gt gt DOWN BARRELI balls ONLY 3 A SHORT RANGE PROJECTILE LAUNCHER Range setting slots Plumb Bob Launcher Parts Ready e Attach the included Trigger String to the hole in the Trigger For example loop the string through the hole and tie the ends together e Always wear safety goggles when you are in a room where a Projectile Launcher is being used e Firmly clamp the base of the Ballistic Pendu
12. other after the collision LEIHO 012 05375C 31 Projectile Launcher Exp 6 Conservation of Momentum Setup 1 Clamp the Projectile Launcher to a sturdy table Mount the Launcher near one end of the table with the Launcher aimed inward toward the table Adjust the angle of the Projectile Launcher to zero degrees so the ball will be launched horizontally onto the table Cover the table with white paper such as butcher paper NOTE The paper must reach the base of the Launcher Fire a test shot on the short range setting to make sure that the ball lands on the table Tape a piece of carbon paper carbon side down over the spot where the ball lands Mount the 2 D Collision Accessory to the front of the Launcher Put a target ball on the post tee of the acces sory Loosen the thumbscrew and rotate the 2 D Collision Accessory slightly to one side The tee must be located so that the launched ball does not rebound into the Launcher but does hit the target ball so that both balls land on the table at the same time Tighten the thumbscrew to hold the accessory in place Load the Launcher and fire a test shot to check that both balls hit the table at the same time Tape a piece of carbon paper on the white paper at each spot where the two balls land on the table Procedure A No Collision Put ball 1 into the Launcher and cock it to the short range setting Do not put a target ball on the
13. target that is at the same height as the muzzle e The scatter pattern of projectiles with the Projectile Launcher is minimized when the Projectile Launcher is securely clamped to a sturdy table Any wobble in the table will show up in the data e The angle of inclination can be determined to within one half of a degree Expectations for the Ballistic Pendulum e Angles reached by the swinging pendulum should be repeatable to within half a degree LEIHO 012 05375C 9 Ballistic Pendulum Projectile Launcher Ballistic Pendulum Theory e Overall error in measurement of ball velocity should not exceed 2 5 exact method or 10 approximate method e NOTE Adjustable leveling feet are not necessary for good results Small deviations from the horizontal will not cause significant error 10 012 05375C DLHO Model No ME 6830 Exp 1 Projectile Motion Exp 1 Projectile Motion Equipment Needed Item Item Projectile Launcher and plastic ball Plumb bob and string Meter stick Carbon paper White paper Sticky tape Purpose The purpose of this experiment is to predict and verify the range of a ball launched at an angle The initial speed of the ball is determined by shooting it horizontally and measuring the range of the ball and the height of the Launcher Theory To predict where a ball will land on the floor when it is shot from the Launcher at some angle above the horizontal it is first necessary to determin
14. 0 d Vo Solving for the angle 0 gives v 0 tan Omax T Since the second derivative is negative for 0 the angle is a maximum To find the initial speed of the ball use the fixed distance x and the maximum height y Solve the y equation for vg and plug in the values for y 0 and x max LEIHO 012 05375C 35 Projectile Launcher Exp 7 Vary the Angle to Maximize the Height Setup 1 Clamp the Projectile Launcher to a sturdy table Mount the Launcher near one end of the table with the Launcher aimed toward a wall about 2 meters from the table 2 Use a vertical board to the protect the wall and cover the board with white paper 3 Fire a test shot to see where the ball hits the board and tape a piece of carbon paper carbon side down at that position Procedure 1 Shoot the ball at various angles and pinpoint which angle gives the maximum height by checking the marks on the white paper Move the carbon paper as necessary 2 Measure the angle that produces the maximum height and record its value in Table 7 1 3 Measure the maximum height and record the value in the Data Table 4 Measure the horizontal distance from the muzzle to the vertical board and record the value 5 Measure the initial height of the ball where it leaves the muzzle and record the value Table 7 1 Data and Results Item Value Measured Angle for Maximum Height Maximum Height Horizontal Distanc
15. 0 rn 57 PASC Om i Projectile Launcher iv 012 05375C I Eno Ballistic Pendulum Projectile Launcher ME 6830 ME 6831 Equipment The ME 6830 Ballistic Pendulum Projectile Launcher includes the following Included Equipment Part Number Ballistic Pendulum and Base unassembled Short Range Projectile Launcher Assembly Plastic Balls 25 mm diameter 3 2 D two dimensional Collision Accessory Ramrod Steel Balls 25 mm diameter 2 Safety Glasses 2 pair Trigger String 45 cm 18 Hex Key Required Equipment 003 05374 003 10550 see ME 6802 see ME 6802 see ME 6802 699 064 699 066 699 067 726 046 Part Number C Clamp Large Recommended Equipment Launcher Spares Kit Time of Flight Accessory Projectile Catcher Accessory Photogate Mounting Bracket Shoot the Target Photogate Head Laser Sight Accessory The ME 6802 Launcher Spares Kit includes Ramrod 2 Plastic Balls 10 pack 2 D Collision Accessory 2 Sights 5 pack Plumb Bob 24 pack Nylon Thread 1 spool and Thumbscrews 10 The ME 6831 Ballistic Pendulum does not include the Short Range Launcher The ME 6831 Ballistic Pendulum includes two steel balls and the hex key SE 7285 Part Number ME 6802 ME 6810 ME 6815 ME 6821A ME 6853 ME 9498A OS 8527A ME 6802 Launcher Spares Kit IZEIA Ballistic Pendulum Projectile Launcher Assembly Assembly The Ballistic Pendulum Projectil
16. 00 772 8700 U S Fax 916 786 7565 Web Www pasco com Email support pasco com For more information about the Ballistic Pendulum Projectile Launcher and the latest revision of this Instruction Manual visit the PASCO web site and enter ME 6830 or ME 6831 into the Search window Limited Warranty For a description of the product warranty see the PASCO catalog Copyright The PASCO scientific 012 05375C Ballistic Pendulum Projectile Launcher Instruction Manual is copyrighted with all rights reserved Permission is granted to non profit educational institutions for reproduction of any part of this manual providing the repro ductions are used only in their laboratories and classrooms and are not sold for profit Reproduction under any other circumstances without the written consent of PASCO scientific is prohibited Trademarks PASCO and PASCO scientific are trademarks or registered trademarks of PASCO scientific in the United States and or in other countries All other brands products or service names are or may be trademarks or service marks of and are used to iden tify products or services of their respective owners For more information visit www pasco com legal LEIHO 012 05375C 57 Ballistic Pendulum Projectile Launcher Technical Support 58 012 05375C IZEIA
17. 40 versus the theoretical value of 39 2 The angle of maximum range decreases with table height 3 The maximum distance increases with table height Exp 4 Projectile Path Analysis 1 Alternately measure your distances from the ground up 52 012 05375C I Ego Model No ME 6830 Teacher s Guide 2 Vertical distances were measured from the ground up for this graph The intercept is the height of the launcher above the ground when done this way M a o M Vertical Distance m O o A oa 0 05 f x 1 181345E 1 x 2 609457E 1 R42 9 997926E 1 0 02 04 06 08 1 12 14 16 18 2 Horizontal Distance Squared m2 3 The slope measuring from the ground is 0 118 for this example Measuring down from the initial height will give the same value except positive In either case the slope is 8 2 2V9 4 The slope calculated here gives us an initial speed of 6 44 m s This compares favorably with the speed calcu lated in experiments and 2 5 Results will vary with this method the point of the activity is that individual measurements are not as accurate as a large number of measurements and a curve fit Questions 1 Yes This tells us that y is a function of x 2 A plot of y versus x would be parabolic instead of linear 3 The projectile moves in a parabolic curve if air resistance is neglected Exp 5 Conservation of Energy Analysis 1 Using the photogate method the initial
18. Vary the Angle to Maximize the Height Equipment Needed Item Item Projectile Launcher and plastic ball Board to protect wall Meter stick or measuring tape Sticky tape White paper large sheet Carbon paper several sheets Plumb bob and string Purpose The purpose of this experiment is to find the launch angle that will maximize the height on a vertical wall for a projectile launched at a fixed horizontal distance from the wall Theory When the ball is shot at an angle at a fixed distance x from a Range target such as a vertical wall the ball hits the wall at a height y given by Initial gees p speed a y yo vosin0 r 3 90 P where y is the initial height of the ball vg is the initial speed of the ball as it leaves the muzzle is the angle of inclination above horizontal g is the acceleration due to gravity and is the time of flight The range is the horizontal distance x between the muzzle of the Launcher and the place where the ball hits given by Initial height x vgcos0 t Solving this equation for the time of flight t gives x vocos Figure 7 1 Maximizing Height Substituting for in the equation for y gives 2 8x y yo xtanO J 2v cos 0 To find the angle O that gives the maximum height y find the first derivative of the equation for y and set it equal to zero Solve for the angle 0 2 2 dy _ xsec26 amp fans seg Bs
