5/24/2007 Collisions ( F.Robilliard) 1

Transcription

1 5/4/007 Collsons ( F.Robllard) 1

2 Interactons: In our earler studes o orce and work, we saw, that both these quanttes arse n the context o an nteracton between two bodes. We wll now look ore closely at such nteractons. Interactons occur between two, or ore, bodes oer an nteral o te. We call the collecton o nteractng bodes, the syste. Here are soe exaples o systes o nteractng bodes - -a galaxy coposed o stars nteractng wth each other gratatonally - the solar syste, consstng o the sun and planets ong under utual gratatonal attracton. - an asterod colldng wth the earth - a rocket engne ejectng uel or thrust. - two cars colldng n a trac accdent - olecules colldng wth the walls o a contaner creatng gas pressure - a proton & an electron, held together by Coulob attracton n the hydrogen ato. - nucle colldng wth each other n a nuclear reacton. 5/4/007 Collsons ( F.Robllard)

3 Collsons: For any such syste, nteracton orces act between the coponent bodes. Assocated wth these nteracton orces, oentu and energy are transerred ro one body n the syste, to another. We are gong to look ore closely at these sultaneous oentu and energy transers. For splcty, we wll ocus on systes o two bodes, that nteract by collson. We a to predct the outcoe o a collson between the bodes. That s - gen the eloctes o the bodes beore the collson, we would lke to predct ther eloctes ater the collson. In theory, ths proble could be soled, by deternng the nstantaneous nteracton orces between the bodes, hence ndng the acceleratons produced, and thence, the nal eloctes. Howeer, ths requres crodetaled knowledge o the non-constant nteracton orces, and s coplex to undertake. An alternate s to use a systes approach. We try to relate the oerall propertes o the syste beore, to those ater, the collson. The propertes n queston here, are oentu, and energy. 5/4/007 Collsons ( F.Robllard) 3

4 Ipulse o a Force: We wll rstly look at syste oentu, by restng Newton s Second Law. oer t Consder a ass,, on whch a orce, F, acts: Say the agntude o the orce, F, ares oer an nteral o te, t, soethng lke - F.dt F Integrate both sdes o (1) - thus F.dt d ( ) ( )...( 1) 5/4/007 Collsons ( F.Robllard) 4 N F oer t F t F.dt d dt t ( )...( ) s called the total pulse o the orce, F, on the ass, oer te nteral t. and () s the total change n oentu o the ass, that s produced by the aryng orce, n te nteral t. Force, F, actng or tny te nteral, dt, wll produce a change n oentu o d()

5 Ipulse-Moentu: oer t F.dt ( )...( ) F Ipulse oer F.dt t s equal F to the area under the F - t graph. t t Consderng the total eect o the orce, F, on the ass,, oer te nteral t: Equaton () says - Total pulse o orce, F, on a ass, Total change n oentu produced () s called the Ipulse-Moentu equaton. It s sply Newton n dsguse. 5/4/007 Collsons ( F.Robllard) 5

6 ( MV) ( )...ro ( ) ( MV) + ( ) 0 Thus Consder a Collson between two asses, and M. As the bodes coe nto contact, each exerts a orce on the other. t F F.dt t.dt Durng a Collson: F orce on M, due to, durng the collson. orce on, due to M, durng the collson. But, by Newton 3, F and are, at eery nstant, equal, and opposte. There s no change n the total oentu o the two asses, due to the collson. F orce t We treat the two asses as a syste. We can see ro the aboe graph, that the total pulse on the syste ( total area under the Force-te graph ), due to collsonal orces, F and, s zero, snce F and are equal, but opposte. Hence the total change n oentu o the syste ust also be zero. F M t 5/4/007 Collsons ( F.Robllard) 6

