Conservation of Momentum II

Transcription

1 Pupose: To veify the pinciples of Consevation of Momentum and Consevation of Enegy in Elastic and Inelastic Collisions, and to exploe Collisions in the Cente of Mass fame. Equipment: Cuved Tack Metal Ball Glass Ball with Tape Plumb Bob Table Clamp Cabon Pape Scatch Pape PoScout USB 200g Balance Mete Stick Tape Theoy: Expeiments involving the collision of objects have played a majo ole in the development of moden physics. The key to the analysis of collision expeiments lies in the consevation of momentum an enegy in situations in which extenal foces ae negligible. Even if the inteaction between colliding bodies is not undestood o extemely complicated, accounting fo the enegy and momentum of the incoming and outgoing paticles allows the expeimente to obtain a geat deal of infomation about the bodies. By such means was the atomic nucleus discoveed, when alpha paticles fied towad an atom bounced back towad the souce, indicating a vey massive cental egion in the atom the nucleus. Moe ecent discoveies, such as that of the top quak, have used analysis of collisions in vey simila ways. Momentum is conseved in collisions in which extenal foces ae negligible. In this expeiment, gavity is pesent, but because it acts in the vetical diection only, the momentum in the hoizontal diection will be conseved. A collision is said to be elastic when the kinetic enegy of the system is conseved, when none of it is conveted into othe foms; othewise the collision is temed inelastic. In today s lab we study both elastic and inelastic collisions between steel and glass balls. In ou tials a pojectile will stike a stationay taget. Ou notation will be: m and u fo the mass and velocity of the pojectile befoe the collision, v fo its velocity afte the collision, and m 2 and v 2 fo the velocity of the taget afte the collision. With these conventions, momentum consevation may be expessed as follows: of 8

2 p initial = p final o Eq. m u = m v + m 2 v 2 Note that since momentum is a vecto, the individual momenta must be added as vectos; also, thee is only one momentum o the left ( initial ) side of the equation, because the taget is stationay (u 2 is zeo). In elastic collisions we have an additional elationship between the quantities, expessing the fact that the kinetic enegy is conseved: KE initial = KE final o Eq. 2 m u 2 2 = m v m v [ elastic only] Note hee that thee ae no aows ove u, v, and v 2. Kinetic enegy, in common with all foms of enegy, is not a vecto, but a scala, without diection it depends on the squae of an object s speed. In today s lab, the pojectile olls down a amp, knocks a stationay taget ball off a suppot, then both balls fall to the floo ad land on cabon pape, leaving maks on scatch pape below. Each ball is in the ai the same amount of time, so the faste a ball goes out hoizontally, the fathe away will be its mak on the pape. Thus, the distances fom the mak below the collision point ae popotional to the balls hoizontal speeds. Logically, the diections of these vectos ae the diections of the velocities. Thus, as shown in Figue, we obtain a pictue in which aows dawn fom the collision point to the points whee the balls land ae popotional to the hoizontal velocities afte the collision. Figue landing point v collision point Top View landing point v 2 2 of 8

3 Expeiment: Pat A: Had Sufaces Collide. We ae going to stat this lab with some pedictions. When the metal pojectile collides with the glass taget, hitting the glass and not the taped side, do you expect the system s hoizontal momentum to be conseved? The vetical momentum? The kinetic enegy? Wite down you pedictions, and explain each. 2. Using a lage table clamp, attach the cuved tack to the edge of a lab table o countetop. You may need two clamps to keep the tack level. Make sue that no pats of the clamps intefee with the olling of the ball. 3. We will use a metal ball as the pojectile, and a glass ball as the taget. Thee is a small squae of tape on the glass ball, which will be used fo the inelastic collision. Using the electonic balance, find the mass of both balls. 4. To make ou calculations easie, we ae going to e-define the units of mass. We will define the pojectile as having a mass of. You must calculate the mass of the glass ball in tems of ou false units. If you can t figue out how to do this, you lab instucto can help you, but TRY IT FIRST! 5. Place the glass taget ball on its suppot, and the metal pojectile ball on the end of the amp. Check to see that the centes of both balls ae at the same height. If they ae not, adjust the suppot scew. 6. Tape togethe 9-2 pieces of scatch pape, enough to cove the entie aea contained by the tajectoies of the balls, including the collision point diectly below the pivot of the suppot am. Tape the pape in place on the floo. 7. Using a plumb bob, mak the point diectly below the pivot of the suppot am, and label it collision point. 8. Befoe placing the cabon pape, make sue that eveything is woking coectly. Set the taget suppot am at oughly a 45 o angle with the amp. Set the taget ball on the suppot. Hold the pojectile against the scew-stop at the top of the amp. When you elease the pojectile, you want to make sue of the following things: Both balls land on the pape The collision is clean, i.e. no seconday collisions, each ball s tajectoy is smooth The balls hit the floo at the same time. You ll need to listen fo this one. Ty launching the balls a couple of times. If they don t land in the same aea each time, ty making the angle of the suppot am a little lage o smalle, until the data is consistent. 3 of 8

