Mach ec Thrusers (Mes) And Over-Uny nergy Producon Proessor merus Jm Woodward CalSae ulleron, Dep. o Physcs 13 November, 2015 We rounely hear a crcsm o MTs based upon an argumen ha clams: a MT s operaed a consan power npu or a sucenly long me, wll acqure enough knec energy o exceed he oal npu energy o operaon. Assumng hs argumen o be correc, crcs asser ha MTs volae energy conservaon as he rao o he acqured knec energy o oal npu energy exceeds uny. Conrary o hs over-uny assumpon, hs argumen s based on lawed physcs and, consequenly, wrong. The ac ha he argumen apples o all smple mechancal sysems (n addon o MTs) should have alered crcs o her msake. Bu ddn. So, a dumb dea ha should have been quckly bured s sll wh us. The purpose o hs essay s o carry ou a long overdue bural. In bre, he over-uny argumen assers ha a consan npu power no a MT wll produce a consan hrus (orce). Ths, n urn, produces a consan acceleraon o any objec o whch he MT s aached. The consan acceleraon produces a lnearly ncreasng velocy o he objec. The knec energy o he objec, however, ncreases as he square o he velocy. Ths means ha a some pon, he knec energy o he objec wll exceed he oal npu energy used o produce he hrus as ha only ncreases lnearly wh me. Crcs hen clamed ha hs purpored behavor consued volaon o energy conservaon and proposed as a aal crque o Mach ec hrusers. Noe, however, ha he argumen apples o all sysems where a consan hrus produced by a consan npu power produces moon. 1
Consder a block o mass M a res on a smooh, level, rconless surace. A me o a orce s appled o he block. The orce s assumed consan. In an nerval d he block acceleraes accordng o Newon s second law: dp dv Ma M (1) d d Where p s he momenum o he block and a s nsananeous acceleraon. v s he velocy o he block wh respec o some chosen rame o reerence. Snce M and are assumed consan, here s no v dm/d erm n he dervave o p, and a s a consan oo. Ths makes v a lnearly ncreasng uncon me: v ( ) a + v (2) o where a s he acceleraon o he block due o he applcaon o he orce. Snce n our smple case o he block and v are always n he same drecon, we can drop he vecor noaon and smply wre: v ( ) a + (3) v o A urher smplcaon s possble we assume ha v o 0, so: v ( ) a (4) To address ssues nvolvng energy, we need a denon o he relaonshp beween work, energy, and moon. Tha denon s: work ds d (5) Where ds s a small (derenal) ncremen o dsance hrough whch he componen o he orce n he drecon o ds acs producng a small (derenal) change n he energy o he block d. The ssue o neres here s he evoluon o he sysem and he power dened as he me rae o change o energy nvolved. To explore hs we derenae he work equaon wh respec o me. Snce s assumed consan, we ge: d d ds v d (6) Mulplyng hrough by d, we ge: d ds v d (7) 2
To ge he oal energy acqured durng an nerval o he applcaon o a consan orce, we smply negrae quaon (5): d v d (8) where he subscrps and denoe nal and nal. As noed above, n hs case v s a lnear uncon o me, so we subsue rom quaon (4) or v n quaon (8) and we urher assume ha and v are n he same drecon so ha we can gnore he do produc. 2 2 2 ( ) a a v d a d 0 (9) 2 2 We see mmedaely ha he acon o a consan orce on he block causes o acqure knec energy ha depends quadracally on he me elapsed rom he ncepon o he acon o he orce. Ths s rue or all mechancal sysems where a consan orce s appled o some objec ha s ree o move under he acon o he orce. I s no a dsncve eaure o he operaon o MTs. So ar hs s all jus elemenary mechancs. We have no ye done anyhng supd or wrong (or boh). As long as we don mess wh he mah, we re OK (and energy conservaon s no volaed). How hen do some argue ha n hs smple sysem and MTs n parcular energy conservaon s volaed? Smple. By dong somehng supd and wrong. In parcular, by akng he gure o mer o a hrus (orce) generaor by denon, he number o Newons o hrus produced per wa o npu power o he hrus generaor and reang as a dynamcal equaon ha can be used o calculae he energy npu o a moor ha acs or some lengh o me; ha s: m (10) P where m s he gure o mer and P he npu power o he moor ha produces he hrus. You mgh hnk ha we can rearrange quaon (10) as ollows: d P (11) d m Ths can be rearranged o: 3
