The Contradiction within Equations of Motion with Constant Acceleration

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1 The Conradicion wihin Equaions of Moion wih Consan Acceleraion Louai Hassan Elzein Basheir (Daed: July 7, 0 This paper is prepared o demonsrae he violaion of rules of mahemaics in he algebraic derivaion of he equaions of moion wih consan acceleraion in Newonian mechanics shows he inconsisency conradicion wihin hese equaions even if we have used he crec calculus derivaion also demonsraes he impac of hese conradicions on Newonian mechanics. THEORETICAL BACKGROUND When he acceleraion is consan (Sraigh-Line Moion, four equaions relae he posiion he velociy v a any ime o he iniial posiion o, he iniial velociy v o (boh measured a ime 0, he acceleraion a: where: v o + v v v o + a ( o + v o + a ( v v o + a( o (3 ( vo + v o ( v av (average velociy (5 And f roaion wih consan angular acceleraion we have ω ω o + α θ θ o + ω o + α ω ω o + α(θ θ o The Derivaion of The Equaions of he Moion wih Consan Acceleraion The simples kind of acceleraed moion is sraighline moion wih consan acceleraion when he - acceleraion a is consan, he average -acceleraion a av f any ime inerval is he same as a. a a av v v (6 Now we le o 0 le be any laer ime. We use he symbol v o f he -velociy a he iniial ime 0; he -velociy a he laer ime is v. Then Eq.(6 becomes a v v o 0 v v o + a (7 Ne well derive an equaion f he posiion as a funcion of ime when he -acceleraion is consan. To do his, we use wo differen epressions f he average - velociy v av during he inerval from 0 o any laer ime. The firs epression comes from he definiion v av (8 which is rue wheher no he acceleraion is consan. We call he posiion a ime 0 he iniial posiion, denoed by o The posiion a he laer ime is simply. Thus f he ime inerval 0 he displacemen is o Eq.(8 gives where: ω o + ω ( ωo + ω θ θ o ω av (average angular velociy v av o ( o 0 o (9 We can also ge a second epression f v av ha is valid only when he -acceleraion is consan, so ha he - velociy changes a a consan rae. In his case he average -velociy f he ime inerval from 0 o is simply Physics College, Universiy of Kharoum P.O. Bo: Zip Code: 3 louaielzein@yahoo.com Young, Hugh D. Sears Zemansky s universiy physics : wih modern physics. 3h ed. p The elecronic version.

2 he average of he -velociies a he beginning end of he inerval: v av v 0 + v (0 (This equaion is no rue if he -acceleraion varies during he ime inerval. We also know ha wih consan -acceleraion, he -velociy v a any ime is given by Eq.(7. Subsiuing ha epression f v ino Eq.(0, we find v av (v o + v o + a v o + a ( Finally, we se Eqs.(9 ( equal o each oher simplify: v o + a o o + v o + a ( I s ofen useful o have a relaionship f posiion, - velociy, (consan -acceleraion ha does no involve he ime. To obain his, we firs solve Eq.(7 f hen subsiue he resuling epression ino Eq.(: v v o a ( v vo o + v o + ( v a a vo a We ransfer he erm o o he lef side muliply hrough by a hen simplify, we ge: v v o + a( o (3 ANALYSIS I. The Violaion of Mahemaical Rules in Algebraic Derivaion of Equaions of Moion wih Consan Acceleraion A he beginning le us menion he inegraion of Eq.(7 evaluae i a o 0 o 0 which implies ha v o 0 also. Then we have v v o + a v d v o d + a d hen o + v o + a ( Afer Evaluaing we obain v a (5 a (6 We have wo differen definiions f average velociy v av as we have been menioned in he basic derivaion of he equaions: Eqs.(9 (0. Evaluae Eq.(9 a 0 he final posiion, 0 he final ime which implies ha he final velociy v is equal o 0, he body acceleraes unifmly o res, hen we obain Consequenly Eq.(0 will becomes v av 0 o 0 o v o (7 v av v o + 0 v o (8 The iniial velociy v o in Eq.(7 is he same iniial velociy v o in Eq.(8 in he basic derivaion hey have been assumed ha boh averages are equal o each oher hey have been equaed as been done on he seps o Eq.(. Then le us also equae Eq.(7 (8 as hey did, we find v o v o (9 Because boh iniial velociies are one-hing ha implies ha he iniial velociy v o is equal o he half-iself which leads o inescapable mahemaical conclusion ha which is no rue. The same mahemaical conclusion can be found if we evaluae o 0 o 0, he body acceleraes unifmly from res, we will find ha he final velociy v is equal o he half-iself: v v (0 In der f us o validly equae Eq.(9 (0, i is mahemaically necessary o balance he equaion ha be done by muliply Eq.(9 by, muliply Eq.(0 by hen we equae hem coninue wih he derivaion of he Eq.(. Then we find Subsiue Eq.(7 we ge v o + v o v o + v o + a o ( o + v o + a ( Because o is he arbirary consan of inegraion, we can rewrie Eq.( as: o + v o + a (3