4 9. Lay down two pieces of cabon pape, one that will ecod the landing of the pojectile, a second to ecod the landing of the taget. Repeat ten uns to get an idea of the expeimental uncetainty. Daw an ellipse containing most of the pojectile landing points, then daw an aow epesenting v fom the collision point to the aveage pojectile v, m v landing point. Do the same fo the taget, v 2. Because, and only because, we have defined the mass of the pojectile as, its momentum m v is epesented by the same vecto as its velocity, v. Accodingly, label this vecto v, m v. Fo Figue 2 now, label the taget velocity simply as v Discuss with you lab patnes how the taget velocity vecto v 2 will elate to the momentum vecto of the taget. Include the esults of this discussion, and any calculations in you lab epot. Label the epesentation of the taget momentum m 2 v 2.. Discuss with you lab patnes what method would allow you to measue the pojectile s initial velocity, u. Once you have decided on a method and checked it with you lab instucto, cay out the pocedue ten times, obtaining an uncetainty ellipse and dawing the vecto labeled u, m u. 2. Hee is the test: is momentum conseved, o not? Add the vectos m v and m 2 v 2 gaphically, labeling the esultant vecto m v + m 2 v Based on you gaphical esults, does momentum consevation as expessed in Eq. hold within the expeiment s uncetainties? Was you pediction bone out? If not, discuss the discepancies. 4. Now fo something weid: in addition to ou convenient choice of mass units, we will define ou time units so that the time to fall fom the collision height, which is the same fo both balls, is, and fo ou distance unit we choose a centimete. Theefoe, a velocity vecto 5 cm long means a speed of 5. Recod the speeds u, v and v 2 in you lab epot. This may seem odd and confusing, but things like this ae done often in science, and you will see that these choices make ou calculations a lot easie! 5. Calculate, in ou false units, the kinetic enegy of the pojectile befoe the collision. Also calculate the kinetic enegies of the pojectile and taget afte the collision. Include you calculations and esults in you lab epot. 6. Does kinetic enegy consevation, as expessed in Eq. 2 hold within the expeiment s uncetainties? A ough idea of the uncetainty in the speeds is a adius of a epesentative uncetainty ellipse. How does this value, as a pecent of the speed, compae with the pecent change in kinetic enegy? Was you ealie pediction bone out? If not, discuss the discepancy. Include calculations of the expeimental uncetainty in you lab epot. 4 of 8

5 Pat B: A Cushioned Collision. In this pat of the expeiment, we tun the glass ball so that the piece of tape is facing the oncoming pojectile. In this case, do you expect the system s (hoizontal) momentum to be conseved? Its kinetic enegy? Biefly explain you answes. 2. Cay out an expeimental pactice un to find whee the taget and pojectile land. If possible, use the same pape as befoe, pehaps using a diffeent colo of pencil to distinguish between the two diffeent collisions. Lay down the two pieces of cabon pape, and cay out ten data uns, dawing in the uncetainty ellipses and velocity vectos. Label the pojectile vecto v, m v, and the taget vectos v 2 and m 2 v Discuss with you lab patnes how u, the initial momentum, and the momentum in this pat compae to those values in the had-sufaces collision. Cay out whateve pocedues you deem necessay to find the initial velocity u and the initial momentum. 4. Add m v and m 2 v 2 gaphically, labeling the esultant m v + m 2 v Does momentum consevation hold within the expeiment s uncetainties? Was you ealie pediction bone out? If not, discuss the discepancy. 6. Calculate the speeds u, v, and v 2. Include the calculations and values in you lab epots. 7. Calculate the kinetic enegies befoe and afte the collision. Include you calculations and esults in you lab epot. 8. Does kinetic enegy consevation, expessed in Eq. 2, hold within the expeiment s uncetainties? Was you ealie pediction bone out? If not, discuss the discepancy. Include calculations of you expeimental uncetainty in you lab epot. Pat C: Collisions in the Fame Being able to visualize how things would look fom an altenative fame of efeence is something quite useful and illuminating in many diffeent physical situations. Pehaps the most impotant altenative efeence fame is the cente-of-mass, o, fame. This is a fame that moves along with the cente of mass of the system. By definition, the position of the cente of mass the aveage position of the mass of a system of N paticles is given by: N N N x = mi xi and y = mi yi o = mi i M M M i= i= i= Fo just two paticles, this becomes: m + m2 2 m = m + 2 Eq. 3 5 of 8

