Invisible Collisions

Transcription

1 Invisible Cllisins Overview: This activity relates an elastic cllisin t the change in a satellite s r spacecraft s speed and directin resulting frm a planetary fly-by, ften called a gravity assist maneuver. Bth hands-n and nline interactive methds are used t explre these tpics. Target Grade Level: 9-12 Estimated Duratin: 2 40-minute sessins Learning Gals: Students will be able t describe the speed f a ball after it cllides elastically with a large bject. In particular, cmpare cases in which the large bject is statinary, is mving tward the ball, r is mving away frm the ball. describe a spacecraft s change f speed and directin as it passes near a planet ( planetary flyby ) in terms f an elastic cllisin with the planet. describe the relatinship between the distance f a spacecraft s clsest apprach t a planet and the resulting change in a spacecraft s speed and directin using an nline mdel. Standards Addressed: Benchmarks (AAAS, 1993) The Nature f Science, 1A: The Scientific Wrld View The Physical Setting, 4G: The Universe Natinal Science Educatin Standards (NRC, 1996) Unifying Cncepts and Prcesses: Evidence, mdels, and explanatin Physical Science: Mtins and frces Principles and Standards fr Schl Mathematics (NCTM, 2000) Representatin Algebra Table f Cntents: Backgrund Page 2 Materials and Prcedure 5 Extensins and Adaptatins 7 References 7 Standards Addressed, detailed 8 Cllisins Student Data Sheet 10 Gravity Assist Simulatr student data sheet 14 Teacher Answer Key 19

2 Backgrund: We are all familiar with the term cllisin as used in ur everyday lives. If yu are unlucky enugh t be riding in a car that hits smething anther car, r perhaps a telephne ple then yu have been in an autmbile cllisin. In physics, thugh, this wrd is used t describe any situatin in which tw bjects traveling thrugh space meet, interact briefly, and then cntinue n their way. Typically, their mtins after the cllisin are different frm their mtins befrehand: they have speeded up, slwed dwn, r their directins f travel have changed. A. B. C. Figure 1. A. Tw cars apprach an intersectin. B. The tw cars cllide. C. The tw cars change bth speed and directin as a result f the cllisin. Such cllisins are all arund us. A bat and ball cllide t prduce a hit; the batted ball is traveling in a different directin, and usually at a different speed, than when it was pitched. Cllisins and Kinetic Energy If tw bjects cllide, what is knwn abut hw their mtins change? Fr example, if we knw their speeds and directins befre the cllisin, what can we say abut their mtins afterward? Answering this questin depends n many details f the situatin, but the matter is greatly simplified if we knw as is the case in a spacecraft flyby that kinetic energy is cnserved. K = ½ m v 2 Figure 2. Kinetic energy K is prprtinal t the prduct f an bject s mass m and the square f its speed v. Kinetic energy is the energy f an bject that depends n its speed, as shwn abve; the greater the speed f an bject, the larger the kinetic energy. When tw bjects cllide, we can cmpare their ttal kinetic energy (the sum f their individual kinetic energies) befre the cllisin, with their ttal kinetic energy after the cllisin. Saying that kinetic energy is cnserved simply means that these tw ttals are the same. A cllisin f this kind is called elastic, and in practice this means that n energy is lst in prcesses such as defrming the bjects bent fenders, fr example, r skidding tires. 2