3 3 Taking he d d we obain v v o + a ( Evaluae Eq.(3 ( a o 0 o 0 which implies ha v o 0, we ge a (5 v a (6 Equaion (3 is no equal o Eq.(, Eq.( is no equal o Eq.(7, if we have equaed hem we will end up wih he same inescapable mahemaical conclusion ha which emphasizes ha hey are no equal. Alhough we have used he same equaion (Eq.(7 in our derivaion we arrived a differen resuls (compare Eq.(3 wih Eq.(. In he firs mehod we have used he calculus (inegraion in he second mehod we have used he algebra, even when we ried o verify he algebraic resul by differeniaing Eq.(3 o arrive o Eq.(7 we failed (compare Eq.( wih Eq.(7. The only way o derive algebraically Eq.( from Eq.(7 using he averages (Eqs.(9 (0 is by violaing he sound mahemaical rules f se up a crec equaion (see Eq.(9 Eq.(0. Due o he previous argumen we will consider his algebraic derivaion as invalid, because he assumpion ha he average rae of change is equal o he arihmeic mean is false assumpion, because afer we crec his assumpion (balancing he equaion (see Eq.( we couldn derive Eq.( from Eq.(7, bu we arrived a a differen resul (see Eq.(3. II. The Conradicion wihin Equaions of Moion wih Consan Acceleraion In he following discussion we are going o depend upon he calculus derivaion of he equaions of moion wih consan acceleraion (Eqs.(7 (, we are going o pu aside he algebraic derivaion because we have been considered i invalid. Also we are going o assume ha o 0, o 0 v o 0, simply we are going o use Eqs.(5 (6. Then we have v a a Solving he equaions f he acceleraion a, we ge a v (7 a (8 Because he moion is under consan acceleraion which implies ha he velociy is no consan, we are going o use he average velociy definiions (Eqs.(9 (0 one a ime ha because hey are no equal o each oher as we have been proofed earlier so we couldn subsiue one definiion in Eq.(7 he oher in Eq.(8. They are no echangeable unless under cerain condiion (see Eq.(. Le us firs rewrie Eqs.(7 (8 in erm of average velociy, hen we have a v av a v av (9 (30 Beginning wih he average rae of change (Eq.(9, we obain v av ( o 0 ( o 0 (3 Subsiue Eq.(3 ino Eqs.(9 (30 hen equaing hem, we find (3 Equaion (3 leads o an inescapable mahemaical conclusion ha which is a conradicion because. Now subsiue Eq.(3 ino Eq.(3 equaing i o Eqs.(9 (30 consecuively, we find (33 Equaion (33 leads o an inescapable mahemaical conclusion ha which is a conradicion because. (3 Equaion (3 leads o an inescapable mahemaical conclusion ha which is a conradicion because.