7 Coparng Beore wth Ater: Snce the total oentu o the syste s constant, we say that oentu has been consered, or the syste. Ths s the prncple o conseraton o oentu, and s usually stated orally as ollows - total ector oentu o syste beore the collson p syste total ector oentu o syste ater the collson syste p Where the Σ s a su oer the syste, p represents ector oentu, ntal, and nal. In usng ths prncple, t s crtcally portant to treat oentu as a ector 5/4/007 Collsons ( F.Robllard) 7

8 Note: We hae assued that the only orces actng on the bodes o the syste are the nteracton orces o the collson. We hae assued that there are NO net external orces, such as rcton, or graty, actng on the syste, whch would change ts oerall oentu. We hae ound, that, one o the bodes loses oentu, then all o that oentu has been transerred, durng the collson, to another body o the syste, so the total ector oentu has not changed. Moentu ust be treated as a ector drecton s portant!. Conseraton o oentu apples not only to colldng blocks, but to other nteractng systes, such as the solar syste, galaxes, olecules o a gas, nteractng eleentary partcles. We now apply conseraton o oentu to the case o two blocks, rstly n a 1-densonal, and then n a -densonal, collson. 5/4/007 Collsons ( F.Robllard) 8

9 1-densonal collson o two asses, and M: We depct the syste, beore, and then ater the collson. u U V M Beore +x u ntal elocty o U ntal elocty o M syste p + u MU p syste M Ater +x nal elocty o V nal elocty o M + MV Note: the negate sgns are due to those oenta beng n the (-x)-drecton Note: we hae ade an assupton about the drecton o eloctes, ater the collson. I our assupton s correct, we wll get a poste soluton, wrong, a negate soluton, or the partcular elocty. We need to nterpret these solutons. 5/4/007 Collsons ( F.Robllard) 9

10 -densonal collson o two asses, and M: u y M x y β α M V x Beore Ater colldes wth a statonary M. M s knocked on at an angle α, whle rebounds at angle β. We need to apply conseraton o oentu n both the x- and y- drectons. syste p In x - drecton : p syste + u.cosβ + MV.cosα... ( 1) In y - drecton : p 0.snβ + MV.snα... Note: the angles α and β wll depend on the detals o how the suraces collded. Note: Equatons (1) and () need to hold sultaneously. 5/4/007 Collsons ( F.Robllard) 10 syste p syste ( )

11 Is Moentu Sucent? Is oentu conseraton sucent to deterne the outcoe o a collson? Let s reconsder the collson o two blocks n 1-denson. Two 1 kg blocks approach each other at 10 /s, along the x-axs. What wll be ther eloctes ater they collde? 10/s 10/s 1kg 1kg Beore +x Experent shows, that there are any possble outcoes, dependng on the detals o the nteracton at the colldng suraces or exaple, whether or not elastc lts hae been exceeded, resultng n peranent deoraton o the suraces. For three representate outcoes, we wll consder the total oentu, p, and also the total energy assocated wth the oton ( total KE), o the colldng blocks, both beore, and ater the collson. 5/4/007 Collsons ( F.Robllard) 11

12 3 Possble Outcoes: kg 1kg Ater +x Σp +(1x10) - (1x10) 0 ΣKE ½ ½ J KE s consered Elastc collson 10/s 10/s 1kg 1kg Beore +x 5 5 1kg 1kg Ater +x Σp +(1x5) - (1x5) 0 ΣKE ½ ½.1.5 5J 75% KE lost Inelastc collson Intal total p & KE: Σp +(1x10) - (1x10) 0 ΣKE ½ ½ J 0 0 1kg 1kg Ater +x Σp +(1x10) - (1x10) 0 ΣKE J All KE lost Perectly nelastc collson Moentu s consered n all three cases. KE only consered or elastc collson. 5/4/007 Collsons ( F.Robllard) 1