3 T build ur understanding abut elastic cllisins, let s cnsider tw bjects that are mving alng the same line, and cllide elastically. Imagine that ne bject vertakes the ther s that they strike ne anther and bunce apart. Then the bject that was vertaken speeds up, while the bject that was vertaking (i.e., initially faster) slws dwn r pssibly even reverses directin. The amunt f slwing dwn r speeding up depends n the relative masses f the tw bjects; if ne is much mre massive than the ther, then its mtin scarcely experiences any change at all. And if the bjects are mving in ppsite directins befre they cllide, and ne is mre massive then the ther, then the lighter bject reverses directin as the result f the cllisin. Such scenaris with ne mass very much larger than the ther are illustrated in Velcities and Cllisins part f the Gravity Assist Simulatr interactive ( Cnnectin t the Gravity Assist Maneuvers Hw des this help us t understand a spacecraft s gravity assist maneuver, such as what tk place in July 2007 when the New Hrizns spacecraft passed near Jupiter n its way tward Plut? Althugh the spacecraft did nt hit Jupiter s surface, it passed clse enugh t feel Jupiter s gravitatinal pull very strngly. We can say, therefre, that it cllided with Jupiter even thugh it did nt tuch the planet. This cllisin had n measurable effect n the mtin f Jupiter, since the planet is enrmusly mre massive than the spacecraft. But it had a prfund effect n the spacecraft's mtin. Jupiter mves rapidly in its rbit abut the Sun, at abut 13.1 km/s. The spacecraft's mtin was at an angle t the planet's rbit that is, partly perpendicular t the rbit, causing it t mve away frm the Sun, and partly parallel, mving in the same directin as Jupiter. The velcity assciated with this secnd (parallel) part f the mtin was initially smaller than 13.1 km/s. S the spacecraft was vertaken by Jupiter, and was "pulled frward" as it passed behind the planet. As a result, the spacecraft was speeded up by the encunter. Figure 2. As yu can see in this rugh simulatin, the New Hrizns spacecraft s mtin is at an angle t Jupiter s rbit as it appraches the July, 2007 flyby. Ntice the speed f the spacecraft is abut 14 km/s. 3

4 Figure 3. As the New Hrizns spacecraft flew past Jupiter, which is traveling frm right t left in this image, the spacecraft was pulled tward Jupiter by the planet s gravity. Yu can see that the path and the speed f the spacecraft were changed by the encunter. This effect can als be bserved in the Gravity Assist Simulatr interactive that is a part f this lessn: Nte that in the Jupiter flyby, bth the directin and the speed f the spacecraft are changed in the encunter. Remember that the gravitatinal attractin between spacecraft and planet is larger the clser the tw cme t each ther. S it shuld nt be surprising that the biggest effect n speed and directin f the spacecraft is bserved in the ptin with the clsest distance f apprach. In additin t bsting a spacecraft s speed and helping it reach the uter planets, flybys can be used t mdify a spacecraft s trajectry in ther ways. Fr example, the animatin includes a Venus flyby, in which a spacecraft passes in frnt f a mving planet and is slwed dwn by the encunter. The MESSENGER spacecraft is currently en rute t the planet Mercury, using a cmplex sequence f flybys invlving Earth (nce), Venus (twice) and Mercury (three times). Each flyby is carefully cnfigured s that after a jurney f six and a half years the MESSENGER spacecraft will be psitined t enter a nearly plar rbit abut the innermst planet, using a minimum amunt f fuel in the prcess. 4

5 Materials: small, dense rubber balls (e.g., superballs) (1 fr each grup) desks r tables (1 area fr each grup) pieces f wd at least the size f a bk r ther handy very hard bjects (NOTE: this bject shuld nt have an absrbent/sft layer like a ping png paddle has) (1 fr each grup) meter sticks (1 per grup) cpies f Cllisins grup data sheet (1 per grup) cpies f Gravity Assist Simulatr data sheet (1 per persn r per grup) cmputers with access t the internet fr either in-class r hmewrk assignment safety gggles (1 pair per student) Prcedure: Brief verview What the teacher will d: The teacher will demnstrate an elastic cllisins fr the class, then walk arund t answer questins while the students are ding the Cllisins activity in small grups. Then the teacher will distribute the Gravity Assist Simulatr data sheet fr students t cmplete as they view the nline Gravity Assist Simulatr in class r as hmewrk. What the students will d: The students will explre different elastic cllisin scenaris in a hands-n activity fllwed by an nline interactive activity. Fr the hands-n Cllisins activity they will drp a ball ( small bject ) nt a table surface r piece f wd ( large bject ). First the large bject is held statinary while the ball cllides with it. Then it is mving fr tw different scenaris. Students recrd their bservatins f these different elastic cllisins in the Cllisins student data sheet. Then they explre similar cllisins and scenaris using an nline interactive. While explring the interactive they recrd their bservatins and answer questins using the Gravity Assist Simulatr data sheet. Advance Preparatin 1. Make cpies and gather materials as indicated in the materials sectin. 2. Arrange fr cmputer time if needed (see 4 under In-class Prcedure ). In-class Prcedure 1. Briefly explain t the class that they will be explring cllisins. Arrange the class with the help f the students t facilitate the hands-n elastic cllisin explratin. Yu may need t mve sme desks aside s each grup can gather arund ne desk r table and drp a ball near the edge f that surface. See the Cllisins grup data sheet fr mre details n hw the students set up their hands-n cllisin explratin. Be sure t emphasize classrm safety rules and use safety gggles with the classrm activity. 2. Assign students t grups f abut 3 r 4 students. Prvide each grup with the fllwing: A. 1 cpy f the Cllisins grup data sheet 5

