Chaper 5 Kinemaics In he firs place, wha do we mean b ime and space? I urns ou ha hese deep philosophical quesions have o be analzed ver carefull in phsics, and his is no eas o do. The heor of relaivi shows ha our ideas of space and ime are no as simple as one migh imagine a firs sigh. However, for our presen purposes, for he accurac ha we need a firs, we need no be ver careful abou defining hings precisel. Perhaps ou sa, Tha s a errible hing I learned ha in science we have o define everhing precisel. We canno define anhing precisel! If we aemp o, we ge ino ha paralsis of hough ha comes o philosophers, who si opposie each oher, one saing o he oher, You don know wha ou are alking abou! The second one sas. Wha do ou mean b know? Wha do ou mean b alking? Wha do ou mean b ou?, and so on. In order o be able o alk consrucivel, we jus have o agree ha we are alking roughl abou he same hing. You know as much abou ime as ou need for he presen, bu remember ha here are some subleies ha have o be discussed; we shall discuss hem laer. Par A: One-Dimensional Moion Inroducion Richard Fenman, The Fenman Lecures on Phsics Kinemaics is he mahemaical descripion of moion. The erm is derived from he Greek word kinema, meaning movemen. In order o quanif moion, a mahemaical coordinae ssem, called a reference frame, is used o describe space and ime. Once a reference frame has been chosen, we can inroduce he phsical conceps of posiion, veloci and acceleraion in a mahemaicall precise manner. Figure 5. shows a Caresian coordinae ssem in one dimension wih uni vecor î poining in he direcion of increasing -coordinae. Figure 5. A one-dimensional Caresian coordinae ssem. Richard P. Fenman, Rober B. Leighon, Mahew Sands, The Fenman Lecures on Phsics, Addison-Wesle, Reading, Massachuses, (963), p. -. /3/7
5. Posiion, Time Inerval, Displacemen Posiion Consider an objec moving in one dimension. We denoe he posiion coordinae of he cener of mass of he objec wih respec o he choice of origin b ( ). The posiion coordinae is a funcion of ime and can be posiive, zero, or negaive, depending on he locaion of he objec. The posiion has boh direcion and magniude, and hence is a vecor (Figure 5.), () = () ˆi. (5..) We denoe he posiion coordinae of he cener of he mass a = b he smbol ( = ). The SI uni for posiion is he meer [m] (see Secion.3). Time Inerval Figure 5. The posiion vecor, wih reference o a chosen origin. Consider a closed inerval of ime [, ]. We characerize his ime inerval b he difference in endpoins of he inerval such ha The SI unis for ime inervals are seconds [s]. Definiion: Displacemen Δ =. (5..) The change in posiion coordinae of he mass beween he imes and is Δ ( ( ) ( )) ˆi Δ( )ˆ i. (5..3) This is called he displacemen beween he imes and (Figure 5.3). Displacemen is a vecor quani. /3/7
Figure 5.3 The displacemen vecor of an objec over a ime inerval is he vecor difference beween he wo posiion vecors 5. Veloci When describing he moion of objecs, words like speed and veloci are used in common language; however when inroducing a mahemaical descripion of moion, we need o define hese erms precisel. Our procedure will be o define average quaniies for finie inervals of ime and hen eamine wha happens in he limi as he ime inerval becomes infiniesimall small. This will lead us o he mahemaical concep ha veloci a an insan in ime is he derivaive of he posiion wih respec o ime. Definiion: Average Veloci The componen of he average veloci, v, for a ime inerval he displacemen Δ divided b he ime inerval Δ, Δ is defined o be v Δ. (5..) Δ The average veloci vecor is hen Δ v() ˆi = v() ˆi. (5..) Δ The SI unis for average veloci are meers per second [m s ]. Insananeous Veloci Consider a bod moving in one direcion. We denoe he posiion coordinae of he bod b ( ), wih iniial posiion a ime =. Consider he ime inerval [, +Δ]. The average veloci for he inerval Δ is he slope of he line connecing he poins (, ( )) and (, ( +Δ)). The slope, he rise over he run, is he change in posiion over he change in ime, and is given b v rise Δ ( +Δ) ( ) = =. (5..3) run Δ Δ /3/7 3