13 Energy: We see that oentu alone does not deterne the outcoe o a collson energy ust also be taken nto account. We know ro our studes o energy, that the prncple o conseraton o energy ust hold or all systes, ncludng colldng ones. Total Energy o syste Total Energy o syste Energy + beore the collson ater the collson losses We wll assue that the only or o energy possessed by colldng bodes, beore the collson, s knetc. Ater the collson, ths energy wll ether stay n the or o KE, or be conerted ( lost ro the syste ) to soe other or o energy, such as heat. Durng the collson, suraces o colldng bodes are deored. I elastc lts are exceeded, peranent deoraton o the suraces results. The deoraton orces do work, and result n a conerson o energy ro the or o KE o the bodes, nto heat. To predct the outcoe o a collson, we need to know how uch energy s conerted to other ors, such as heat, and consequently lost to the syste. Elastc Collsons: Inelastc Collsons: no KE lost. soe, or all, o the KE s lost 5/4/007 Collsons ( F.Robllard) 13

14 For all collsons - Conseraton o Energy: Total KE o syste beore the collson Σ ( KE) Σ( KE) + Elost Total KE o syste ater the collson + Energy losses ntal; nal We need to know the energy losses, n order to use ths equaton I the collson s elastc, the energy losses are zero, all energy stays n the or o knetc, and the conseraton o energy becoes - Total KE o syste beore the collson Σ Total KE o syste ater the collson ( KE) Σ( KE) (We hae the conseraton o KE.) 5/4/007 Collsons ( F.Robllard) 14

15 A collson between two bodes: Suary: u U M Beore +x s constraned by V M Ater +x u ntal elocty o U ntal elocty o M nal elocty o V nal elocty o M Conseraton o Moentu: syste p p syste + u MU + MV Assung no external orces change the oerall oentu o the syste Conseraton o Energy: Σ 1 ( KE) Σ( KE) u + 1 MU + E 1 lost + 1 MV + E lost where E lost 0 the collson s Elastc. Note: that oentu s a ector whose drecton ust be consdered, whereas energy s a scalar, and KE s always poste. 5/4/007 Collsons ( F.Robllard) 15

16 To Sole Collson Probles: 1. Draw a labeled dagra o beore, and ater. Assue a poste x-drecton ( and y-drecton). 3. Wrte conseraton o oentu equaton 4. Wrte the conseraton o energy equaton, losses are known. 5. Sole the equatons or unknowns. 6. Interpret your soluton. 5/4/007 Collsons ( F.Robllard) 16

17 Exaple 1: An arrow, o ass 0.5 kg, s red nto a statonary, reely hangng sandbag, o ass 10 kg, causng t to swng orward wth a elocty o /s. Fnd the elocty o pact o the arrow. u M Beore rest +x M The collson s nelastc and we do not know how uch energy was lost. We can only use oentu conseraton. V Ater +x Thus syste p p + u syste ( + M) V + M u V /s The arrow s elocty was 4 /s. 5/4/007 Collsons ( F.Robllard) 17

18 Exaple : A 3 kg body ong wth a elocty o /s akes a 1-densonal, elastc collson wth a statonary ass o 9 kg. Fnd the eloctes o the two asses ater the collson. u/s rest 3kg M9kg Beore +x V 3kg M9kg Ater +x We assue drectons o the bodes, ater the collson In ths proble, we wll need to use both oentu, and energy conseraton. Moentu: + syste ( 3)( ) p Σ( KE) Σ( KE) V p syste - + 3V... ( 1) Collson s elastc: 1 4 ( 3) V V... ( ) 5/4/007 Collsons ( F.Robllard) 18

19 - + 3V...(1) 4 + 3V.. () Exaple : Sole (1) and () sultaneously or and V: (1): 3 V -.(3) (3)->(): 4 (3 V-) + 3 V 4 9V 1 V V 0 1 V (V - 1) thus: V 0 or +1.(4) (4)->(3): - or +1 correspondngly The two solutons are llustrated. Clearly Ater1 s the sae stuaton as Beore and represents the stuaton where no collson took place, and s thus nald. Ater represents the ald soluton. u/s rest 3kg M9kg Beore +x /s V0 3kg M9kg Ater1 +x 1/s 1/s 3kg M9kg Ater +x 5/4/007 Collsons ( F.Robllard) 19