19. asure from the point of contact to each of the dots made by Ball 2 Find the average of the five lengths Draw a straight line from the point of contact through the center of the group of dots made by Ball 2 6 Measure the angle from the centerline to the straight line for Ball 2 Use this angle and the average length of the line for Ball 2 to calculate the x component for Ball 2 and the y component for Ball 2 Record the values 7 Add the x momentum for Ball 1 and the x momentum for Ball 2 and record the result in Table 6 1 as Final X momentum 8 Calculate the initial kinetic energy of Ball 1 and the sum of the kinetic energy of Ball 1 and Ball 2 after the collision 9 Calculate the percent differences Table 6 1 Data for the Elastic Collisions Item Value Item Value Percent difference Initial x momentum Final x momentum Ball 1 Ball 1 Ball 2 Final y momentum Final y momentum Ball 1 Ball 2 Initial kinetic energy Final kinetic energy Ball 1 Ball 1 Ball 2 LEIHO 012 05375C 33 Projectile Launcher Exp 6 Conservation of Momentum C Inelastic Collision Draw straight lines from the point of contact spot to the dots made by the inelastic collision shot There should be two lines Measure from the point of contact to each of the dots made by the inelastic collision shot Measure the angle from the centerline to the straight line for
20. e Initial Height Calculated Initial Soeed Calculated Angle for Maximum Height Percent Difference between Angles Analysis 1 Calculate the initial speed by solving the y equation for vg and substituting the values for Ypa Omay and x from Table 7 1 2 Calculate the angle for maximum height using the initial speed calculated in step 1 and the horizontal distance from the wall to the launcher 3 Calculate the percent difference between the measured angle and the calculated Difference A B 100 angle Let A be one of the angles and B be the other angle 2 Questions 1 For the angle that gives the maximum height when the ball hits the wall has it already reached the peak of its trajectory 2 For what distance from the wall would the height be maximized for a launch angle of 45 What would the maximum height be in this case 36 012 05375C TPA LIH o Model No ME 6830 Exp 8 Projectile Velocity Approximate Method Exp 8 Projectile Velocity Approximate Method Equipment Needed Item Item Ballistic Pendulum Projectile Launcher Steel ball C clamp Mass balance String Purpose The muzzle velocity of the projectile launcher is determined by launching the ball into the ballistic pendulum and observing the angle to which the pendulum swings As derived earlier in the manual the equation for the velocity of the ball is approximately M 2gR 1 cos where M is t
21. e White paper Carbon paper Photogate Head ME 9498A 2 optional Photogate Mounting Bracket ME 6821A optional Use the Photogates and Photogate Mounting Bracket with a PASCO Interface or Timer to measure the initial speed of the ball directly see Experiment 2 Purpose final position The purpose of this experiment is to confirm that the initial kinetic energy of a projectile shot straight up is transformed into an equal amount of gravitational h potential energy Theory The total mechanical energy of a projectile is the sum of its gravitational poten initial y tial energy and its kinetic energy In the absence of friction total mechanical position energy is conserved When a projectile is shot straight up the initial gravitational potential energy GPE can be defined as zero The initial kinetic energy KE depends on the mass m of the projectile and the initial speed vg KE mn When the projectile reaches its maximum height h the speed of the projectile is zero and therefore the kinetic energy is zero The gravitational potential energy depends on the mass of the projectile and the height Figure 5 1 Conservation GPE mgh of Energy where g is the acceleration due to gravity If friction in the form of air resistance is ignored the initial kinetic energy should equal the final gravitational potential energy The initial speed of the projectile must be dete
22. e Ballistic Pendulum upright Clamp the Ballistic Pendulum Projec tile Launcher to a sturdy table or other horizontal surface Mount the Launcher near one end of the table and aimed away from the table LEIHO 012 05375C 11 Projectile Launcher Exp 1 Projectile Motion 2 Adjust the angle of the Projectile Launcher to zero degrees so the ball will by launched horizontally Part A Determining the Initial Horizontal Speed of the Ball 1 Puta plastic ball in the Projectile Launcher and use the ramrod to cock it at the long range position Fire one shot to locate where the ball hits the floor At that point tape a piece of white paper to the floor Place a piece of carbon paper carbon side down on top of the white paper and tape it in place e When the ball hits the carbon paper on the floor it will leave a mark on the white paper 2 Fire ten shots 3 Measure the vertical distance from the bottom of the ball as it leaves the barrel to the floor Record this distance in the Data Table Launch Position of Ball The Launch Position of Ball in the barrel is marked on the label on the side of the Launcher 4 Use a plumb bob to find the point on the floor that is directly beneath the release point on the barrel Measure the horizontal distance along the floor from the release point to Bottom the leading edge of the piece of white paper Record the distance in the Data Table JIER a al 5 Carefully remove
23. e Launcher arrives in a custom made package and some assembly is required The package has several cut outs for the Ballistic Pendulum and ramrod base upright and Projectile Launcher safety glasses and miscellaneous small parts including a hex key Allen wrench used for assembly Assemble the Base Unscrew the thumbscrew to temporarily Axle remove the Projectile Launcher from the upright Use the included hex key and the two socket head screws to attach the base to Angle Indicator the upright The screws are coated with a Upright strong adhesive that activates when they are screwed into place Mount the Ballistic Pendulum To attach the Ballistic Pendulum to the upright unscrew the axle from the yoke The Ballistic Pendulum has a hinge at the top of the rod with a hole through it Line up the hole in the hinge with the axle hole in the yoke and screw the axle back into place Note that the long pin that extends from either side of the Ballistic Pendulum rod should be behind the angle indicator a _ Hex key Assemble the Base Angle Indicator m i m u f A Long pin a m E Mount the Ballistic Pendulum Introduction The PASCO Ballistic Pendulum Projectile Launcher has been designed for ballistic pendulum and projectile motion experiments and demonstrations The only addition equipment required is a C clamp for mounting the base of the Ballistic Pendulum to a table or sturdy horizon
24. e Projectile Launcher 000 0c c eee eee eee 4 Ballistic Pendulum Theory so0ccdseseseadetaraggcedbesepececeneasecaada wea 6 Installation of the Optional Photogate Bracket 00 0 0c eee eee eae 8 Installation of the Two Dimensional Collision Attachment 2005 9 Expectations for the Projectile Launcher 2 20 0 9 Expectations for the Ballistic Pendulum 0 2 0 0000 9 EXPERIMENTS 1 Projectile Motion 3 aseterosreee ecsetn wis e205 acd ane eee eee Mere Ba ee acd acces ee ee erarerr ae 11 2 Projectile Motion Using Photogates 00 cee eee 15 3 Projectile Range versus Angle 0002 20 eee eee 19 4 Projectile Fam rr 23 5 Conservation Of Energy os asuwdecweos a A Geeek eons 27 6 Conservation of Momentum 0 2 000 0 31 7 Vary Angle to Maximize Height 2 00 eee 35 8 Projectile Velocity Approximate Metho d 0 37 9 Projectile Velocity Exact Method 0 cee eens 39 DEMONSTRATIONS 10 Do 30 and 60 Launch Angles give the Same Range 0 43 11 Simultaneously Fire Two Balls Horizontally at Different Speeds 45 12 Shoot through Hoops 2 2 2 2 00 cee ene 47 13 Elastic and Inelastic Collisions 00 2000 ee ees 49 T achers GQUIGG cc wet ehde se eeseee vide dee Cheeedebeeueee4onee ct eyed eedee 51 Technical Support Warranty and Copyright 2 000 020