6 B. the piece f wd/paddle/ther large bject C. a small rubber ball (e.g., a superball) D. a meter stick 3. Students shuld fllw the instructins n the Cllisins grup data sheet by explring the three elastic cllisin scenaris (1. ball cllides with a statinary large bject, 2. ball and large bject are mving in ppsite directins, 3. ball and large bject are mving in the same directin and ball vertakes large bject). They shuld answer the questins n the student data sheet as they prceed thrugh the explratry. Yu may wish t demnstrate the first scenari in frnt f the class, r walk the students thrugh the first trial. 4. Either prvide in-class cmputer time r assign as hmewrk the Gravity Assist Simulatr student data sheet, in which students answer questins related t the nline gravity assist interactive. Nte: if prviding in-class cmputer time yu may want t assign students t grups f tw. **This data sheet can be used fr assessment.** URL fr the Gravity Assist Simulatr interactive: 6

7 Extensins and Adaptatins: Fr seeing impaired students, ask ne f their team members t describe the set-up and changes in speed fr all f the scenaris in the hands-n activity. We have included lng descriptins in the nline interactive. Access them by clicking n the Learn Mre tab in the upper right crner f each scenari, and then selecting D-Link. References: Frm the New Hrizns website: (specifically see pages 2 and 9) Frm the MESSENGER website: An advanced explanatin: A basic explanatin: 7

8 Standards: Natinal Science Educatin Standards (NRC, 1996) Cntent Standards: 9-12 Unifying Cncepts and Prcesses, CONTENT STANDARD: Evidence, mdels, and explanatin Physical Science, CONTENT STANDARD: Mtins and frces Benchmarks (AAAS, 1993) Chapter 1. The Nature f Science 1A: The Scientific Wrld View Grades 9 thrugh 12 Chapter 4. The Physical Setting 4G: The Universe Scientists assume that the universe is a vast single system in which the basic rules are the same everywhere. The rules may range frm very simple t extremely cmplex, but scientists perate n the belief that the rules can be discvered by careful, systematic study. Grades 9 thrugh 12 Gravitatinal frce is an attractin between masses. The strength f the frce is prprtinal t the masses and weakens rapidly with increasing distance between them. Principles and Standards fr Schl Mathematics (NCTM, 2000) Representatin Algebra Grades K thrugh 12 Use representatins t mdel and interpret physical, scial, and mathematical phenmena Create and use representatins t rganize, recrd, and cmmunicate mathematical ideas 8

9 Grades 9 thrugh 12 Analyze change in varius cntexts: apprximate and interpret rates f change frm graphical and numerical data 9

10 Cllisins Student Data Sheet grup members: Recall, in an elastic cllisin the ttal kinetic energy is the same befre the cllisin as it is afterward. N energy is used t defrm the bjects permanently in the cllisin (such as car fenders are bent during an autmbile cllisin). If the ttal kinetic energy remains the same, can the speed f the bjects in the cllisin change? In the fllwing activity we will explre elastic cllisins invlving a small bject (e.g., a super ball) and a large bject (e.g., a piece f wd). Pay clse attentin t the speeds during the three scenaris belw. Instructins: 1. Fr each f the fllwing scenaris yu will be drpping a ball frm the same height, 50 cm. In tw f them yu will als be mving a large bject (e.g., a piece f wd) in sme directin while the ball is mving. The meter stick shuld be placed with 0 m n the surface f the table. 2. Read the instructins fr the ball, the meter stick, and the large bject befre attempting each scenari. Try each scenari a few times and then recrd yur results belw. 3. This diagram shuld help yu set up fr the activity (nte: S1 is fr Scenari 1, S2 is fr Scenari 2, etc.): One persn shuld hld the meter stick perpendicular t the surface and near the edge f the desk S1 S3 S2 The ball shuld be drpped frm the same height in each scenari Scenari 1: Here yu will investigate what happens when the ball cllides with a larger bject that is nt mving. 10