Le s see wha happens o he average veloci as we shrink he size of he ime inerval. The slope of he line connecing he poins (, ( )) and (, ( + Δ )) approaches he slope of he angen line o he curve ( ) a he ime (Figure 5.4). Figure 5.4 Graph of posiion vs. ime showing he angen line a ime. In order o define he limiing value for he slope a an ime, we choose a ime inerval [, +Δ]. For each value of Δ, we calculae he average veloci. As Δ, we generae a sequence of average velociies. The limiing value of his sequence is defined o be he -componen of he insananeous veloci a he ime. Definiion: Insananeous Veloci The -componen of insananeous veloci a ime is given b he slope of he angen line o he curve of posiion vs. ime curve a ime : Δ ( +Δ) ( ) d v ( ) lim v = lim = lim. (5..4) The insananeous veloci vecor is hen Δ Δ Δ Δ Δ d Eample : Deermining Veloci from Posiion v() = v () ˆi. (5..5) Consider an objec ha is moving along he -coordinae ais represened b he equaion () b = + (5..6) /3/7 4
where is he iniial posiion of he objec a =. We can eplicil calculae he -componen of insananeous veloci from Equaion (5..4) b firs calculaing he displacemen in he -direcion, Δ = ( +Δ) ( ). We need o calculae he posiion a ime + Δ, Then he insananeous veloci is ( +Δ ) = + b( +Δ ) = + b( + Δ +Δ ). (5..7) + b ( + Δ +Δ) + b ( +Δ) ( ) v () = lim = lim Δ Δ Δ Δ. (5..8) This epression reduces o v() = lim b+ bδ. (5..9) Δ The firs erm is independen of he inerval Δ and he second erm vanishes because he limi as Δ of Δ is zero. Thus he insananeous veloci a ime is v () = b. (5..) In Figure 5.5 we graph he insananeous veloci, v ( ), as a funcion of ime. Figure 5.5 A graph of insananeous veloci as a funcion of ime. 5.3 Acceleraion /3/7 5
We shall appl he same phsical and mahemaical procedure for defining acceleraion, he rae of change of veloci. We firs consider how he insananeous veloci changes over an inerval of ime and hen ake he limi as he ime inerval approaches zero. Average Acceleraion Acceleraion is he quani ha measures a change in veloci over a paricular ime inerval. Suppose during a ime inerval Δ a bod undergoes a change in veloci Δ v= v( +Δ) v( ). (5.3.) The change in he -componen of he veloci, hen Δ, for he ime inerval [, +Δ] is v Definiion: Average Acceleraion Δ v = v ( +Δ) v ( ). (5.3.) The -componen of he average acceleraion for he ime inerval o be Δ is defined ˆ Δ v ˆ ( v( +Δ) v( )) ˆ Δv a = a ˆ i i = i = i. (5.3.3) Δ Δ Δ The SI unis for average acceleraion are meers per second squared, [m s ]. Insananeous Acceleraion On a graph of he -componen of veloci vs. ime, he average acceleraion for a ime inerval Δ is he slope of he sraigh line connecing he wo poins (, v( )) and ( +Δ, v( +Δ )). In order o define he -componen of he insananeous acceleraion a ime, we emplo he same limiing argumen as we did when we defined he insananeous veloci in erms of he slope of he angen line. Definiion: Insananeous Acceleraion. The -componen of he insananeous acceleraion a ime is he limi of he slope of he angen line a ime of he graph of he -componen of he veloci as a funcion of ime, ( v( + Δ) v( )) Δv dv a( ) lim a = lim = lim Δ Δ Δ Δ Δ d. (5.3.4) The insananeous acceleraion vecor is hen /3/7 6