20 Soe Rocket Scence: As a nal applcaton o the conseraton o oentu, we look at the acceleraton o a rocket n space we do soe rocket scence! To accelerate orward, the rocket ejects uel partcles backward. The rocket accelerates orward because o the reacton orce, orward, on the rocket. Howeer, the acceleraton o the rocket depends on the rocket s ass, whch ncludes the ass o the on-board uel. As uel s ejected, howeer, the total ass o rocket and uel becoes less. Thereore, een wth a constant thrust orward, the acceleraton o the rocket s NOT unor. Say a rocket, o total ass, ejects uel partcles o ass, d, at an ejecton elocty, e, relate to the rocket. + d d d Beore e Ater Say the ejecton o the uel, causes the rocket s elocty to ncrease ro, to ( + d) 5/4/007 Collsons ( F.Robllard) 0

21 +y The Change n Velocty: +y + d d d Beore +x e Ater +x + so Conseraton o oentu: p p ( + d) d ( + d - e ) + ( + d) d d [ snce d.d 0] thence syste d e syste - d e... ( 1) "-"s needed n ( 1 ), snce d s a decrease ( neg. ), whereas d s an ncrease ( pos. ). Veloctes ust be relate to the xy axes. But e s, by denton, the elocty o the uel, relate to the rocket. (See Ater pcture, aboe.) A space journey conssts o three stages - acceleraton, cruse, and deceleraton. We wll calculate the uel or the acceleraton, then double t or the deceleraton. No uel s used to cruse. 5/4/007 Collsons ( F.Robllard) 1

22 Intal elocty a The Rocket Equaton: Fnal elocty Say the rocket starts wth an ntal elocty, and accelerates, oer a perod o te, to a nal elocty. Integrate - e d e (1) - e - : ln d [ ln ( )]... ( ) Equaton () s the rocket equaton. We see that the total change n elocty o the rocket, or a gen aount o uel used, s proportonal to the ejecton elocty, e, o the uel partcles. Thereore hgh uel ejecton eloctes are desrable. Also, as would be expected, the ore uel used, the greater the elocty change. Once e s set, and the elocty,, n the cruse stage o a space trp (whch deternes the te o the trp), has been decded, the rocket equaton allows us to calculate the total uel ass needed or the trp. 5/4/007 Collsons ( F.Robllard)

23 The rocket thrust s easly ound: F e Rocket Thrust: d Thrust F dt d e dt ( Newton )...( 3) 5/4/007 Collsons ( F.Robllard) 3 F e d dt usng (1) Rocket thrust s proportonal to the ejecton elocty, and to the rate o ejecton, o uel. Exaple: Fnal elocty: Fuel partcles are ejected ro the thrusters o a space ehcle at 5 k/s, relate to the ehcle. Startng ro rest, what elocty wll the ehcle reach, ater hal o ts total ass has been ejected at a rate o kg/s, as uel partcles, by the thrusters. Fnd the thrust. - e ln...( ) Thrust: d F ro ( 3 ) 3 3 ( 5x10 ) ln 3.5x10 /s 1 e dt 3 ( 5x10 ) 1x10 10 kn thrust 4 N

24 Suary: When bodes nteract, such as n collsons, oentu and energy are transerred ro one body to another. I no oentu enters, or leaes, a syste o bodes, the total ector oentu o the syste reans constant - prncple o conseraton o oentu. I no energy enters, or leaes, a syste o bodes, the total energy o the syste reans constant - prncple o conseraton o energy. Knowng the ntal state o a syste, we can nd the nal state o that syste, wthout hang to analyse the oent-to-oent hstory o the nteracton process tsel. Fnally we were able to apply these deas to soe rocket scence. 5/4/007 Collsons ( F.Robllard) 4

25 5/4/007 Collsons ( F.Robllard) 5