11 The large bject: place the large bject r piece f wd (labeled with S1 in abve diagram) n the surface f the desk and parallel with the surface ( flat ). The meter stick: place the meter stick n tp f the large bject fr this scenari and perpendicular t the surface as in the diagram abve. The ball: drp the ball frm 50 cm nt the large bject. Befre yu begin, predict hw yu think the interactin between the ball and the large bject will change the speed and the bunce height f the ball: Nw perfrm the activity. Observe hw high the ball bunces after clliding with the large bject. Try t bserve the speed f the ball as it appraches the large bject (initial speed), and then immediately after it has hit the large bject and is traveling in the ppsite directin (final speed). Repeat this several times. A. Hw des the height frm which it was drpped cmpare with the height t which it bunced? B. Hw des the initial speed f the ball cmpare with the final speed (smaller than, larger than, r the same as)? Scenari 2: In this scenari yu will investigate what happens when the ball and the larger bject are mving in ppsite directins (i.e., tward ne anther). The large bject: begin with the large bject parallel with the surface f the desk, but ff t the side and belw the desk surface by abut 10 cm. (Psitin S2 in the diagram abve). Yu will mve this bject up just as the ball appraches s that they cllide at abut the same height as the bttm f the meter stick (r edge f the desk). The meter stick: place the meter stick n and perpendicular t the surface f the desk, as in the diagram abve. It shuld be near the edge f the desk clsest t the large bject. The ball: Drp the ball frm 50 cm nt the surface f the large bject, which will be mving tward it. 11

12 This time yu will drp the ball tward the large bject, hwever the large bject will mve tward the ball as it appraches. Befre yu begin, predict hw yu think the interactin between the ball and the large bject will change the speed and the bunce height f the ball: Nw perfrm the activity. Again bserve the initial speed f the ball as it appraches the large bject and the final speed and height after the bject and ball have cllided. Try this several times and then recrd yur bservatins: C. Hw des the height frm which it was drpped cmpare with the height t which it bunced? D. Hw des the initial speed f the ball cmpare with the final speed f the ball (smaller than, larger than, r the same as)? Scenari 3: In this scenari yu will investigate what happens when the ball vertakes the large bject because the ball and the large bject are mving in the same directin and the ball is mving faster than the large bject. The large bject: begin with the large bject abve parallel t the surface f the desk, but ff t the side and abut 10 cm abve the surface f the desk (psitin S3 in the diagram). Yu will mve this bject dwn just as the ball appraches s that they cllide at abut the same height as the bttm f the meter stick (r edge f the desk). The meter stick: place the meter stick n and perpendicular t the surface f the desk, as in the diagram abve. It shuld be near the edge f the desk clsest t the large bject. The ball: Drp the ball frm 50 cm nt the surface f the large bject, which will be mving away frm it. Again the large bject will be mving, but this time it will be mving away frm the ball as it appraches. Nte that t achieve this, the large bject must be mving slwer than the ball. Befre yu begin, predict hw yu think the interactin between the ball and the large bject will change the speed and the bunce height f the ball: 12

13 Nw perfrm the activity. Observe the initial speed f the ball as it appraches the large bject and the final speed and height after the bject and ball have cllided. E. Hw des the height frm which it was drpped cmpare with the height t which it bunced? F. Hw des the initial speed f the ball cmpare with the final speed f the ball (smaller than, larger than, r the same as)? In scenari 1 the large bject remained statinary during the cllisin, whereas in the ther tw scenaris the large bject was mving either tward r away frm the ball at the pint f impact. G. Hw did the relatinship between the initial and final speeds change in these three scenaris? Let s cmpare just Scenari 2 and Scenari 3. (Recall in Scenari 2 the ball and the large bject are mving tward each ther; the large bject began beneath the table surface and was pushed up t cllide with the ball. In Scenari 3 the ball and the large bject are mving in the same directin; the large bject began abve the table surface and was mving dwn as it and the ball cllided.) H. Hw did the final speeds f the ball in these tw scenaris cmpare and why? 13