25. e launcher Questions 1 Is there another way to measure the muzzle velocity that you could use to check your results You may want to use that second method and compare the two answers 2 What sources of error are there in this experiment How much do these errors affect your result 3 It would simplify the calculations if kinetic energy were conserved in the collision between the ball and the pendulum see Ballistic Pendulum Theory in the Introduction What percentage of the kinetic energy is lost in the collision between the ball and pendulum Would it be valid to assume that energy was conserved in that collision 4 How does the angle reached by the pendulum change if the ball is not caught by the ball catcher You may test this by turning the pendulum around so the ball strikes the back of the ball catcher Is there more energy or less energy transferred to the pendulum 38 012 05375C TPA LIH o Model No ME 6830 Exp 9 Projectile Velocity Exact Method Exp 9 Projectile Velocity Exact Method Equipment Needed Item Item Ballistic Pendulum Projectile Launcher and steel ball Ruler C clamp Mass balance String Stopwatch Purpose The muzzle velocity of the projectile launcher is determined by launching the ball into the ballistic pendulum and observing the angle to which the pendulum swings As derived earlier in the manual the equation for the velocity of the ball is approximately vy _ PIMgR 1
26. e of white paper to the floor Place a piece of carbon paper car bon side down on top of the white paper and tape it in place When the ball hits the carbon paper it will leave a mark on the white paper Fire ten shots Measure the vertical distance from the bottom of the ball as it leaves the barrel to the floor Record this dis tance in the Table 5 1 Use the distance to calculate the time of flight and record it The Launch Position of Ball in the barrel is marked on the label on the side of the Launcher Use a plumb bob to find the point on the floor that is directly beneath the release point on the barrel Measure the horizontal distance along the floor from the release point to the leading edge of the piece of white paper Record the distance in Table 5 1 Carefully remove the carbon paper Measure from the leading edge of the white paper to each of the ten dots and record these distances in Table 5 1 Find the average of the ten distances and record it Using the horizontal distance and the time of flight calculate the initial speed of the ball Record the speed Table 5 1 Item Value Item Value Vertical distance Calculated time of flight Horizontal distance to edge of paper Initial speed 28 012 05375C LLEI O Model No ME 6830 Exp 5 Conservation of Energy Table 5 1 Trial Distance Trial Distance 1 6 2 7 3 8 4 9 5 10 Average
27. e ranges are the same Change the angle to 45 and shoot the ball again to show that the ball now lands further away missing the box Ask the question What other pairs of angles will have a common range Will 20 and 70 have the same range Will 35 and 55 have the same range This demonstration can be done for any two angles that add up to 90 44 012 05375C 7 Wefe Model No ME 6830 Exp 11 Demo Simultaneous Shots at Different Speeds Exp 11 Demo Simultaneous Shots at Different Speeds Equipment Needed ltem Projectile Launcher 2 and plastic ball 2 Purpose The purpose of this demonstration is to confirm that regardless of the initial speed of projectiles fired horizontally the projectiles will hit the floor at the same time Theory Two projectiles are shot horizontally from the same height y The muzzle speeds of the two projectiles are differ ent The vertical and horizontal motions of a projectile are independent of each other The horizontal distance x trav elled by the projectile depends on the initial speed vg and the time of flight t The distance x vol The time of flight depends on the vertical distance that the projectile falls r f 8 where g is the acceleration due to gravity Since the vertical distance is the same fore each projectile the time of flight is the same for each projectile Setup 1 Vo short Voltong Clamp two Projectile La
28. e the initial speed muzzle velocity of the ball That can be determined by shooting the ball horizontally from the Launcher and measuring the vertical and horizontal distances that the ball travels The initial speed can be used to calculate where the ball will land when the ball is shot at an angle above the hori zontal e NOTE For rest results see the notes on Repeatable Results in the Introduction Initial Horizontal Speed For a ball shot horizontally with an initial speed vy the horizontal distance travelled by the ball is given by x vof where is the time the ball is in the air Neglect air friction The vertical distance of the ball is the distance it drops in time f given by a y 58 The initial speed can by determined by measuring x and y The time of flight t of the ball can be found using r f 8 and the initial horizontal speed can be found using vg IR Initial Speed at an Angle To predict the horizontal range x of a ball shot with an initial speed vg at an angle O above the horizontal first predict the time of flight from the equation for the vertical motion 1 2 t z t y Yo vosin 58 where y is the initial height of the ball and y is the position of the ball when it hits the floor In other words solve the quadratic equation for and then use x vg cos t where vy cos is the horizontal component of the initial speed Setup 1 Put the Launcher in the top position on th
29. e thumbscrews and C clamp and adjust the angle and position of the Launcher to align the centers of both sights on your target Tighten the thumbscrews and C clamp when the Launcher is aimed Load e To load a ball in the Launcher when its mounted on the low position either hold the Ballistic Pendulum out of the way or rotate the pendulum until the rod is horizontal Bore Sights and it catches in the component clip on the underside of the yoke e Place a ball in the muzzle of the Launcher NOTE Always cock the piston with a ball in the piston You may damage the piston if you use the ramrod without a ball in the piston e Remove the ramrod from its storage place on the edge of the upright While looking through the range setting slots on the top side of the Launcher push the ball down the barrel with the ramrod until the trigger catches the edge of the piston at the desired range setting The trigger will click into place e When the yellow indicator tape on the piston is visible in the mid dle range setting slot the piston is in the SHORT RANGE position When the indicator tape on the piston is visible in the next range setting slot fourth from the muzzle the piston is in the NG rs MEDIUM RANGE position and when the tape is visible in the last range setting slot the piston is in the LONG RANGE position e Remove the ramrod and return it to the storage place on the edge of the upright e When the Proj
30. ease the efficiency of the energy transfer in the col lision Try it The Projectile Velocity A pproximate Method experiment uses the approximate equation for velocity M aa 2gR 1 cos 9 What is the value of velocity when you use this equation Is there a significant difference between these two calculated values What factors would increase the difference between these two results How would you build a ballistic pendulum so that the approximate equation gave better results i7 Wrefe 012 05375C 41 Projectile Launcher Exp 9 Projectile Velocity Exact Method 42 012 05375C 7 Wefe Model No ME 6830 Exp 10 Demo Do 30 and 60 Give the Same Range Exp 10 Demo Do 30 and 60 Give the Same Range Equipment Needed Item Item Projectile Launcher and steel ball Box to make landing area same height as muzzle Purpose The purpose of this demonstration is to confirm that the range of a ball launched at 30 is the same as one launched at 60 if the ball lands at the same height from which it was launched Theory The range is the horizontal distance x between the muzzle of the Launcher and the place where the projectile lands given by x vg cos 0 t where v is the initial speed of the ball as it leaves the muzzle is the launch angle above horizontal and is the time of flight If the ball lands on a target that is at the same height as the level of the muzzle of the la
31. ectile Launcher is loaded the yellow indicator tape is visible through one of the range setting slots on the upper side of the barrel Never look down the barrel To check whether the Launcher is loaded look through the range setting slots on the barrel Shoot e Before shooting the ball make certain that no one is in the way e To shoot the ball pull straight up on the trigger string that is attached to the trig ger You only need to pull about one centimeter e The trigger will automatically return to its initial position after you release the string Maintenance and Storage e The Ballistic Pendulum Projectile Launcher does not need any special mainte nance Do not oil the Launcher Et To store the Launcher in the least amount of space align the barrel vertically One way is to mount it in one of the two vertical slots Tighten the thumbscrews to hold the Launcher in place LEIHO 012 05375C 5 Ballistic Pendulum Projectile Launcher Ballistic Pendulum Theory Ballistic Pendulum Theory Overview The ballistic pendulum is a classic method of determining the velocity of a projectile It is also a good demonstra tion of many of the basic principles of physics The ball is fired into the ballistic pendulum which then swings up a measured amount From the height reached by the pendulum you can calculate its gravitational potential energ
