Relative velocity in one dimension


 Archibald Hensley
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1 Connexions module: m Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies peraining o moion are characerisically relaive in naure. The measuremens, describing moion, are subjec o he sae of moion of he frame of reference wih respec o which measuremens are made. Our day o day percepion of moion is generally earh's view a view common o all bodies a res wih respec o earh. However, we encouner occasions when here is percepible change o our earh's view. One such occasion is raveling on he ciy rains. We nd ha i akes lo longer o overake anoher rain on a parallel rack. Also, we see wo people alking while driving separae cars in he parallel lane, as if hey were saionary o each oher!. In erms of kinemaics, as a maer of fac, hey are acually saionary o each oher  even hough each of hem are in moion wih respec o ground. In his module, we se ourselves o sudy moion from a perspecive oher han ha of earh. Only condiion we subjec ourselves is ha wo references or wo observers making he measuremens of moion of an objec, are moving a consan velociy (We shall learn aferward ha wo such reference sysems moving wih consan velociy is known as inerial frames, where Newon's laws of moion are valid.). The observers hemselves are no acceleraed. There is, however, no resricion on he moion of he objec iself, which he observers are going o observe from dieren reference sysems. The moion of he objec can very well be acceleraed. Furher, we shall sudy relaive moion for wo caegories of moion : (i) one dimension (in his module) and (ii) wo dimensions (in anoher module). We shall skip hree dimensional moion hough wo dimensional sudy can easily be exended o hree dimensional moion as well. 1 Relaive moion in one dimension We sar here wih relaive moion in one dimension. I means ha he individual moions of he objec and observers are along a sraigh line wih only wo possible direcions of moion. 1.1 Posiion of he poin objec We consider wo observers A and B. The observer A is a res wih earh, whereas observer B moves wih a velociy v BA wih respec o he observer A. The wo observers wach he moion of he poin like objec C. The moions of B and C are along he same sraigh line. noe: I helps o have a convenion abou wriing subscriped symbol such as v BA. The rs subscrip indicaes he eniy possessing he aribue (here velociy) and second subscrip indicaes he eniy wih respec o which measuremen is made. A velociy like v BA shall, herefore, mean velociy of B wih respec o A. Version 1.8: Oc 3, :46 am GMT5 hp://creaivecommons.org/licenses/by/2.0/
2 Connexions module: m The posiion of he objec C as measured by he wo observers A and B are x CA and x CB as shown in he gure. The observers are represened by heir respecive frame of reference in he gure. Posiion Figure 1 Here, x CA = x BA + x CB 1.2 Velociy of he poin objec We can obain velociy of he objec by diereniaing is posiion wih respec o ime. As he measuremens of posiion in wo references are dieren, i is expeced ha velociies in wo references are dieren, because one observer is a res, whereas oher observer is moving wih consan velociy. and v CA = x CA v CB = x CB Now, we can obain relaion beween hese wo velociies, using he relaion x CA = x BA + x CB and diereniaing he erms of he equaion wih respec o ime : x CA = x BA + x CB
3 Connexions module: m v CA = v BA + v CB Relaive velociy Figure 2 The meaning of he subscriped velociies are : v CA : velociy of objec "C" wih respec o "A" v CB : velociy of objec "C" wih respec o "B" v BA : velociy of objec "B" wih respec o "A" Example 1 Problem : Two cars, sanding a disance apar, sar moving owards each oher wih speeds 1 m/s and 2 m/s along a sraigh road. Wha is he speed wih which hey approach each oher? Soluion : Le us consider ha "A" denoes Earh, "B" denoes rs car and "C" denoes second car. The equaion of relaive velociy for his case is : v CA = v BA + v CB Here, we need o x a reference direcion o assign sign o he velociies as hey are moving opposie o each oher and should have opposie signs. Le us consider ha he direcion of he velociy of B is in he reference direcion, hen
4 Connexions module: m Relaive velociy Figure 3 Now : v BA = 1 m/s and v CA = 2 m/s. v CA = v BA + v CB 2 = 1 + v CB v CB = 2 1 = 3 m/s This means ha he car "C" is approaching "B" wih a speed of 3 m/s along he sraigh road. Equivalenly, i means ha he car "B" is approaching "C" wih a speed of 3 m/s along he sraigh road. We, herefore, say ha he wo cars approach each oher wih a relaive speed of 3 m/s. 1.3 Acceleraion of he poin objec If he objec being observed is acceleraed, hen is acceleraion is obained by he ime derivaive of velociy. Diereniaing equaion of relaive velociy, we have : v CA = v BA + v CB v CA = v BA + v CB The meaning of he subscriped acceleraions are : a CA = a BA + a CB a CA : acceleraion of objec "C" wih respec o "A" a CB : acceleraion of objec "C" wih respec o "B"
