Chapter 8. Linear Momentum, Impulse, and Collisions

Transcription

1 Chapte 8 Lnea oentu, Ipulse, and Collsons 8. Lnea oentu and Ipulse The lnea oentu p of a patcle of ass ovng wth velocty v s defned as: p " v ote that p s a vecto that ponts n the sae decton as the velocty vecto v. One can show that ewton s second law of oton, F a, ay be wtten n tes of lnea oentu.

2 F a dv d ( v) dp That s, dp F The net foce F actng on a patcle s equal to the te ate of change of ts lnea oentu. Law of Consevaton of Lnea oentu When the net extenal foce actng on a syste s zeo, then the lnea oentu of the syste eans constant (o conseved). Ths s so because f F 0, dp then 0 p constant. The Ipulse - oentu Theoe The pulse J of the net extenal foce F actng on a patcle dung a te nteval t s equal to the change s lnea oentu p of the patcle dung that nteval. 2

3 J " # p t f % $F F ave #t t 0 ( ) that s, J #" p % $ o % & F ave "t Ipulse equals the aea unde the foce vesus te cuve. 3

4 8.2 Consevaton of Lnea oentu When the net extenal foce actng on a syste s zeo, then the lnea oentu of the syste eans constant (o conseved). Ths s so because f F 0, dp then 0 p constant. 8.3 and 8.4 oentu Consevaton and Collsons A collson s an event dung whch two objects coe close to each othe and nteact by eans of foces. 4

5 A. Elastc Collsons An elastc collson s one n whch the knetc enegy of the syste s the sae befoe and afte the collson. That s, the knetc enegy s conseved. Consde the followng elastc collson n one denson: () consevaton of lnea oentu (f F 0) eans that A v A + whch n ths case yelds: B v B A v Af + B v Bf A( va ) + B ( vb ) A( vaf ) + B ( vbf ) () consevaton of knetc enegy gves: 5

6 AvA + BvB AvAf whch ay be e-wtten fo densonal collsons (usng consevaton of lnea oentu) as v Af v Bf ( v v ) A B 2 B v 2 Bf B. Inelastc Collsons An nelastc collson s one fo whch the knetc enegy of the syste s not conseved. C. Copletely (Pefectly) Inelastc Collsons A copletely nelastc collson s one fo whch the colldng bodes stck togethe and ove as a unt afte the collson. 6

7 Collsons n Two Densons If F ext 0, then the two coponents of the lnea oentu of the syste ae conseved. % $ P # " P x y const const Often tes, t s ease to solve fo the coponents of the fnal veloctes than to solve dectly fo the agntudes of the fnal veloctes. 8.5 The Cente of ass The cente of ass (C) of a syste of patcles oves as f all the ass of the syste wee concentated at that pont. 7

8 A. Syste consstng of dscete patcles The x-coodnate x of the C of a syste of dscete patcles s gven by: x x + x x x whee x s the x-coodnate of the th patcle. Also, the y- coodnate of the cente of ass of the syste of dscete patcles s equal to y y + 2 y2 + 3 y y The poston vecto of the cente of ass has coodnated ( x, y ) and equals xˆ + y ˆ j o 8

9 B. Cente of ass of an Extended Object x x d " x d x d y y d " y d y d whee s the ass. 9

10 C. oton of a Syste of Patcles A. Poston Vecto of a syste of dscete patcles: B. Velocty V of the C of a syste of patcles: V d d v V v C. Total Lnea oentu P of syste of patcles: P V P " v v P p 0

11 D. Lnea Acceleaton a of syste of patcles: a a d V dv a a E. et Extenal Foce on syste of patcles: Re-aange last foula to obtan: Fext a " Fext a ote that f F dv ext 0, then a 0 0 " V const " P V const That s, the net lnea oentu of a syste of patcles eans constant (s conseved) f no net extenal foce acts on the syste. If an solated syste consstng of two o oe patcles s ntally at est, then ts cente of ass C eans at est unless acted upon by a net extenal foce.

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13 8.6 Rocket Populson The opeaton of a ocket depends on the law of consevaton of lnea oentu appled to a syste, whee the syste s the ocket and the ejected fuel. As a ocket oves n fee space (a vacuu) ts lnea oentu changes when soe of ts ass s eleased n the fo of ejected gases. The ocket s thus acceleated as a esult of the push o thust fo the exhaust gases. The pocess epesents the nvese of a copletely nelastc collson; that s, lnea oentu s conseved, but the knetc enegy of the syste s nceased at the expense of the enegy stoed n the fuel of the ocket. Let ass of the ocket plus fuel (-d) ass of ejected fuel v ex exhaust speed of the ejected fuel elatve to the ocket. Thus v fuel v - v ex speed of ejected fuel. v speed of ocket befoe ejectng the fuel 3

14 Afte the ocket ejects fuel of ass ( d), the speed of the ocket nceases to v + dv. Applyng the law of consevaton of lnea oentu to ths syste, P befoe P afte v ( + d) ( v + dv) + ("d)( v " v ex ) 0 dv + d dv + v ex d neglect te ddv because t s a poduct of two sall quanttes and thus t s uch salle than the othe tes, dv "v ex d ntegate both sdes of the equatons (note the change to duy vaables) so that v f d v d " # " $v ex # " v o f o # v f " v o v ex ln & o % ( $ f ' 4

15 Coents:. The ncease n the speed of the ocket s popotonal to the exhaust speed v ex. 2. Thust on the ocket s a dv d. ote v ex that the thust nceases as the ate of change of ass nceases (.e., as the bun ate nceases). 3. We have assued thoughout ths analyss that the ocket s n gavty-fee oute space. Howeve, gavty ust be taken nto account when a ocket s launched fo the suface of a planet. (see poble 8.2). 5