32. ed The table gives typical results range Angle Predicted Range Actual Range Percent Error 30 5 22 5 19 0 57 45 5 30 5 16 2 64 60 4 35 4 23 2 87 39 5 39 5 31 1 48 NOTE The maximum angle is not 45 in this case The range at 60 is not equal to the range at 30 This is because the initial height of the ball is not the same as the impact point The maximum range for this setup with the Launcher 1 15 m above ground level was calculated to be at 39 This was verified experimentally 2 Answers will vary depending on the method for estimating the precision The primary source of error is ignor ing the effect of air resistance Exp 2 Projectile Motion Using Photogates Except for the method of determining initial speed this experiment is identical to experiment 1 LEIHO 012 05375C 51 Ballistic Pendulum Projectile Launcher Teacher s Guide Exp 3 Projectile Range Versus Angle Procedure e Shooting off a level surface 4 5 4 Angle degrees e Shooting off a table 6 10 20 30 40 50 60 70 80 90 Angle degrees o e NOTE The curves show the calculated ranges in each case The data points are the actual measured ranges Questions 1 Ona level surface the maximum range is at 45 For a non level surface the angle of maximum range depends on the initial height of the projectile For the experimental setup with an initial height of 1 15 m the maximum range is at
33. efore the ball has already reached the peak and is on its way down 2 Solve the equation for maximum angle to determine x va ve 0 0 tanO max See A 2 Yo 2 ele 2 2 Yo 2 Yo Yo y Yor 5 Vota Yo yY Yo Exp 8 Projectile Motion Approximate Method Procedure 1 The exact mass used is not critical Pick a value that gives a fairly large swing for best results 2 With the steel ball and extra masses on the pendulum the balance point will be somewhere on the ball catcher itself This makes it difficult to use string but it is relatively easy to find the center of mass by balancing the pendulum on a straightedge 3 The angle reached by the pendulum should not vary by more than one degree between successive trials LEIHO 012 05375C 55 Ballistic Pendulum Projectile Launcher Teacher s Guide Calculations e Use the equations given in the theory section for the approximate method Questions 1 The best other method of measuring velocity is described in the first part of experiment 1 2 The greatest source of error is the equation used This is an approximate equation based on the assumption that the masses involved are point masses The amount of effect this equation has on the results will depend on the exact geometry of the pendulum and ball and should be between five and eight percent 3 Typically 70 of the kinetic energy of the ball is lost It is not valid to assume that kinetic energy is con
34. et board Meter stick or measuring tape Sticky tape Graph paper Carbon paper White paper The target board should be as tall as the distance from the muzzle to the floor Purpose The purpose of this experiment is to determine how the vertical distance a projectile drops is related to the hori zontal distance the projectile travels when the projectile is launched horizontally Theory The range is the horizontal distance x between the muzzle of the Launcher and the place where the projectile hits given by x vot where vg is the initial speed of the projectile as it leaves the muzzle and f is the time of flight If the projectile is launched horizontally the time of flight of the projectile will be X i 0 The vertical distance y that the projectile falls during time t is given by lt P y 58 where g is the acceleration due to gravity Substituting for in the second equation gives 2 2 2 2 2v 8 A plot of y versus x will give a straight line with a slope equal to 5 2v Setup 1 Clamp the Projectile Launcher T to a sturdy table or other hori A zontal surface Mount the Launcher near one end of the table with the Launcher aimed away from the table J m Se A ir a a 2 Adjust the angle of the Projec j tile Launcher to zero degrees so the ball will be launched horizontally 3 Fire a test shot on medium range to determine the initial 1 position of
35. f about 84 5 In the inelastic case the angle will be less than in the elastic case The exact angle will depend on the degree of inelasticity which will depend on the type and amount of tape used Exp 7 Vary the Angle to Maximize the Height Procedure 1 You should be able to measure the angle of maximum height to within 2 2 Measure the distance to the front edge of the ball 3 Measure the initial height to the center of the ball Analysis 1 The initial speed should be close to the initial speed determined by other methods You may wish to determine the initial speed by the method in experiment 1 and use that value in your calculations for the rest of the experiment 2 Measured value and calculated vale should agree to within 3 Questions 1 The ball will have passed its peak by the time it reaches the wall To show this take the derivative of y with respect to x 2 SX y yot xtanO 5 5 2vgcos Onax 2 dy gx ss tanO max ae VgCos 0 max 54 012 05375C TPA LIH o Model No ME 6830 Teacher s Guide e substitute 2 1 dx x 2 EX max 2 Vo 1 VgCoOs tan EX may cos tan 9 g b 2 e Substitute a b e and simplify 2 2 4 22 dy _ vo Z Vo x vo 8 X max 7 7 g 7 2 2 dx EX max 8X max 8X max VOR an o fa 22 Yot 8 Xmax dy Vo Vor gx 2 dx 8X max 8X max Vo e When x Xaine the value of this derivative is negative dy _8 max dx 2 x Vo e Ther
36. f flight t angle 0 and initial speed Vg to predict the horizontal distance range x vg cos 8 f Record the predicted range 5 Draw a line across the middle of a white piece of paper and tape the paper on the floor so the line is at the pre dicted horizontal distance Cover the white paper with carbon paper and tape the carbon paper in place 6 Use a plumb bob to find the point on the floor that is directly beneath the release point on the barrel Measure the horizontal distance along the floor from the release point to the leading edge of the piece of white paper Record the distance in the Data Table 7 Shoot the ball ten times 8 Carefully remove the carbon paper and measure from the leading edge of the white paper to each of the ten dots Record these distances in the Data Table and find the average Calculate and record the total horizontal distance distance to paper plus average distance from edge of paper to dots Angle above horizontal Horizontal distance to edge of paper Calculated time of flight Predicted range 16 012 05375C 7 Wefe Model No ME 6830 Exp 2 Projectile Motion Using Photogates Data Table Part B Table 2 2 Confirm the Predicted Range Trial Distance 1 2 3 9 10 Average Total Distance Analysis 1 Calculate the percent difference between the predicted theoretical distance A and the ac
37. ght by unscrewing and removing the axle Use a mass balance to find the mass of the pendulum and ball together Record this value as M in Table 9 1 4 Measure the mass of the ball alone and record this as m LEIHO 012 05375C 39 Projectile Launcher Exp 9 Projectile Velocity Exact Method 5 Tie a loop in a piece of string and hang the pendulum horizontally from the loop See Figure 9 1 With the ball latched in position in the ball catcher adjust the position of the pendulum in the loop until the pendulum balances Measure the distance from the pivot point to this balance point and record the distance as Rp String loop a gt en Figure 9 1 Setup e NOTE It may be easier to balance the pendulum on the edge of a ruler or similar object 6 Measure the distance between the pivot point and the center of the ball Record this as R 7 Remove the launcher so that the pendulum can swing freely With the ball in the ball catcher give the pendu lum an initial displacement of 5 or less Using the stopwatch time how long it takes to go through ten oscil lations Divide this time by the number of oscillations and record the result as T in Table 9 1 8 Calculate the value of I and record it in Table 9 1 9 Reattach the launcher and load it Set the angle indicator to an angle one or two degrees less than the angle reached in step 2 This will nearly eliminate the friction against t
38. he mass of the pendulum and ball combined m is the mass of the ball g is the acceleration due to gravity R m is the distance from the pivot point to the center of mass of the pendulum and 0 is the angle reached by the pendulum after the collision Setup 1 Put the Projectile Launcher on the Ballistic Pendulum upright at the level of the ball catcher the low posi tion Make sure that the pendulum can hang vertically without touching the launcher 2 Clamp the Ballistic Pendulum base to a sturdy table Make sure that the clamp does not interfere with the pen dulum swing Procedure 1 Latch the pendulum at 90 so it is out of the way temporarily and then load the projectile launcher with the steel ball Allow the pendulum to hang freely and move the angle indicator to zero degrees 2 Fire the launcher and record the angle reached If you want to do the experiment with a lower or higher angle add to or remove mass from the pendulum Repeat these test measurements until you are satisfied with the mass of the pendulum 3 Once you have chosen the mass to use for your experiments remove the pendulum from the upright by unscrewing and removing the axle Use a mass balance to find the mass of the pendulum and ball together Record this value as M in Table 8 1 4 Measure the mass of the ball alone and record this as m e Tie a loop in a piece of string and hang the pendulum horizontally from the loop See Figure 8 1 With the ba