In Figure 5.6 we illusrae his geomerical consrucion. a() = a () ˆi. (5.3.5) Figure 5.6 Graph of veloci vs. ime showing he angen line a ime. Since veloci is he derivaive of posiion wih respec o ime, he -componen of he acceleraion is he second derivaive of he posiion funcion, a = dv = d. (5.3.6) d Eample : Deermining Acceleraion from Veloci Le s coninue Eample, in which he posiion funcion for he bod is given b = + (/ ) b, and he -componen of he veloci is v = b. The -componen of he insananeous acceleraion a ime is he limi of he slope of he angen line a ime of he graph of he -componen of he veloci as a funcion of ime (Figure 5.5) a d Δ Δ Δ Δ d dv v ( + Δ) v ( ) b + b Δ b = = lim = lim = b. (5.3.7) Noe ha in Equaion (5.3.7), he raio Δv/ Δ is independen of Δ, consisen wih he consan slope of he graph in Figure 5.5. 5.4 Consan Acceleraion /3/7 7
Le s consider a bod undergoing consan acceleraion for a ime inerval Δ = [, ]. When he acceleraion a is a consan, he average acceleraion is equal o he insananeous acceleraion. Denoe he -componen of he veloci a ime = b v, v( = ). Therefore he -componen of he acceleraion is given b a Δv v() v = a = = Δ,. (5.4.) Thus he veloci as a funcion of ime is given b v () = v + a. (5.4.), When he acceleraion is consan, he veloci is a linear funcion of ime. Veloci: Area Under he Acceleraion vs. Time Graph In Figure 5.7, he -componen of he acceleraion is graphed as a funcion of ime. Figure 5.7 Graph of he -componen of he acceleraion for a consan as a funcion of ime. The area under he acceleraion vs. ime graph, for he ime inerval Δ = =, is Using he definiion of average acceleraion given above, Area( a, ) a. (5.4.3) Area( a, ) a =Δ v = v( ) v,. (5.4.4) Displacemen: Area Under he Veloci vs. Time Graph In Figure 5.8, we graph he -componen of he veloci vs. ime curve. /3/7 8
Figure 5.8 Graph of veloci for a consan as a funcion of ime. The region under he veloci vs. ime curve is a rapezoid, formed from a recangle and a riangle and he area of he rapezoid is given b Subsiuing for he veloci (Equaion (5.4.)) ields Area( v, ) = v, + ( v( ) v,). (5.4.5) Area( v, ) v ( v a v ) v a =, +, +, =, +. (5.4.6) Figure 5.9 The average veloci over a ime inerval. We can hen deermine he average veloci b adding he iniial and final velociies and dividing b a facor of wo (see Figure 5.9). v = ( v ( ) + v, ). (5.4.7) The above mehod for deermining he average veloci differs from he definiion of average veloci in Equaion (5..). When he acceleraion is consan over a ime /3/7 9
inerval, he wo mehods will give idenical resuls. Subsiue ino Equaion (5.4.7) he -componen of he veloci, Equaion (5.4.), o ield ( ) v = ( v() + v, ) = ( v, + a) + v, = v, + a. (5.4.8) Recall Equaion (5..); he average veloci is he displacemen divided b he ime inerval (noe we are now using he definiion of average veloci ha alwas holds, for non-consan as well as consan acceleraion), he displacemen is equal o Δ = v. (5.4.9) () Subsiuing Equaion (5.4.8) ino Equaion (5.4.9) shows ha displacemen is given b Δ () = v = v,+ a. (5.4.) Now compare Equaion (5.4.) o Equaion (5.4.6) o conclude ha he displacemen is equal o he area under he graph of he -componen of he veloci vs. ime, Δ () = v,+ a = Area( v, ), (5.4.) and so we can solve Equaion (5.4.) for he posiion as a funcion of ime, () v a = +, +. (5.4.) Figure 5. shows a graph of his equaion. Noice ha a = he slope ma be in general non-zero, corresponding o he iniial veloci componen v,. Figure 5. Graph of posiion vs. ime for consan acceleraion. 5.5 Inegraion and Kinemaics /3/7
Change of Veloci as he Inegral of Non-consan Acceleraion When he acceleraion is a non-consan funcion, we would like o know how he - componen of he veloci changes for a ime inerval Δ = [, ]. Since he acceleraion is non-consan we canno simpl mulipl he acceleraion b he ime inerval. We shall calculae he change in he -componen of he veloci for a small ime inerval Δi [ i, i + ] and sum over hese resuls. We hen ake he limi as he ime inervals become ver small and he summaion becomes an inegral of he -componen of he acceleraion. For a ime inerval Δ = [, ], we divide he inerval up ino N small inervals Δi [ i, i + ], where he inde i =,,..., N, and, N. Over he inerval Δ i, we can approimae he acceleraion as