5 Connexions module: m a BA : acceleraion of objec "B" wih respec o "A" Bu we have resriced ourselves o reference sysems which are moving a consan velociy. This means ha relaive velociy of "B" wih respec o "A" is a consan. In oher words, he acceleraion of "B" wih respec o "A" is zero i.e. a BA = 0. Hence, a CA = a CB The observers moving a consan velociy, herefore, measure same acceleraion of he objec. As a maer of fac, his resul is characerisics of inerial frame of reference. The reference frames, which measure same acceleraion of an objec, are inerial frames of reference. 2 Inerpreaion of he equaion of relaive velociy The imporan aspec of relaive velociy in one dimension is ha velociy has only wo possible direcions. We need no use vecor noaion o wrie or evaluae equaion of relaive velociies in one dimension. The velociy, herefore, can be reaed as signed scalar variable; plus sign indicaing velociy in he reference direcion and minus sign indicaing velociy in opposie o he reference direcion. 2.1 Equaion wih reference o earh The equaion of relaive velociies refers velociies in relaion o dieren reference sysem. v CA = v BA + v CB We noe ha wo of he velociies are referred o A. In case, A denoes Earh's reference, hen we can convenienly drop he reference. A velociy wihou reference o any frame shall hen mean Earh's frame of reference. v C = v B + v CB v CB = v C v B This is an imporan relaion. This is he working relaion for relaive moion in one dimension. We shall be using his form of equaion mos of he ime, while working wih problems in relaive moion. This equaion can be used eecively o deermine relaive velociy of wo moving objecs wih uniform velociies (C and B), when heir velociies in Earh's reference are known. Le us work ou an exercise, using new noaion and see he ease of working. Example 2 Problem : Two cars, iniially 100 m disan apar, sar moving owards each oher wih speeds 1 m/s and 2 m/s along a sraigh road. When would hey mee? Soluion : The relaive velociy of wo cars (say 1 and 2) is : v 21 = v 2 v 1 Le us consider ha he direcion v 1 is he posiive reference direcion. Here, v 1 = 1 m/s and v 2 = 2 m/s. Thus, relaive velociy of wo cars (of 2 w.r. 1) is : v 21 = 2 1 = 3 m / s This means ha car "2" is approaching car "1" wih a speed of 3 m/s along he sraigh road. Similarly, car "1" is approaching car "2" wih a speed of 3 m/s along he sraigh road. Therefore, we can say ha wo cars are approaching a a speed of 3 m/s. Now, le he wo cars mee afer ime :
6 Connexions module: m = Displacemen Relaive velociy = = 33.3 s 2.2 Order of subscrip There is sligh possibiliy of misundersanding or confusion as a resul of he order of subscrip in he equaion. However, if we observe he subscrip in he equaion, i is easy o formulae a rule as far as wriing subscrip in he equaion for relaive moion is concerned. For any wo subscrips say A and B, he relaive velociy of A (rs subscrip) wih respec o B (second subscrip) is equal o velociy of A (rs subscrip) subraced by he velociy of B (second subscrip) : v AB = v A v B and he relaive velociy of B (rs subscrip) wih respec o A (second subscrip) is equal o velociy of B (rs subscrip) subraced by he velociy of A (second subscrip): v BA = v B v A 2.3 Evaluaing relaive velociy by making reference objec saionary An inspecion of he equaion of relaive velociy poins o an ineresing feaure of he equaion. We need o emphasize ha he equaion of relaive velociy is essenially a vecor equaion. In one dimensional moion, we have aken he libery o wrie hem as scalar equaion : v BA = v B v A Now, he equaion comprises of wo vecor quaniies ( v B and v A ) on he righ hand side of he equaion. The vecor v A is acually he negaive vecor i.e. a vecor equal in magniude, bu opposie in direcion o v A. Thus, we can evaluae relaive velociy as following : 1. Apply velociy of he reference objec (say objec "A") o boh objecs and render he reference objec a res. 2. The resulan velociy of he oher objec ("B") is equal o relaive velociy of "B" wih respec o "A". This concep of rendering he reference objec saionary is explained in he gure below. In order o deermine relaive velociy of car "B" wih reference o car "A", we apply velociy vecor of car "A" o boh cars. The relaive velociy of car "B" wih respec o car "A" is equal o he resulan velociy of car "B".