d d (12) m Snce and m are boh consans, quaon (12) negraes o: ( ) 0 (13) m m Snce boh quaons (9) and (13) gve he derence beween he nal and nal energes o he objec on whch he moor s mouned, mgh seem ha we can smply equae he expressons or hese energy derences. Tha s: m a 2 2 (14) ha: Now we have done somehng supd and wrong. Ths amouns o he asseron a 2 m 2 (15) whch s obvously wrong. or some values o, he coecen o 2 on he rgh hand sde o quaon (15) (a consan by he way) may make hs equaon vald. [Tha s, can be reaed as a smple quadrac equaon and solved by he usual echnques.] As a connuous evoluon equaon, however, s nonsense. Bu hs s he mahemacs o hose who make he over uny energy conservaon volaon argumen abou he operaon o MTs. The real queson here s how could anyone, havng done hs calculaon or s equvalen, hnk ha hey had made a proound dscovery abou anyhng? [Or MTs n parcular?] Aer all, s unversally known ha energy conservaon s no volaed n classcal mechancs. I won aemp an answer o he oregong rheorcal queson. Bu s worh ponng ou he lkely source o he error. I s he velocy. In general, velocy s no an nvaran quany as depends on he moon o he observer as well as any velocy ascrbed o moon n some oher reerence rame. The prncple o relavy precludes snglng ou any parcular observer as prvleged over all ohers. The reedom o choce o reerence rame s a double-edged sword. I allows you o choose a rame n whch calculaons can be sgncanly smpled because some veloces are zero or example. Bu can also be roublesome as can lead o argumens based on veloces n one rame ha are very deren n oher rames. In he case o he over-uny argumen, hs means ha an accelerang orce can be producng over-uny behavor n one rame o reerence, and no be dong hs n anoher reerence rame. lemenary physcal nuon ells you ha should be one or he oher n all neral rames o reerence. 4
An example o an analogous case ha rounely comes up n elemenary mechancs s he rocke equaon. Tha s, he applcaon o Newon s second law o he case o a rocke. In hs case, quaon (1) reads: dp dv dm dm M + v Ma + v (16) d d d d The second erm on he rgh hand sde ha quanes he magnude o he nsananeous momenum lux n he exhaus plume s requred o balance he changng momenum produced by he orce o he propellan on he rocke s combuson chamber as s burned. Tha s, Ma. v s he velocy o he jus ejeced exhaus plume wh respec o he rocke moor. Gven he orm o Newon s second law as saed n quaons (1) and (16), even compeen physcss have come o beleve ha v dm/d s a orce, jus as Ma s a orce. Bu v dm/d sn lke an Ma orce. Ths s usually llusraed n elemenary physcs exs wh problems/examples lke: a ralway car moves along a smooh, level, sragh, rconless rack wh consan velocy. A ple o sand on he bed o he car s allowed o all hrough a hole n he loor o he car. Does he speed o he car relave o he arh (whch can be aken o have eecvely nne mass) change as he sand alls? A colleague who monors he pedagogcal leraure ells me ha people rounely mess hs up and ha a nervals o ve o en years, arcles or blog commens addressng hs ssue rounely appear. And, alas, ha even hose auned o he sublees o he ssue make msakes n handlng. In he case o a rocke moor, he hng o observe s ha here s one nvaran velocy nvolved: ha o he exhaus plume wh respec o he moor. All observers, rrespecve o her own moons, agree on boh he magnude and drecon o hs velocy. And s he velocy ha yelds momenum conservaon. An argumen based on an ncorrec applcaon o Newon s second law o MTs was advanced as a crcsm o Mach ecs by an Oak Rdge scens many years ago. I s deal wh on pages 77 and 127 o Makng Sarshps and Sargaes: he Scence o Inersellar Propulson and Absurdly Bengn Wormholes. I wll no be dscussed urher here. To wrap hs up, we ask: s possble o do a correc calculaon o he sor ha crcs dd ha does no lead o wrong predcons o he volaon o energy conservaon? By payng aenon o he physcs o he suaon, yes, such a calculaon s possble. We ake quaons (9) and (13) as he negraons or he consan orce work equaon and he gure o mer equaon respecvely. We know ha, sarng rom 0, we le he negraon nerval ge very large, he work equaon negral wll rs equal and hen exceed he energy calculaed by he gure o mer equaon. So we requre ha be sucenly small ha hs obvous volaon o energy conservaon does no happen. Should all o he npu power be ransormed no knec energy, we would choose he posve roo o he soluon o quaon (15). I some o he power ends up as, or example, hea, hen a smaller value o would oban. We hen choose he value o or he me derenal ha or all nervals o be summed o ge he energes or he wo 5
mehods. Tha s, we noe wha should be obvous physcs or hs suaon: he energes added o he wo sums n every derenal me nerval are always n he same rao as hey are n he very rs nerval because he only nvaran velocy ha exss n hs case s he one o nsananeous res a he ouse o each nerval. I hs prescrpon he only one ha makes physcal sense n he crcumsances s ollowed, no energy conservaon volaon ollows rom he calculaon. And elemenary mechancs s no hreaened by an obvously wrong calculaon. 6