6 A question cental to undestanding things in the fame is: What is the velocity of the cente of mass, with espect to the lab oom? Taking a time deivative of both sides of Eq. 3 we have: mv + m2v2 mv + m2v2 v = = Eq. 4 m + m2 M whee v and v 2 may epesent the pojectile and taget afte the collision. But what of the velocity of the cente of mass befoe the collision? We can use the same geneal fom: v mu + m2 0 mu = = M M Eq. 5 And now we see on one of the easons why the cente-of-mass fame is so special: if momentum is conseved, as expessed in Eq., the numeatos of Eqs. 4 and 5 ae equal; so the velocity of the cente of mass is the same afte the collision as befoe, and thus can have nothing to do with the actual details of the collision! Finally, the geneal method fo elating velocities in two diffeent fames of efeence, A and B, is as follows. If v ib is the velocity of object i elative to fame B and v AB the velocity of fame B elative to fame A, then the velocity of object i elative to fame A is given by: v ia = v + v [ Fame A and Fame B ] ib AB In today s lab, fame A is the lab oom, so the v ia ae the velocities v i as you have aleady dawn them (the A is undestood), and fame B is the cente-of-mass fame, so that v BA is v, and we have: v i = v + v [ Lab oom fame and fame] Eq. 6 i In othe wods, the velocities v i go fom the tip of the v of the balls elative to the fame ae vectos that vecto to the tip of the v i vectos.. Guided by Eq. 5, daw in and label as v any calculations in you lab epot. the velocity of the cente of mass. Include 2. Without actually dawing it, how would a vecto epesenting v but guided by Eq. 4 appea on you pape? Would it bea out the claim that the velocity of the cente of mass has nothing to do with the details of the collision, and why? 6 of 8

7 3. Now, in the appopiate colos, daw in and label as v i the velocities of the balls afte the collision elative to the fame. Thee will be fou such velocities, two fo each collision. (Remembe: The velocities v i go fom the tip of the v vecto to the tip of the v i vectos.) 4. Daw, in the appopiate colos, any new vectos necessay to epesent m v v and add these labels at the appopiate places fo each collision. m 2 2 and 5. The cente-of-mass fame is special! Accoding to you diagams, what does the momentum in the fame appea to be afte the collisions? Answe fo each of the collisions. 6. What should be the momentum in the fame befoe the collisions? 7. Add, in the appopiate colos, a u and a u2 to each collision. Just as fo the afte-collision velocities, these go fom the tip of the v vecto to the tip of the u and u 2 vectos (and u 2 is zeo). Then add and label the befoe-collision momentum vectos mu and m2u2. Is you claim of Step 6 bone out? 8. Just by looking at the velocity vectos in the fame befoe and afte the collisions i.e., without calculations can you einfoce qualitatively you ealie findings about kinetic enegy consevation in the two collisions? 9. Using Eq. 6 applied to each of the two masses, then inseting into Eq. 4, pove you claim in Step 5. Results: Wite at least one paagaph descibing the following: what you expected to lean about the lab (i.e. what was the eason fo conducting the expeiment?) you esults, and what you leaned fom them Think of at least one othe expeiment might you pefom to veify these esults Think of at least one new question o poblem that could be answeed with the physics you have leaned in this laboatoy, o be extapolated fom the ideas in this laboatoy. This lab was adapted fom Physics 9A Lab Manual, The Staff of the Physics Depatment, Univesity of Califonia at Davis, of 8

8 Clean-Up: Befoe you can leave the classoom, you must clean up you equipment, and have you instucto sign below. How you divide clean-up duties between lab membes is up to you. Clean-up involves: Completely dismantling the expeimental setup Removing tape fom anything you put tape on Dying-off any wet equipment Putting away equipment in pope boxes (if applicable) Retuning equipment to pope cabinets, o to the cat at the font of the oom Thowing away pieces of sting, pape, and othe detitus (i.e. you wate bottles) Shutting down the compute Anything else that needs to be done to etun the oom to its pistine, pe lab fom. I cetify that the equipment used by has been cleaned up. (student s name),. (instucto s name) (date) 8 of 8