7 Connexions module: m Relaive velociy Figure 4 This echnique is a very useful ool for consideraion of relaive moion in wo dimensions. 2.4 Direcion of relaive velociies For a pair of wo moving objecs moving uniformly, here are wo values of relaive velociy corresponding o wo reference frames. The values dier only in sign no in magniude. This is clear from he example here. Example 3 Problem : Two cars sar moving away from each oher wih speeds 1 m/s and 2 m/s along a sraigh road. Wha are relaive velociies? Discuss he signicance of heir sign. Soluion : Le he cars be denoed by subscrips 1 and 2. Le us also consider ha he direcion v 2 is he posiive reference direcion, hen relaive velociies are :
8 Connexions module: m Relaive velociy Figure 5 v 12 = v 1 v 2 = 1 2 = 3 m / s v 21 = v 2 v 1 = 2 ( 1 ) = 3 m / s The sign aached o relaive velociy indicaes he direcion of relaive velociy wih respec o reference direcion. The direcions of relaive velociy are dieren, depending on he reference objec. However, wo relaive velociies wih dieren direcions mean same physical siuaion. Le us read he negaive value rs. I means ha car 1 moves away from car 2 a a speed of 3 m/s in he direcion opposie o ha of car 2. This is exacly he physical siuaion. Now for posiive value of relaive velociy, he value reads as car 2 moves from car 1 in he direcion of is own velociy. This also is exacly he physical siuaion. There is no conradicion as far as physical inerpreaion is concerned. Imporanly, he magniude of approach whaever be he sign of relaive velociy is same.
9 Connexions module: m Relaive velociy Figure Relaive velociy.vs. dierence in velociies I is very imporan o undersand ha relaive velociy refers o wo moving bodies no a single body. Also ha relaive velociy is a dieren concep han he concep of "dierence of wo velociies", which may perain o he same or dieren objecs. The dierence in velociies represens dierence of nal velociy and iniial velociy and is independen of any order of subscrip. In he case of relaive velociy, he order of subscrips are imporan. The expression for wo conceps viz relaive velociy and dierence in velociies may look similar, bu hey are dieren conceps. 2.6 Relaive acceleraion We had resriced ou discussion up o his poin for objecs, which moved wih consan velociy. The quesion, now, is wheher we can exend he concep of relaive velociy o acceleraion as well. The answer is yes. We can aach similar meaning o mos of he quaniies  scalar and vecor boh. I all depends on aaching physical meaning o he relaive concep wih respec o a paricular quaniy. For example, we measure poenial energy (a scalar quaniy) wih respec o an assumed daum. Exending concep of relaive velociy o acceleraion is done wih he resricion ha measuremens of individual acceleraions are made from he same reference. If wo objecs are moving wih dieren acceleraions in one dimension, hen he relaive acceleraion is equal o he ne acceleraion following he same working relaion as ha for relaive velociy. For example, le us consider han an objec designaed as "1" moves wih acceleraion "a 1 " and he oher objec designaed
10 Connexions module: m as "2" moves wih acceleraion " a 2 " along a sraigh line. Then, relaive acceleraion of "1" wih respec o "2" is given by : a 12 = a 1 a 2 Similarly,relaive acceleraion of "2" wih respec o "1" is given by : a 21 = a 2 a 1 3 Worked ou problems Example 4: Relaive moion Problem :Two rains are running on parallel sraigh racks in he same direcion. The rain, moving wih he speed of 30 m/s overakes he rain ahead, which is moving wih he speed of 20 m/s. If he rain lenghs are 200 m each, hen nd he ime elapsed and he ground disance covered by he rains during overake. Soluion : Firs rain, moving wih he speed of 30 m/s overakes he second rain, moving wih he speed of 20 m/s. The relaive speed wih which rs rain overakes he second rain, v 12 = v 1 v 2 = = 10 m/s. The gure here shows he iniial siuaion, when faser rain begins o overake and he nal siuaion, when faser rain goes pas he slower rain. The oal disance o be covered is equal o he sum of each lengh of he rains (L1 + L2) i.e = 400 m. Thus, ime aken o overake is :
11 Connexions module: m The oal relaive disance Figure 7: rain. The oal relaive disance o cover during overake is equal o he sum of lenghs of each = = 40 s. In his ime inerval, he wo rains cover he ground disance given by: s = 30 x x 40 = = 2000 m. Exercise 1 (Soluion on p. 12.) In he quesion given in he example, if he rains ravel in he opposie direcion, hen nd he ime elapsed and he ground disance covered by he rains during he period in which hey cross each oher. 4 Check your undersanding Check he module iled Relaive velociy in one dimension (Check your undersanding) 1 o es your undersanding of he opics covered in his module. 1 "Relaive velociy in one dimension(applicaion)" <hp://cnx.org/conen/m14035/laes/>
12 Connexions module: m Soluions o Exercises in his Module Soluion o Exercise 1 (p. 11) v 12 = v 1 v 2 = 30 ( 20 ) = 50 m/s. The oal disance o be covered is equal o he sum of each lengh of he rains i.e = 400 m. Thus, ime aken o overake is : = = 8 s. Now, in his ime inerval, he wo rains cover he ground disance given by: s = 30 x x 8 = = 400 m. In his case, we nd ha he sum of he lenghs of he rains is equal o he ground disance covered by he rains, while crossing each oher.
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