a consan, a( i). Then he change in he - componen of he veloci is he area under he acceleraion vs. ime curve, Δv v ( ) v ( ) = a ( ) Δ + E i (5.5.), i i+ i i i where Ei is he error erm (see Figure 5.a). Then he sum of he changes in he - componen of he veloci is i= N Δ v, i = v v = + v 3 v + + v N = v N i= ( ( ) ( )) ( ( ) ( )) ( ( ) ( )). (5.5.) In his summaion pairs of erms of he form ( v v ) overall sum becomes ( ) ( ) = sum o zero, and he v () v () = Δv i= N, i. (5.5.3) i= Subsiuing Equaion (5.5.) ino Equaion (5.5.3), i= N i= N i= N. (5.5.4) v () v () = Δ v = a ( ) Δ + Ei, i i i i= i= i= We now approimae he area under he graph in Figure 5.a b summing up all he recangular area erms, i= N Area ( a, ) = a ( ) Δ. (5.5.5) N i i i= /3/7
Figures 5.a and 5.b Approimaing he area under he graph of he -componen of he acceleraion vs. ime Suppose we make a finer subdivision of he ime inerval Δ = [, ] b increasing N, as shown in Figure 5.b. The error in he approimaion of he area decreases. We now ake he limi as N approaches infini and he size of each inerval Δ approaches zero. For each value of N, he summaion in Equaion (5.5.5) gives a value for A rea N( a, ), and we generae a sequence of values {Area ( a, ), Area ( a, ),..., Area ( a, )}. (5.5.6) N The limi of his sequence is he area, Area( a, ), under he graph of he -componen of he acceleraion vs. ime. When aking he limi, he error erm vanishes in Equaion (5.5.4), i i= N Ei = N i = lim. (5.5.7) Therefore in he limi as N approaches infini, Equaion (5.5.4) becomes i= N i= N i= N, (5.5.8) v ( ) v () = lim a ( ) Δ + lim E = lim a ( ) Δ = Area( a, ) i i i i i N N N i= i= i= and hus he change in he -componen of he veloci is equal o he area under he graph of -componen of he acceleraion vs. ime. Definiion: Inegral of acceleraion /3/7
The inegral of he -componen of he acceleraion for he inerval [, ] is defined o be he limi of he sequence of areas, A rea ( a, ), and is denoed b N = i= N = a ( ) d lim a ( ) Δ = Area( a, ). (5.5.9) i i Δi i = Equaion (5.5.8) shows ha he change in he componen of he veloci is he inegral of he -componen of he acceleraion wih respec o ime. = v () v () = a ( ) d. (5.5.) = Using inegraion echniques, we can in principle find he epressions for he veloci as a funcion of ime for an acceleraion. Inegral of Veloci We can repea he same argumen for approimaing he area A rea( v, ) under he graph of he -componen of he veloci vs. ime b subdividing he ime inerval ino N inervals and approimaing he area b i= N Area ( a, ) = v ( ) Δ. (5.5.) N i i i= The displacemen for a ime inerval Δ = [, ] is limi of he sequence of sums Area ( a, ), N i= N Δ = () () = lim v( i) Δ i. (5.5.) N i = This approimaion is shown in Figure 5.. /3/7 3
Figure 5. Approimaing he area under he graph of he -componen of he veloci vs. ime. Definiion: Inegral of Veloci The inegral of he -componen of he veloci for he inerval [, ] is he limi of he sequence of areas, Area ( a, ), and is denoed b N = i= N = v ( ) d lim v ( ) Δ = Area( v, ). (5.5.3) i i Δi i = The displacemen is hen he inegral of he -componen of he veloci wih respec o ime, = Δ = () () = v( ) d. (5.5.4) = Using inegraion echniques, we can in principle find he epressions for he posiion as a funcion of ime for an acceleraion. Eample: Le s consider a case in which he acceleraion, a ( ), is no consan in ime, a () = b + b + b (5.5.5) The graph of he -componen of he acceleraion vs. ime is shown in Figure 5.3 /3/7 4
Figure 5.3 A non-consan acceleraion vs. ime graph. Le s find he change in he -componen of he veloci as a funcion of ime. Denoe he iniial veloci a = b v, v ( = ). Then, = = 3 b b (), = ( ) = ( o + + ) = + + 3 = =. (5.5.6) v v a d b b b d b The -componen of he veloci as a funcion in ime is hen 3 b b v() = v, + b+ +. (5.5.7) 3 Denoe he iniial posiion b ( = ). The displacemen as a funcion of ime is he inegral () = v ( ) d. (5.5.8) = = Use Equaion (5.5.7) for he -componen of he veloci in Equaion (5.5.8) o find = 3 3 4 b b b b b () = v, + b + + d = v,+ + +. 3 6 = (5.5.9) Finall he posiion is hen /3/7 5