39. he pendulum caused by the angle indicator since the pendulum will only move the angle indicator for a short distance 10 Fire the launcher and record the angle reached by the pendulum in Table 9 1 Repeat several times setting the angle indicator to an angle one or two degrees less than the previous angle reached by the pendulum each time Table 9 1 M m Rem R T I Table 9 1 Data Item Value 8 02 83 94 95 Average 0 Muzzle velocity Calculations 1 Find the average angle reached by the pendulum and record the value in Table 9 1 2 Calculate the muzzle velocity of the ball fired from the projectile launcher 40 012 05375C DLHO Model No ME 6830 Exp 9 Projectile Velocity Exact Method Questions 1 Is there another way to measure the muzzle velocity that you could use to check your results You may want to use that second method and compare the two answers What sources of error are there in this experiment How much do these errors affect your result It would simplify the calculations if kinetic energy were conserved in the collision between the ball and the pendulum see Ballistic Pendulum Theory in the Introduction What percentage of the kinetic energy is lost in the collision between the ball and pendulum Would it be valid to assume that energy was conserved in that collision Does increasing the mass of the pendulum increase or decr
40. icted theoretical distance A and the actual average dis tance B when shot at an angle A B x100 A B 2 2 Estimate the precision of the predicted range How many of the final 10 shots landed within this range 12 012 05375C I Ego Model No ME 6830 Exp 1 Projectile Motion Data Table A Determine the Initial Speed Vertical distance Calculated time of flight Initial speed Horizontal distance to edge of paper Trial Distance 9 10 Average Total Distance Data Table B Predict the Range Angle above horizontal Calculated time of flight Horizontal distance to edge of paper Predicted range Trial Distance 9 10 Average Total Distance IZEIA 012 05375C 13 Projectile Launcher Exp 1 Projectile Motion Notes 14 012 05375C TPA Yee Model No ME 6830 Exp 2 Projectile Motion Using Photogates Exp 2 Projectile Motion Using Photogates Equipment Needed Item Item Projectile Launcher and plastic ball Plumb bob and string Photogate Head ME 9498A 2 Photogate Mounting Bracket ME 6821A PASCO Interface or Timer PASCO Data acquisition software Meter stick Carbon paper White paper Sticky tape See the PASCO web site at www pasco com for information about PASCO interfaces timers and data acquisition
41. istance between the photogates is 0 10 m 10 cm 4 Plug the photogates into an interface or a timer Procedure Part A Determining the Initial Speed of the Ball 1 Puta plastic ball in the Projectile Launcher and use the ramrod to cock it at the long range position 2 Setup the data acquisition software or the timer to measure the time between the ball blocking the two photo gates 3 Shoot the ball three times and calculate the average of these times Record the data in Data Table 2 1 LEIHO 012 05375C 15 Projectile Launcher Exp 2 Projectile Motion Using Photogates 4 Calculate the initial speed of the ball based on the 0 10 m distance between the photogates Record the value Data Table Part A Table 2 1 Determine the Initial Speed Trial Time 1 2 3 Average Time Initial Speed Part B Predicting the Range of a Ball Shot at an Angle 1 Keep the angle of the Projectile Launcher at the original angle above horizontal 2 Measure the vertical distance from the bottom of the ball as it leaves the barrel to the floor Record this distance in Data Table 2 2 taune e The Launch Position of Ball in the barrel is marked on the label on the side of the Launcher Position of Ball 3 Use the vertical distance the angle and the initial speed to calculate the time of flight Record the value I2 Bottom y Yo vgsin t 5gt 5 of ball 4 Use the time o
42. jectile to reach the peak of its trajectory At the peak the vertical speed is zero so v 0 vpsin 8t eak where vy is the initial speed of the projectile Solving for the time gives an expression for the total time of flight as vosin 2theak g For the case in which the projectile is launched at an angle above horizontal from a table onto the floor the time of flight is found using the equation for vertical motion 1 2 yo 0 t lt gt y Ygt vgsin 58 where y is the initial height of the projectile in the Launcher and y is the vertical position of the ball when it hits the floor x Figure 3 2 Shooting from a table Setup 1 Put the Launcher in the top position on the Ballistic Pendulum upright Clamp the Ballistic Pendulum Projec tile Launcher to a sturdy table or other horizontal surface Mount the Launcher near one end of the table but aim it toward the center of the table rather than away from the table LEIHO 012 05375C 19 Projectile Launcher Exp 3 Projectile Range versus Angle 2 Adjust the angle of the Projectile Launcher to 10 degrees 3 Puta plastic ball into the Projectile Launcher and cock it to the medium or long range setting e Note In general the experiment will not work as well on the short range setting because the muzzle speed is more variable with the change in angle 4 Fire one shot to locate where the ball hits Place M a box or
43. ll latched in position in the ball catcher adjust the position of the pendulum in the loop until the pendulum balances Measure the distance from the pivot point to this balance point and record the distance as Rpp String loop i Figure 8 1 Setup LEIHO 012 05375C 37 Projectile Launcher Exp 8 Projectile Velocity Approximate Method NOTE It may be easier to balance the pendulum on the edge of a ruler or similar object 5 Reattach the pendulum to the upright making sure that it is facing the right way Be sure that the angle indica tor is in front of the long pin of the pendulum 6 Load the launcher and then set the angle indicator to an angle one or two degrees less than the angle reached in step 2 This will nearly eliminate the friction against the pendulum caused by the angle indicator since the pendulum will only move the angle indicator for a short distance 7 Fire the launcher and record the angle reached by the pendulum in Table 8 1 Repeat several times setting the angle indicator to an angle one or two degrees less than the previous angle reached by the pendulum each time Table 8 1 M m R cm Table 8 1 Data Item Value Average 9 Muzzle velocity Calculations 1 Find the average angle reached by the pendulum and record the value in Table 8 1 2 Calculate the muzzle velocity of the ball fired from the projectil
44. lum is thus Ta R Mgsin For small angles O sin O 9 so if you make this substitution and solve for a you get Ge Mk I This angular equation is in the same form as the equation for linear simple harmonic motion k 2 Q x 0 x m So if you compare the two equations linear and angular you can see that the pendulum exhibits simple harmonic motion and that the square of the angular frequency cw for this motion is a MgR Solving for gives the desired result p MBRem _ MgR T 2 7 2 0 4n where T is the period of the pendulum e NOTE You used a small angle approximation to find the equation for 7 but J does not depend on 0 This means that you must measure the period T using small angle oscillations Once you have calculated J with that period you may use that value of J regardless of the amplitude reached during other parts of the experiment Installing the Optional Photogate Bracket ME 6821A The Photogate Bracket is an optional accessory for mounting one or two photogates on the Projectile Launcher to measure the muzzle speed of the ball e Prepare the Photogate Bracket by loosening the thumbscrew near the end of the bracket Leave the square nut in place on the end of the thumbscrew Use the smaller 0 75 in thumbscrews that are stored on the bottom side of the bracket to mount one or two photogates to the bracket e Align the square nut of the bracket with the T shaped slot on the bot
45. lum to a sturdy table or other surface e Mount the Projectile Launcher on the Ballistic Pendulum base Mount the Launcher to the lower two holes in the base if you intend to shoot horizontally at the ball catcher of the Ballistic Pendulum e Use the hole and curved slot near the top of the base when you want to adjust the Launcher s launch angle Note For this configuration the Launcher should be mounted on the back side of the Ballistic Pendulum base Muzzle Launcher on high position w 4 012 053750 IZEIA Model No ME 6830 ME 6831 General Operation of the Projectile Launcher Aim e Ifyou have the Launcher mounted on the top position you can adjust the angle of inclination above the hori zontal 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 e You can bore sight through the barrel at a target such as the ME 6853 Shoot The Target Look through the back end of the barrel when the Launcher is not loaded There are two tripod three spoke sights inside the barrel one at the end of the barrel and one at the end of the piston about midway in the barrel Each sight has a sighting hole at its center Loosen th