14 Directins: Navigate t the Gravity Assist Simulatr, name: **T navigate within this Simulatr, d nt use the back brwser buttn. Instead, use the navigatin within the Simulatr, such as Main Menu buttn in the upper left crner r navigatins ptins prvided t the left and right f the Start buttn in the middle f the screen.** Directins: After selecting Part 1, yu will see fur pssible scenaris acrss the bttm f the screen. These scenaris are similar t thse frm the in-class explratin yu cmpleted. G thrugh all f the scenaris and answer the questins belw. Ntice that yu will be answering sme questins BEFORE bserving each f the scenaris. 1. BEFORE bserving the scenari, predict hw yu think the mtin f the bject will change and hw the final speed will cmpare with the initial speed. Recrd yur answer here: 2. AFTER bserving the scenari, hw did the mtin f the bject change and hw did the final speed cmpare with the initial speed? Was this what yu had predicted? Nte: t prceed t the next scenari click n the scenari menu buttn t the left f the replay buttn in the middle f the screen 14

15 3. BEFORE bserving the scenari, predict hw yu think the mtin f the bject will change and hw the final speed will cmpare with the initial speed. Recrd yur answer here: 4. AFTER bserving the scenari, hw did the mtin f the bject change and hw did the final speed cmpare with the initial speed? Was this what yu had predicted? *Again, use the scenari menu buttn 5. BEFORE bserving the scenari, predict hw yu think the mtin f the bject will change and hw the final speed will cmpare with the initial speed. Recrd yur answer here: 6. AFTER bserving the scenari, hw did the mtin f the bject change and hw did the final speed cmpare with the initial speed? Was this what yu had predicted? 7. BEFORE bserving the scenari, predict hw yu think the mtin f the bject will change and hw the final speed will cmpare with the initial speed. Recrd yur answer here: 8. AFTER bserving the scenari, hw did the mtin f the bject change and hw did the final speed cmpare with the initial speed? Was this what yu had predicted? 15

16 9. Yu did nt try a scenari like this ne in the hands-n prtin f the activity. If yu had, yu wuld have placed the large bject ABOVE the ball (e.g. at abut 60 cm). Then, just as the ball was released, the large bject wuld begin t fllw the ball as it drpped. The large bject wuld be mving FASTER than the ball, s it wuld catch up with the ball and they wuld cllide. If yu had perfrmed this scenari, d yu think the ball wuld speed up r slw dwn as a result f the cllisin? Directins: After returning t the Main Menu (upper left crner) and selecting Part 2, yu will see three flyby distances. In this interactive the planet remains statinary as the spacecraft flies past frm these three different distances. Observe all three and then answer the questins belw. 1. After bserving all three scenaris, hw des the path r trajectry f the spacecraft differ in the three different trajectries? 2. What frce is acting n the spacecraft t change its mtin? 3. In which trajectry (i.e. distance frm the planet) is the frce greatest? Hw d yu knw? 16

17 If New Hrizns is the fastest spacecraft ever launched, why des it need a bst r gravity assist frm Jupiter? Plut is s far away that withut the gravity assist frm Jupiter, the New Hrizns spacecraft wuldn t reach the Plut system until 2018 at the earliest. Instead, Jupiter s gravity accelerated the spacecraft by an additinal 4 kilmeters per secnd (9,000 miles per hur) t pass thrugh the Plut system in July Nt nly des this save precius time, but it requires less fuel, which is bth heavy and cstly. Directins: After returning t the Main Menu (upper left crner) and selecting Part 3, yu can see what wuld happen if the New Hrizns spacecraft culd fly past Jupiter at three different distances. Yu can chse ne distance at a time r all three tgether. Select replay t chse anther distance. 1. Which trajectry changed the speed f the spacecraft the mst? 2. Let s relate this t the scenaris yu bserved in Part 1 with the ping-png ball, previusly. D yu cnclude this is mst like scenari 3 (small bject vertakes larger bject) r like scenari 4 (larger bject vertakes small bject)? (Nte: g back t the main menu and select Part 1 t refresh yur memry!) 3. Select the link at the bttm f the page and see the real New Hrizns trajectry frm Earth t Plut, with a gravity assist frm Jupiter alng the way! In the previus animatin (Part 3) yu culd see that the spacecraft passed just behind Jupiter in its rbit. What wuld happen if the spacecraft were t pass just in frnt f the planet? Directins: After returning t the Main Menu and selecting Part 4, yu can see what wuld happen if a spacecraft were t pass in frnt f the planet Venus, fr example. Again there are three distances that yu can watch independently r tgether. Use replay t chse a different distance. 17