4 Now we swich o he arihmeic average (Eq.(0, hen we obain v av (v v v (35 Subsiue Eq.(35 ino Eqs.(9 (30 hen equaing hem, we find v v (36 Again, Eq(36 leads o an inescapable mahemaical conclusion ha which is a conradicion because. Now subsiue Eq.(35 ino Eq.(3 equaing i o Eqs.(9 (30 consecuively, we find (v av (37 Again, Eq(37 leads o an inescapable mahemaical conclusion ha which is a conradicion because. (v av Equaion (38 leads o an inescapable mahemaical conclusion ha which is a conradicion because. Similarly he equaions of Roaion wih Consan Angular Acceleraion have eacly he same conradicions he whole previous analysis also hold rue f i. II. The Effec of The Conradicion on Newonian Mechanics Assuming ha o 0, o 0 which implies v o 0. We are going o limi he discussion o he average rae of change which simply mean ha v av v /.. Kineic Energy The Wk-Energy Theem From Eq.(3 we have a v v o ( o When we muliply his equaion by m equae o he ne fce F, we find F ma m v v o ( o F ( o mv mv o (39 Which is he wk-energy heem, where is he definiion of kineic energy. K mv Evaluae Eq.(39 a o 0 v o 0, we obain F mv (0 Now when we muliply Eq.(7 by mv, we find (ma(v mv Equaing Eq.(0 (, we ge F mv ( mv mv Again, ha leads o he inescapable mahemaical conclusion ha which is a conradicion because. And when we muliply Eq.(8 by m, we find (ma m (38 F mv (

5 5 Now equaing Eq.(0 (, we ge mv mv Again, ha leads o he inescapable mahemaical conclusion ha which is a conradicion because. Then we obain (ma m ( F m ( v v v. Impulse-Momenum Theem F mv The definiion of he impulse-momenum heem is: J p po J mv mv o (3 Evaluae Eq.(3 a v o 0, hen we obain J mv ( Now when we muliply Eq.(7 by m, we find (ma mv F mv Equaing Eq.( (5, we ge J mv (5 mv mv (6 Which is free from conradicion se as eample of conradicion-free equaion. When we muliply Eq.(8 by m, we find (ma m F mv Now equaing Eq.( (7, we ge J mv (7 mv mv Again, ha leads o he inescapable mahemaical conclusion ha which is a conradicion because. Equaing Eq.( (8, we ge J mv (8 mv mv Again, ha leads o he inescapable mahemaical conclusion ha which is a conradicion because. Ohers eample of conradicions could be found in he same way one of hem is he escape velociy. CONCLUSION We conclude ha hree differen definiions have been assigned o one physical quaniy in he equaions of moion wih consan acceleraion, ha quaniy is he consan acceleraion (eiher in sraigh-line moion roaional moion. Also we conclude ha every equaion from he equaions of moion wih consan acceleraion has a differen definiion o acceleraion consequenly conradicions have been arisen beween hese equaions eiher in he equaions of sraigh-line moion roaional moion. Also we conclude ha he algebraic proof ha has been used o derive he equaions of moion wih consan acceleraion which involve using differen ypes of fmulae o evaluae he average velociy has been shown o be invalid violaing he sound mahemaical rules f se up crec equaions. Also we conclude ha he heems fmulae ha have been derived from he equaions of moion wih consan acceleraion inheris he same discrepancies ha he equaions of moion wih consan acceleraion have which leads o differen definiions o he same physical quaniies ha have been used in Newonian mechanics. Now muliply Eq.(3 by m afer evaluaing i a o 0 v o 0 solving f he acceleraion a.

Week 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem)

Week 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem) Week 1 Lecure Problems, 5 Wha if somehing oscillaes wih no obvious spring? Wha is ω? (problem se problem) Sar wih Try and ge o SHM form E. Full beer can in lake, oscillaing F = m & = ge rearrange: F =