3 4 b b b () = + v,+ + +. 6 (5.5.) 5.6 Free Fall An imporan eample of one-dimensional moion (for boh scienific and hisorical reasons) is an objec undergoing free fall. Suppose ou are holding a sone and hrow i sraigh up in he air. For simplici, we ll neglec all he effecs of air resisance. The sone will rise and fall along a line, and so he sone is moving in one dimension. Galileo Galilei was he firs o definiivel sae ha all objecs fall owards he earh wih a consan acceleraion, laer measured o be of magniude g = 9.8 m s o wo significan figures (see Secion 3.3). The smbol g will alwas denoe he magniude of he acceleraion a he surface of he earh. (We will laer see ha Newon s Universal Law of Graviaion requires some modificaion of Galileo s saemen, bu near he earh s surface his saemen holds.) Le s choose a coordinae ssem wih he origin locaed a he ground, and he -ais perpendicular o he ground wih he -coordinae increasing in he upward direcion. Wih our choice of coordinae ssem, he acceleraion is consan and negaive, a = g = (5.6.) () 9.8m s. When we ignore he effecs of air resisance, he acceleraion of an objec in free fall near he surface of he earh is downward, consan and equal o 9.8 m s. Of course, if more precise numerical resuls are desired, a more precise value of be used (see Secion 3.3). Equaions of Moions g mus We have alread deermined he posiion equaion (Equaion (5.4.)) and veloci equaion (Equaion (5.4.)) for an objec undergoing consan acceleraion. Wih a simple change of variables from, he wo equaions of moion for a freel falling objec are and () v g = +, (5.6.) v () = v g, (5.6.3), /3/7 6
where is he iniial posiion from which he sone was released a =, and v, is he iniial -componen of veloci ha he sone acquired a = from he ac of hrowing. Par B: Two-Dimensional Moion 5.7 Inroducion o he Vecor Descripion of Moion in Two and Three Dimensions So far we have inroduced he conceps of kinemaics o describe moion in one dimension; however we live in a mulidimensional universe. In order o eplore and describe moion in his universe, we begin b looking a eamples of wo-dimensional moion, of which here are man; planes orbiing a sar in ellipical orbis or a projecile moving under he acion of uniform graviaion are wo common eamples. We will now eend our definiions of posiion, veloci, and acceleraion for an objec ha moves in wo dimensions (in a plane) b reaing each direcion independenl, which we can do wih vecor quaniies b resolving each of hese quaniies ino componens. For eample, our definiion of veloci as he derivaive of posiion holds for each componen separael. In Caresian coordinaes, in which he direcions of he uni vecors do no change from place o place, he posiion vecor r () wih respec o some choice of origin for he objec a ime is given b The veloci vecor v ( ) r() = () ˆi+ () ĵ. (5.7.) a ime is he derivaive of he posiion vecor, d() ˆ d() v() = i ˆ v () ˆ () d + j d i + v ˆj, (5.7.) where v ( ) ( ) / d d and v( ) d( ) / d denoe he - and -componens of he veloci respecivel. The acceleraion vecor veloci vecor, a ( ) is defined in a similar fashion as he derivaive of he dv ()ˆ () dv a() = i ˆ a () ˆ (), d + j d i + a ˆj (5.7.3) where a ( ) dv ( ) / d and a ( ) dv ( ) / d denoe he - and -componens of he acceleraion. /3/7 7