46. nd record it in Table 5 3 2 Calculate the final gravitational potential energy and record it in Table 5 3 3 Calculate the percent difference between the initial kinetic energy and the final gravitational potential energy and record it in Table 5 3 KE GPE 109 KE GPE 2 Table 5 3 Results Item Value Maximum height of ball Mass of ball Initial Kinetic Energy Final Potential Energy Percent difference Questions 1 How does the initial kinetic energy compare to the final gravitational potential energy 2 How does friction in the form of air resistance affect the result for the conservation of energy 3 When the Launcher is cocked it has elastic potential energy If energy is conserved how should the elastic potential energy compare to the initial kinetic energy 30 012 05375C TPA LIH o Model No ME 6830 Exp 6 Conservation of Momentum Exp 6 Conservation of Momentum Equipment Needed Item Item Projectile Launcher and 2 plastic balls 2 D Collision Accessory Meter stick or measuring tape Sticky tape White paper large sheet Carbon paper 2 or 3 sheets Protractor Plumb bob and string Purpose The purpose of this experiment is to confirm that momentum is conserved for elastic and inelastic collisions in two dimensions Theory A ball is shot toward another ball that is initially at rest resulting in a collision after which the two balls move in different directions
47. other horizontal surface at that location Figure 3 3 Shooting to a level surface so the ball will hit the top of the box at the same level as the muzzle of the launcher Procedure Part A Shooting to a Level Surface 1 Fire one shot to locate where the ball hits the top of the box Tape a piece of white paper on the box at this location Tape a piece of carbon paper carbon side down on top of the white paper e When the ball hits the carbon paper it will leave a mark on the white paper underneath 2 Fire five shots 3 Use a measuring tape to measure the horizontal distance from the muzzle to the leading edge of the paper If a measuring tape is not available use a plumb bob to find the point on the table that is directly beneath the release point on the barrel and measure the distance along the table from the muzzle to the leading edge of the paper Record the distance in the Data Table 4 Carefully remove the carbon paper Measure from the leading edge of the paper to each of the five dots and record these distances in the Data Table 5 Increase the launch angle by 10 degrees and repeat all the steps 6 Keep repeating for angles up to and including 80 degrees the complementary angle of 10 degrees Table 3 1 Shooting to a Level Surface Angle 10 20 30 40 50 60 70 80 1 S c 2 2 3 a T 4 5 N 5 So amverace a I Average Paper distance To
48. rmined in order to calculate the initial kinetic energy To calculate the initial speed vo of a projectile fired horizontally the horizontal distance travelled by the projectile is x vot where is the time that the projectile is in the air The vertical distance that projectile drops in time f is given by _1 2 y s t The initial speed of the projectile can be calcu lated by measuring x and y and using y to calcu late the time t The time of flight of the projectile can be found using t E s X A Ng Figure 5 2 Find the initial speed and then the initial speed can be found using Yo sl 7 Wrefe 012 05375C 27 Projectile Launcher Exp 5 Conservation of Energy Setup 1 Clamp the Projectile Launcher to a sturdy table or other horizontal surface Mount the Launcher near one end of the table with the Launcher aimed away from the table 2 Point the Launcher straight up and fire a test shot on medium range to make sure that the ball doesn t hit the ceiling If it does use the short range setting for this experiment or put the Launcher closer to the floor 3 Adjust the angle of the Projectile Launcher to zero degrees so the ball will be launched horizontally Procedure Part A Determine the Initial Speed without photogates 1 Put the plastic ball into the Launcher and cock it to the medium range setting Fire one shot to locate where the ball hits the floor At that position tape a piec
49. sec x vot em Y Yo 1 2 gt cm 0 1 0 2 0 3 0 4 0 5 3 Lay the two meter stick on the floor in a straight line away from the Launcher Remove the back mounting screw from the Launcher base so that the back of the Launcher can rotate upward Look through the Launcher at the end of the two meter stick Adjust the end of the stick until the end is aligned with the sites in the Launcher and the stick is along the path of the ball when it is fired 4 Starting at the muzzle of the Launcher measure off each set of x and y distances and place a ring clamp on a stand at each position corresponding to one tenth of a second see Figure 10 1 5 Shoot the ball through the rings 6 Ask the class What shape of curve is formed by the rings What is the path of the projectile Figure 12 1 Demonstration setup y 48 012 05375C LLH Model No ME 6830 Exp 13 Demo Elastic and Inelastic Collisions Exp 13 Demo Elastic and Inelastic Collisions Equipment Needed Item Projectile Launcher and plastic or steel ball Purpose The purpose of this demonstration is to show the difference in kinetic energy transfer between an elastic collision and an inelastic collisions Theory The amount of kinetic energy transferred between colliding objects depends on the elasticity of the collision By reversing the pendulum of the Ballistic Pendulum so the ball bounces off instead of being caught
50. served Exp 9 Projectile Motion Exact Method Procedure 1 The exact mass is not critical Pick a value that gives a fairly large swing for best results 2 With the steel ball and extra masses on the pendulum the balance point will be somewhere on the ball catcher itself This makes it difficult to use string but it is relatively easy to find the center of mass by balancing the pendulum on a straightedge 3 Measure this period as exactly as possible using the smallest measurement angle that is practical 4 The angle reached by the pendulum should not vary more than one degree between successive trials Calculations e Use the equations given in the theory section for the approximate method Questions 1 The best other method of measuring velocity is described in the first part of experiment 1 2 Sources of error include friction and measurement error 3 Typically 70 of the kinetic energy of the ball is lost It is not value to assume that kinetic energy is con served 4 The energy transfer is less efficient when there is a larger difference in the masses involved 5 The exact method will give results that are typically within 2 5 of the actual value 56 012 05375C 2 Wiele Model No ME 6830 Technical Support Technical Support For assistance with any PASCO product contact PASCO at Address PASCO scientific 10101 Foothills Blvd Roseville CA 95747 7100 Phone 916 786 3800 worldwide 8
51. software Purpose The purpose of this experiment is to predict and verify the range of a ball launched at an angle Photogates are used to determine the initial speed of the ball Theory To predict where a ball will land on the floor when it is shot from the Launcher at some angle above the horizontal it is first necessary to determine the initial speed muzzle velocity of the ball The speed can be determined by shooting the ball and measuring a time using photogates To predict the range x of the ball when it is shot with an initial speed at an angle 0 above the horizontal first predict the time of flight using the equation for the vertical motion 1 2 t z t y ygt vgsin 28 where y is the initial height of the ball and y is the position of the ball when it hits the floor Solve the quadratic equation to find the time t Use x vg cos 8 to predict the range e NOTE For best results see the notes on Repeatable Results in the Introduction Setup 1 Put the Launcher in the top position on the Ballistic Pendulum upright Clamp the Ballistic Pendulum Projec tile Launcher to a sturdy table or other horizontal surface Mount the Launcher near one end of the table aimed away from the table 2 Adjust the angle of the Projectile Launcher to an angle between 30 and 60 degrees and record the angle 3 Attach the photogate mounting bracket to the Launcher and attach two photogates to the bracket Check that the d
52. speed of the ball was found to be 4 93 m s for the short range launcher at the medium range setting The ball mass was 9 6 g so the total kinetic energy was 0 117 J 2 The ball reached an average height of 1 14 m above the muzzle The gravitational potential energy was 0 107 J 3 The energy difference was 8 5 of the original kinetic energy 4 NOTE It seems unlikely that this much energy is lost due to air resistance especially when you consider the extraordinarily good results for labs 3 and 4 It is likely that the error here enters the calculation in the actual measurements of initial velocity and maximum height LEIHO 012 05375C 53 Ballistic Pendulum Projectile Launcher Teacher s Guide Exp 6 Conservation of Momentum in Two Dimensions Setup e If possible use medium range setting instead of the short range setting The medium range setting gives more predictable results than the short range setting Analysis e Results for the x component of momentum should be within 5 of initial values The total y component should be very small compared to the x component Questions 1 Momentum is conserved on both axes 2 Kinetic energy is nearly conserved in the elastic collision There is some loss of energy which indicates that the collision is not perfectly elastics 3 Momentum is conserved for the inelastic collision but kinetic energy is not conserved 4 The angle should be nearly 90 Trials had angles o