18 1. In this animatin, did the spacecraft speed up r slw dwn as a result f passing Venus? 2. Why d yu think the speed f the spacecraft changed in this way? 3. Let s relate this t the scenaris yu bserved in Part 1 with the ping png ball, previusly. D yu cnclude this is mst like scenari 3 (small bject vertakes larger bject) r like scenari 4 (larger bject vertakes small bject)? 4. The MESSENGER spacecraft is currently en rute t the planet Mercury, using a cmplex sequence f flybys invlving Earth (nce), Venus (twice) and Mercury (three times). Each flyby is carefully cnfigured s that after a jurney f six and a half years the MESSENGER spacecraft will be psitined t enter a nearly plar rbit abut the innermst planet. Again, select the link at the bttm f the page t see the very cmplicated trajectry that the MESSENGER spacecraft had t take t enter int rbit arund the planet Mercury! 18

19 Answer Key fr Teachers Cllisins Student Data Sheet: Scenari 1 A. The ball shuld appear t have the same r similar initial and final speeds. B. The ball shuld bunce t abut the height frm which it was drpped (abut 50 cm). Scenari 2 C. The ball shuld be mving faster after the cllisin; the final speed shuld be greater than the initial speed. D. The ball shuld bunce t a height greater than 50 cm; bunce height greater than drpped height. Scenari 3 E. The ball shuld be mving slwer after the cllisin; the final speed shuld be less than the initial speed. F. The ball shuld nt bunce as high as 50 cm; bunce height is less than the drpped height. Cmparisns G. Scenari 1: the initial and final speeds were abut the same Scenari 2: final speed greater than initial speed Scenari 3: initial speed greater than final speed H. In scenari 2 the final is greater than the initial because the large bject is mving tward the ball when they cllide s it pushes the ball (like a ball and a bat). In scenari 3 the large bject is mving away frm the ball when they cllide, s it causes the ball t slw dwn (like when a catcher pulls the mit back when catching a baseball). Gravity Assist Simulatr Part 1 1. their predictins will vary; hpefully initial and final speeds will be the same 2. initial and final speeds were the same. 19

20 Part 2 Part 3 Part 4 3. predictin: final speed shuld be greater 4. bserve: final speed is greater than the initial speed 5. predictin: initial speed is greater than final speed 6. bserve: initial speed is greater than final speed 7. predict: final speed is greater than initial speed 8. bserve: final speed is greater than initial speed 9. the ball wuld speed up as a result f this type f cllisin. 1. the clser the spacecraft is t the planet the mre it bends as a result f the flyby 2. The gravitatinal frce frm the planet is acting n the spacecraft t change its mtin. 3. The gravitatinal frce is greatest clsest t the planet, s in the 1 millin km trajectry the frce is the greatest. We knw this because the speed increases the mst in this trajectry and the path is changed the mst. 1. In the clsest trajectry (1 millin km) the speed f the spacecraft changed the mst. 2. It is mst like scenari 4 (larger bject vertakes small bject) because the spacecraft (small bject) speeds up as a result f the encunter. If yu think f the mtin f the spacecraft in terms f vectrs, the large bject (Jupiter) is mving faster than the spacecraft in the left t right r hrizntal cmpnent f the vectr. 1. The spacecraft slwed dwn as a result f the encunter with Venus. 2. Since the spacecraft passed right in frnt f the planet, the gravitatinal frce frm the planet pulled the planet tward it, thereby slwing the spacecraft dwn. 3. This is mst like scenari 3 (small bject vertakes large bject). The small bject r spacecraft is traveling faster than the planet and passes right in frnt f it. The larger bject (planet) then slws the spacecraft dwn. 20