5.8 Projecile Moion A special case of wo-dimensional moion occurs when he verical componen of he acceleraion is consan and he horizonal componen is zero. Then he complee se of equaions for posiion and veloci for each independen direcion of moion are given b ˆ ˆ (, ) ˆ r() = () i+ () j = + v i + + v,+ a ˆ j, (5.8.) v() = v () ˆi+ v () ˆj = v ˆi + v + a ˆj, (5.8.) ( ),, a() = a () ˆ () ˆ ˆ i+ a j= a j. (5.8.3) Consider he moion of a bod ha is released wih an iniial veloci heigh h above he ground. Two pahs are shown in Figure 5.4. v a a Figure 5.4 Acual orbi and parabolic orbi of a projecile The doed pah represens a parabolic rajecor and he solid pah represens he acual orbi. The difference beween he pahs is due o air resisance. There are oher facors ha can influence he pah of moion; a roaing bod or a special shape can aler he flow of air around he bod, which ma induce a curved moion or lif like he fligh of a baseball or golf ball. We shall begin our analsis b neglecing all influences on he bod ecep for he influence of gravi. We shall choose coordinaes wih our -ais in he verical direcion wih ĵ direced upwards and our -ais in he horizonal direcion wih î direced in he direcion ha he bod is moving horizonall. We choose our origin o be he place where he bod is released a ime =. Figure 5.5 shows our coordinae ssem wih he posiion of he bod a ime and he coordinae funcions ( ) and ( ). /3/7 8
Figure 5.5 A coordinae skech for parabolic moion. The coordinae funcion ( ) represens he disance from he bod o he origin along he -ais a ime, and he coordinae funcion ( ) represens he disance from he bod o he origin along he -ais a ime. The -componen of he acceleraion, a = g, (5.8.4) is a consan and is independen of he mass of he bod. Noice ha because we chose our posiive -direcion o poin upwards. a < ; his is Since we are ignoring he effecs of an horizonal forces, he acceleraion in he horizonal direcion is zero, a = ; (5.8.5) herefore he -componen of he veloci remains unchanged hroughou he fligh. Kinemaic Equaions of Moion The kinemaic equaions of moion for he posiion and veloci componens of he objec are () = + v,, (5.8.6) v () = v, (5.8.7), () v g = +,, (5.8.8) /3/7 9
v () = v g. (5.8.9), Iniial Condiions In hese equaions, he iniial veloci vecor is v = v ˆ i+ v ĵ. (5.8.) (),, Ofen he descripion of he fligh of a projecile includes he saemen, a bod is projeced wih an iniial speed v a an angle θ wih respec o he horizonal. The vecor decomposiion diagram for he iniial veloci is shown in Figure 5.6. The componens of he iniial veloci are given b v = v cosθ, (5.8.), v = v sinθ. (5.8.), Figure 5.6 A vecor decomposiion of he iniial veloci Since he iniial speed is he magniude of he iniial veloci, we have ( ) / v = v + v. (5.8.3),, The angle θ is relaed o he componens of he iniial veloci b, an v. θ = v, (5.8.4) The iniial posiion vecor appears wih componens r = ˆi+ ˆj. (5.8.5) /3/7
Noe ha he rajecor in Figure 5.6 has = =, bu his will no alwas be he case, as in he analsis below Orbi equaion So far our descripion of he moion has emphasized he independence of he spaial dimensions, reaing all of he kinemaic quaniies as funcions of ime. We shall now eliminae ime from our equaion and find he orbi equaion of he bod. We begin wih Equaion (5.8.6) for he -componen of he posiion, () = + v, (5.8.6) and solve Equaion (5.8.6) for ime as a funcion of ( ), () v =. (5.8.7), The verical posiion of he bod is given b Equaion (5.8.8), () = + v, g. (5.8.8) We hen subsiue he above epression, Equaion (5.8.7) for ime ino our equaion for he -componen of he posiion ielding This epression can be simplified o give () () = +. (5.8.9) () v, g v, v, v g () = + () () () +. (5.8.) ( ) (, v, v, This is seen o be an equaion for a parabola b rearranging erms o find ) g g v, v, g () = () + () + +. (5.8.) v, v, v, v, v, The graph of ( ) as a funcion of ( ) is shown in Figure 5.7. /3/7