53. square nut The bar has a post and you can balance a second ball on the post in front of the muzzle When the launched ball collides with the second ball they experience a two dimensional 2 D collision Assembly To assemble the Collision Accessory insert the thumb screw through the hole in the plastic bar and screw the square nut onto the thumbscrew Leave the square nut loose on the thumbscrew until you install the Collision Acces sory onto the Launcher To install the Collision Accessory onto the Launcher slide the square nut into the T shaped slot on the bottom side of the barrel Adjust the position of the Collision Acces sory and then tighten the thumbscrew Place a ball on the top of the post loosen the thumbscrew slightly and rotate the Collision Accessory to one side or the other until the ball on the post is in a place where it will be hit by the launched ball at the angle that you want Expectations for the Projectile Launcher e The muzzle speed will vary slightly with angle The difference between muzzle speed when shot horizontally versus vertically can be between zero to eight percent depending on the range setting e Although the muzzle end of the Projectile Launcher does not change height with angle it is about 30 centime ters 12 inches above table level If you desire to show that projectiles fired with the same muzzle speed but at complementary angles will have the same range you need to shoot to a horizontal
54. tal distance Part B Shooting Off the Table 1 Turn the Projectile Launcher so it will launch the ball to the floor 20 012 05375C I Ego Model No ME 6830 Exp 3 Projectile Range versus Angle 2 Repeat the procedure and record the data in the Data Table Table 3 2 Shooting off the Table Angle 10 20 30 40 50 60 70 80 8 2 1 S 2 n T 3 5 4 N 5 lt Average Paper distance Total distance Analysis 1 Find the average of the five distances in each case and record the results in the Data Tables 2 Add the average distance to the distance from the Launcher to the leading edge of the white paper to get the total distance range in each case Record the results in the Data Tables 3 For each Data Table plot the range versus the angle and draw a smooth curve through the points Questions 1 From the graph what angle give the maximum range for each case 2 Is the angle for the maximum range greater or less for shooting off the table 3 Is the maximum range further when the ball is shot off the table or on the level Notes i7 Wrefe 012 05375C 21 Projectile Launcher Exp 3 Projectile Range versus Angle 22 012 05375C IZEIA Model No ME 6830 Exp 4 Projectile Path Exp 4 Projectile Path Equipment Needed Item Item Projectile Launcher and plastic ball Movable vertical targ
55. tal surface The features of the Ballistic Pendulum include e Reliable Ball Catcher Mechanism The sensitive spring loaded barb type catch on the pendulum will catch balls with a large range of speeds In addition the catcher holds the ball in line with the pendulum rod for best accuracy e Removable Pendulum All moving parts of the pendulum may be removed so that the mass and center of mass can be measured accurately In addition the pendulum can be reversed so that elastic collisions can be compared to inelastic collisions 2 012 05375C 2 Wiele Model No ME 6830 ME 6831 Introduction e Variable Mass Pendulum The pendulum includes masses that can be removed so that the pendulum can be used with lightweight balls over a wide range of speeds Leave the masses on the pendulum when you use heavyweight balls The features of the Projectile Launcher include e Launch at Any Angle Balls can be launched from any angle from zero to ninety degrees measured from hor izontal zero degrees The angle is easily adjusted using thumbscrews and the built in protractor and plumb bob give an accurate way to measure the angle of inclination e Three Range Settings Each version of Projectile Launchers has three range settings The Short Range Pro jectile Launcher ranges are approximately 1 2 m 3 m and 5 m when the launch angle is 45 The Long Range Projectile Launcher ranges are approximately 2 5 m 5 m and 8 m The Long Range Launcher
56. tee Shoot the ball straight ahead and repeat the procedure five times Elastic Collision Use two balls Load Ball linto the Launcher at the short range setting Place Ball 2 on the tee of the 2 D Collision Accessory Shoot Ball 1 so it collides with the target ball Ball 2 Repeat the procedure five times Inelastic Collision Use two balls Load Ball linto the Launcher at the short range setting Put a small loop of sticky tape sticky side out on Ball 2 and place it on the tee Orient the tape side of Ball 2 so that it will be struck by the launched ball Ball 1 causing an inelastic colli sion Fire a test shot to locate where the two balls hit the table Tape a piece of carbon paper to the white paper Shoot Ball 1 and if the two balls stick together but miss the carbon paper relocate the carbon paper and shoot once more 32 012 05375C LLEI O Model No ME 6830 Exp 6 Conservation of Momentum e Since the tape does not produce the same inelastic collision each time it is only useful to record this collision once 5 Use a plumb bob to locate on the paper the spot directly below the point of contact of the two balls Mark this spot on the paper as the point of contact spot Carefully remove the carbon paper from the white paper Analysis The time of flight for each shot is the same because the vertical distance for each shot is the same Therefore the horizontal length of each pa
57. th is proportional to the speed of the ball Since the masses are the same the horizontal length of each path is also proportional to the momentum of the ball A No Collision 1 Draw straight lines from the point of contact spot to each of the dots made by the no collision shots 2 Measure each straight line and record the length Find the average of the five lengths and record the length as the initial x momentum in Table 6 1 and Table 6 2 For example if the length is 65 cm record 65 as the value for the initial x momentum but do not include any units B Elastic Collision 1 Draw a straight line from the point of contact through the center of the group of dots made by the no colli sion shots This is the center line from which all of the angles will be measured 2 Draw straight lines from the point of contact spot to each of the dots made by the elastic collision shots There should be five lines on each side of the center line 3 Measure from the point of contact to each of the dots made by Ball 1 Find the average of the five lengths Draw a straight line from the point of contact through the center of the group of dots made by Ball 1 4 Measure the angle from the centerline to the straight line for Ball 1 Use this angle and the average length of the line for Ball 1 to calculate the x component for Ball 1 and the y component for Ball 1 Record the values 5 Me
58. the carbon paper and measure from the leading edge of the white paper to each of the ten dots Record these distances in the Data Table and find the aver age Calculate and record the total horizontal distance distance to paper plus average distance from edge of paper to dots 6 Using the vertical distance y and the total horizontal distance x calculate the time of flight t and the initial horizontal speed of the ball vg Record the time and speed in the Data Table Part B Predicting the Range of a Ball Shot at an Angle 1 Adjust the angle of the Projectile Launcher to an angle between 30 and 60 degrees Record this angle in the second Data Table 2 Using the initial speed and vertical distance from the first part of this experiment calculate the new time of flight and the new horizontal distance based on the assumption that the ball is shot at the new angle you have just selected Record the predictions in the second Data Table 3 Draw a line across the middle of a white piece of paper and tape the paper on the floor so that the line on the paper is at the predicted horizontal distance from the Projectile Launcher Cover the white paper with carbon paper carbon side down and tape the carbon paper in place 4 Shoot the ball ten times 5 Carefully remove the carbon paper Measure the distances to the ten dots and record the distances in the sec ond Data Table Analysis 1 Calculate the percent difference between the pred
59. the vertical target Figure 4 1 Launcher setup board Place the target board on the floor so that the ball hits the board near the bottom See Figure 4 1 LEIHO 012 05375C 23 Projectile Launcher Exp 4 Projectile Path 4 Cover the target board with white paper Tape carbon paper over the white paper Procedure 1 Measure the vertical height from the floor to the muzzle and record the height in the Table 4 1 Mark this height on the target 2 Measure the horizontal distance from the muzzle of the Launcher to the target board and record it in the Data Table 3 Shoot the ball 4 Move the target board about 10 to 20 cm closer to the Launcher 5 Repeat steps 2 through 4 until the height of the ball when it strikes the target board is about 10 to 20 cm below the height of the muzzle Data Table 4 1 Height of Muzzle Table 4 1 x y Data Horizontal x Vertical y x Analysis 1 On the target board measure the vertical distances from the muzzle level mark down to the ball marks and record them in Table 4 1 2 Calculate x2 for all the data points and record them in the Data Table 3 Plot a graph of y versus x2 and draw the best fit light through the data points 4 Calculate the slope of the graph and record it in Table 4 2 5 From the slope of the graph calculate the initial speed of the ball as it leaves the muzzle Record the initial speed in Table 4 2
60. thod Begin with the potential energy of the pendulum at the top of its swing after the collision with the ball APE MgAh where M is the combined mass of the pendulum and ball g is the acceleration due to gravity and Ah is the change in height Substitute for the change in height Ah R 1 cos APE Mgk 1 cos where R is the distance from the pivot point to the center of mass of the pendulum ball system This potential energy is equal to the kinetic energy immediately after the collision 1 2 K aM where vp is the speed of the speed of the pendulum just after collision The momentum of the pendulum just after the equation is P M Vp which you can substitute into the previous equation to give 2 KE 2M Solving this equation for the pendulum momentum gives P J2M KE p 6 012 05375C PA SCO Model No ME 6830 ME 6831 Ballistic Pendulum Theory This momentum equal to the momentum of the ball just before the collision P mv b b Setting these two equations equal to each other and replacing KE with our known potential energy gives mv mgr cos Solve this for the ball s velocity and simplify to get M n 2gR 1 cos 9 Exact Method The potential energy is found in a way identical to the way shown previously APE MgR_ 1 cos For the kinetic energy you can use the equation for angular kinetic energy instead of linear kinetic
61. tom of the Projectile Launcher barrel and slide the nut into the Launcher slot until the photogate nearest to the bar parre rel is as close to the muzzle as possible without blocking the photogate beam Tighten the bracket thumbscrew to secure the bracket in place Photogate Mounting Bracket Photogate Install the Optional Photogate Bracket 8 012 05375C PA SCO Model No ME 6830 ME 6831 Ballistic Pendulum Theory Repairing the Plumb Bob If the string breaks that holds the plumb bob on the protractor of the Launcher replace it with an equal length of nylon thread such as the thread included in the ME 6802 Launcher Spares Kit Make sure that the replace ment string is long enough so that when the Launcher is inclined at an angle of 50 the string extends well below the corner of the Launcher Carefully thread the replacement string through the small hole at the vertex of the pro tractor and tie a triple knot at that end of the string To put the plumb bob onto the string thread the string through the hole in the center of the plumb bob and tie a triple knot in that end of the string Make the string long enough Plumb bob Thread through the hole Installing the 2 D two dimensional Collision Accessory Tie a triple knot in the end Repairing the Plumb Bob Introduction Projectile The 2 D two dimensional Collision Accessory is a plastic Launcher bar with a thumbscrew and
62. tual average dis tance B when shot at an angle 1 Estimate the precision of the predicted range How many of the final 10 shots landed within this range A B A B 2 x100 IZEIA 012 053750 17 Projectile Launcher Exp 2 Projectile Motion Using Photogates Notes 18 012 05375C DLHO Model No ME 6830 Exp 3 Projectile Range versus Angle Exp 3 Projectile Range versus Angle Equipment Needed Item Item Projectile Launcher and plastic ball Plumb bob and string Meter stick or measuring tape Box to make landing area same elevation as muzzle Graph paper Carbon paper White paper Sticky tape Purpose The purpose of this experiment is to determine how the range of the ball depends on the launch angle The angle that gives the greatest range is determined for two cases for shooting on level ground and for shooting off a table Theory The range is the horizontal distance x between the muzzle of the Launcher and the place where the projectile hits given by x vg cos 0 t where v is the initial speed of the projectile as it leaves the muzzle 0 is the launch angle above horizontal and f is the time of flight See the figure X Figure 3 1 Shooting on a level surface For the case in which the projectile hits on a surface that is the same level as the level of the muzzle of the Launcher the time of flight of the projectile will be twice the time it takes for the pro
63. uncher the time of flight of the ball will be twice the time it takes the ball to reach the peak of its trajectory when its vertical component of speed reaches zero 2vgsin t 2t peak 7 where g is the acceleration due to gravity Substituting for t in the equation for x gives and f is the time of flight 2 2vosin cos 8 KS and using a trigonometry identity gives 2 vosin20 8 X The ranges for the angles 30 and 60 are the same since sin 60 sin 120 Setup 1 Clamp the Projectile Launcher to a sturdy table Mount the Launcher near one end of the table with the Launcher aimed toward the middle of the table 2 Adjust the angle of the Launcher to 30 3 Put the steel ball into the Launcher and cock it to the medium range or long range setting e NOTE In general this demonstration will not work as well on the short range setting because the muzzle speed is more variable with the change in angle 4 Fire a test shot to see where the ball hits Place the box in front of that location so that the next ball will hit the top of the box Procedure 1 Shoot the ball at 30 to demonstrate that the ball lands on the box LEIHO 012 05375C 43 Projectile Launcher Exp 10 Demo Do 30 and 60 Give the Same Range Change the angle of the Launcher to 60 and shoot the ball again Call attention to the fact that the ball again lands on the box confirming that th
64. unchers adjacent to each other on a sturdy table Mount the Launchers near one end of the table with the Launchers aimed away from the table so the balls will land on the floor 2 Adjust the angle of the Launcher to 0 so lt Ashen i the balls will fire horizontally i Xiong Figure 9 1 Shots fired simultaneously Procedure 1 Put aplastic ball into each Launcher Cock one Launcher to the short range setting and clock the other Launcher to the long range setting 2 Ask the class to be quiet and listen for the balls striking the floor e NOTE If there is only one click that means that the balls hit the floor simultaneously 3 Put both trigger release strings in the same hand and pull them at the same time so that the balls are launched simultaneously 4 After the balls hit the floor ask the class if they heard one click or two LEIHO 012 05375C 45 Projectile Launcher Exp 11 Demo Simultaneous Shots at Different Speeds 46 012 05375C 7 Wefe Model No ME 6830 Exp 12 Demo Shooting Through Hoops Exp 12 Demo Shooting Through Hoops Equipment Needed Item Item Projectile Launcher and plastic ball Ring clamp on stand 5 Photogate Head ME 9498A 2 optional Photogate Mounting Bracket ME 6821A optional Meter stick Two meter stick Purpose The purpose of this demonstration is to confirm that the part of a projectile is parabolic Theory The range is the horizontal distance x
65. y The gravitational potential energy is equal to the kinetic energy of the pendulum at the bottom of the swing just after the collision with the ball You cannot equate the kinetic energy of the pendulum after the collision with the kinetic energy of the ball before the swing since the collision between ball and pendulum is inelastic and kinetic energy is not conversed in inelas tic collisions Momentum is conserved in all forms of collisions so you know that the momentum of the ball before the collision is equal to the momentum of the pendulum after the collision Once you know the momentum of the ball and the ball s mass you can determine the initial velocity There are two ways of calculating the velocity of the ball The first method called the approximate method assumes that the pendulum and the ball together act as a point mass located at their combined center of mass This method does not take rotational inertia into account It is somewhat quicker and easier than the second method called the exact method but not as accurate The second method exact method uses the actual rotational inertia of the pendulum in the calculations The equa tions are slightly more complicated and it is necessary to take more data in order to find the moment of inertia of the pendulum but the results are generally better Please note that the subscript cm used in the following equations stands for center of mass Approximate Me

PASCO Specialty & Mfg. ME-6830, User's Manual

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PASCO Specialty &amp; Mfg. ME-6830, User's Manual

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PASCO Specialty & Mfg. ME-6830, User's Manual