Figure 5.7 The parabolic orbi Noe ha a an poin ( (), ()) along he parabolic rajecor, he direcion of he angen line a ha poin makes an angle θ wih he posiive -ais as shown in figure 5.7. This angle is given b d θ = an, (5.8.) d where d / d is he derivaive of he funcion ( ) ( ( )) = a he poin ( (), ()). The veloci vecor is given b d() ˆ d() v() = i ˆ v () ˆ () d + j d i + v ˆj (5.8.3) The direcion of he veloci vecor a a poin ( (), () ) can be deermined from he componens. Le φ be he angle ha he veloci vecor forms wih respec o he posiive -ais. Then v () d / d d φ = = = v() d / d d an an an. (5.8.4) Comparing our wo epressions we see ha φ = θ ; he slope of he graph of ( ) vs. ( ) a an poin deermines he direcion of he veloci a ha poin. We canno ell from our graph of ( ) how fas he bod moves along he curve; he magniude of he veloci canno be deermined from informaion abou he angen line. /3/7
If, as in Figure 5.6, we choose our origin a he iniial posiion of he bod a =, hen = and =. Our orbi equaion, Equaion (5.8.) can now be simplified o g, () = () (). v +, v, v (5.8.5) Secion C: Non-Consan Acceleraion We have seen an eample where he acceleraion of an objec was a given non-consan funcion of ime, Equaion (5.5.5). In man phsical siuaions he force on an objec will be modeled as depending on he objec s veloci. We have alread seen saic and kineic fricion beween surfaces modeled as being independen of he surfaces relaive veloci. Common eperience (swimming, hrowing a Frisbee) ells us ha he fricion force beween an objec and a fluid can be a complicaed funcion of veloci. Indeed, hese complicaed relaions are an imporan par of such opics as aircraf design. 5.9 Fricion Force as a Linear Funcion of Veloci A reasonable model for he fricional force on an objec m moving a low speeds hrough a viscous medium is F fricion = γ m v (5.9.) where γ is a consan ha depends on he properies (densi, viscosi) of he medium and he size and shape of he objec. Noe ha γ has dimensions of inverse ime, [ ] [ ] dim Force M L T dim[ γ ] = = dim[mass] dim veloci M L T T =. (5.9.) The minus sign in Equaion (5.9.) indicaes ha he fricional force is direced agains he objec s veloci (relaive o he fluid). In a siuaion where F fricion is he ne force, Equaion (5.9.) reduces o a = γ v. (5.9.3) The acceleraion has no componen perpendicular o he veloci, and in he absence of oher forces will move in a sraigh line, bu wih varing speed. Denoe he direcion of his moion as he -direcion, so ha Equaion (5.9.3) becomes a dv d v = = γ. (5.9.4) /3/7 3
Equaion (5.9.4) is now a differenial equaion. For our purposes, we ll creae an iniialcondiion problem b specifing ha he iniial -componen of veloci is v ( = ) = v. Two mehods of solving his problem, boh used b phsiciss, will be presened here. The firs is he use of an ansaz ; from Equaion (5.9.4) we would epec a graph of v as a funcion of ime o sar a v wih a negaive slope, and since v is decreasing we epec he slope o decrease. There are man such funcions. Wha we do for an ansaz is o pick a specific funcional form ha has he desired shape and see how, if a all, he funcion can be made o saisf Equaion (5.9.4) and he iniial condiion v ( = ) = v. Two such funcions are considered: v v = v e / τ = v, + / τ (5.9.5) where τ and τ are consans wih dimensions of ime ha ma have o be deermined. The wo funcions v and v are ploed in Figure 5.8 below. For Ploing purposes, he verical scale is he raio v/ v or v/ v and he horizonal scale is = / τ. The upper plo (he green plo if viewed in color) is v and he lower (red) is v. I should be clear ha boh plos have he desired qualiaive proper of decreasing wih decreasing slope. We sill need o see which of eiher of he epressions in (5.9.5) saisf boh Equaion (5.9.4) and he iniial condiion v ( = ) = v. Boh saisf he iniial condiion, and indeed he leading facor could be changed as desired o mach an iniial condiion. Performing he differeniaions, dv d dv d = v e = v / τ τ τ = v = τ ( + / τ ) vτ v. (5.9.6) A mahemaical assumpion, esp. abou he form of an unknown funcion, which is made in order o faciliae soluion of an equaion or oher problem. Oford English Dicionar. In oher words, an inspired guess. /3/7 4
Thus, v as given in he second epression in (5.9.5) canno be a soluion o (5.9.4). We see ha v will be a soluion if we choose τ = / γ, wih he resul v = v e γ. (5.9.7) Figure 5.8 Plos of he rial funcions v Noe ha even hough he chosen form for did no work for his problem, we see ha if we encounered a similar problem wih he magniude of he fricional force proporional o he square of he speed, we ve go ha one solved, if we remember where we did i. (Such a dependence of fricion on speed is known as Newonian Damping, so we suspec i s worh knowing.) The differenial equaion in (5.9.4) is known as a separable equaion, in ha he equaion ma be rewrien as The inegraion in his case is simple, leading o dv d = ln[ v ] = γ. (5.9.8) v d d ln [ v ] = γ + ln[ v ] v v = e γ, (5.9.9) /3/7 5
reproducing he resul of Equaion (5.9.7) I should be noed ha he resul in Equaion (5.9.8) is someimes obained b crossmulipling he epression in (5.9.4) o obain dv v = γ d (5.9.) and hen inegraing boh sides wih respec o he respecive inegraion variables, o obain he same resul. This is indeed equivalen o making a change of variables in he calculus wording. 5. Linear Fricion wih Gravi A common eension of he above eample is o have an objec falling hrough he same viscous medium, subjec o gravi bu no oher forces. Taking he posiive -direcion o be downward, he equaion of moion becomes a dv g = =. (5..) d γ v Noe ha he epression in Equaion (5..) is valid for he verical veloci direced upwards ( v < ) or downwards ( v > ). Unlike he previous eample, here is a preferred direcion; we epec ha in he limi of long imes and no oher forces, an objec would evenuall fall sraigh down. We will use he epecaion o simplif our mehods of soluion, saring wih an ansaz ha assumes a erminal veloci (acuall, a erminal speed) ; he erminal veloci is ha for which he acceleraion given b v erm Equaion (5..) is zero, verm = g/ γ (noe ha v erm has dimensions of veloci). A his poin, i helps o rewrie Equaion (5..) as dv d = γ ( v erm v ) (5..) If we have, as before, an iniial-value problem, in his case he iniial condiion being v () = v, our rial soluion will be one ha has v () = v bu which approaches v erm for large imes. From our previous eperience, we suspec ha a funcion involving an eponenial will be more likel o lead o success han a raional funcion. So, our rial funcion will be Performing he differeniaion, / 3 ( ) v = v + v v e τ. (5..3) 3 erm erm /3/7 6
dv d 3 = ( v verm) e τ 3 = τ 3 ( v v ) 3 erm / τ3 (5..4) and we see ha v is a soluion o he problem wih he choiceτ 3 = / γ ; 3 ( ) v v v v e γ = erm + erm. (5..5) A plo of he raio v / verm as a funcion of = γ is shown in Figure 5.9, for he four differen iniial condiions v = verm v = ver m, v = and v = v e rm. Noe ha he las of hese (he blue plo if seen in color) is a siuaion where he objec is iniiall moving upward. Figure 5.9 Falling objecs wih fricion The success of our ansaz suggess a more direc echnique. In Equaion (5..), make he subsiuion u = v v. Recognizing ha du / d = dv / d, ha equaion becomes erm du d = γ u (5..6) /3/7 7
and he iniial condiion becomes u ( = ) = v ver m. This has been reduced o a problem done previous (he previous secion) and we can jus quoe ha resul; ( ) v v v v e γ erm = erm (5..7) and we re done. If we hadn seen o do his, Equaion (5..) is sill separable, dv d = ln verm v v v d d erm = γ. (5..8) Inegraing and eponeniaing, including he iniial condiion, γ ln[ ] ( ), ln v v = + v v erm erm v v = v v e γ erm erm (5..9) reproducing he previous resul. In he resul, he behavior of he soluion should be checked in he large and small limis of ime. From he graphs, or from he analic form, v v as. For small imes / γ, consider he soluion epressed as erm γ ( ) v = v e + v e erm g γ = ( e ) + v e γ g γ ( γ ) = g+ v + v γ γ (5..) o zero order in γ, as epeced. /3/7 8