Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.

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1 Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in aoms, coninuing wih hermal moion of molecules and ending up wih moion of sars and galaxies a he scale of he universe iself. Classical mechanics sudies moion of objecs a every-day scale. Even hough classical mechanics requires correcions o describe moion a very small aomic scale as well as a very large scale of universe, bu i sill serves as he basis for descripion of moion a hese levels oo. Think wha does i mean o sudy moion? Wha are he basic seps in his process, wha sors of quesions one has o answer abou moion? The lab exercise may help you o answer hese quesions. Classical mechanics can be furher subdivided ino several pars. The par of classical mechanics which deals wih classificaion and comparison of moion is called kinemaics (from he Greek word kinema, meaning "moion"). The main quesion of kinemaics is How he moion akes place? However, we canno sudy moion wihou undersanding is reasons. This is why mechanics is no limied o kinemaics only. The oher par of mechanics, which sudies causes of moion and deals wih quesion Why moion akes place? is called dynamics. Dynamics is based on he hree Newon s laws. We will discuss hese laws laer during his semeser. Now we shall sar our sudy of kinemaics. The descripion of moion is someimes difficul and challenging ask. You have jus seen ha while performing he lab exercise. Since physics is quaniaive science, we shall inroduce quaniaive characerisics of moion in such a way ha hey can be measured experimenally. Then we shall examine relaions beween hose characerisics. Based on you experience in he lab, hink wha quaniies (variables) one may need in order o be able o describe moion. For oday we shall resric our aenion sudying moion along he sraigh line (one-dimensional moion). However, his moion can occur in any direcion. I could be a car moving along he sraigh par of he highway, or i could be King Kong falling sraigh down from he op of he Empire Sae Building. In boh cases we will only consider objecs moving as a whole. This means ha moion of such an objec is very much like moion of a paricle. All he pars of he objec are moving in he same

2 direcion and a he same rae. King Kong is jus falling down wihou any roaion around his own axis and every porion of he car moves in a same way.. Posiion, Disance and Displacemen To describe moion firs we have o locae he objec in space. This can only be done relaive o some reference poin. The choice of his reference poin depends on he problem. For insance, for he car i can be he place where i sared moving, for King Kong i can be eiher he op of he Empire Sae Building or he ground level. The reference poin serves as he origin (zero poin) of he coordinae sysem. Since oday we are alking abou one-dimensional moion, our coordinae sysem only has one axis. Then we have o choose posiive direcion in which coordinae will increase. This choice depends on our decision. For my firs example, i is naural, bu no required o choose posiive direcion o be he same as direcion of he car's moion. Bu even if he car urns back afer raveling some ime, we sill have o keep he same posiive direcion of he coordinae axis; we canno change he rules during he game. I seems ha in he example abou King Kong, i does no make any difference o have posiive direcion eiher going upwards or downwards. So, now we can define posiion of he objec by is coordinae. If we call he axis o be he x-axis, hen posiion of he objec is defined by is x-coordinae. If he objec moves from is posiion a some x o anoher posiion a x, we shall call he change of is posiion x displacemen x x x (-.) If he objec moves in he posiive direcion of axis x, hen x is also posiive. If i moves in he negaive direcion, hen x x and displacemen x x and is displacemen is negaive. So, displacemen has no jus he magniude x bu also i has he direcion. Physical quaniies which have boh magniude and direcion are called vecors. This means ha displacemen is a vecor. In fac, displacemen only depends on he final and original posiion of he objec bu no on he disance he objec has raveled. For insance, if a car goes from Abilene o Dallas and hen back o he same place in Abilene, is displacemen is zero, while i raveled quie a long way. In he case of he moion along he sraigh line direcion of displacemen can be shown by is sign. In general, however, i is no enough o say ha displacemen is posiive or negaive. Moreover, you canno even say wheher or no i is posiive or negaive, if he displacemen vecor is no direced along he axis of coordinae sysem.

3 In conras o displacemen, he disance raveled by he objec is equal o he oal lengh of he pah raveled and i does no depend on he final and original posiions bu i raher depends on he pah iself. I is a scalar. However, i is also measured in unis of lengh.. Average Velociy and Average Speed Our reference frame is no compleed ye. This is because every even occurs no only somewhere in space bu also somewhere in ime. So, o obain he full physical descripion of moion, we have o upgrade our coordinae sysem by adding a measuring device o coun ime. Again, we have o choose a reference poin, he original momen in ime, when we sar our observaion. In my examples i can be he momen, when he car sars from res or when King Kong sars falling down from he op of he building. Now we can see no only how far he objec goes, bu also how fas i is. This can be presened graphically as he dependence of he posiion on ime x b g, exacly as you jus have done in he lab when measuring posiions of he objecs as funcions of ime. Look a he graphs you go here and ry o inerpre hem, ry o recall wha sor of moion hey describe. x (m) (s) Here, we have a similar graph for he moion of a car. I sars from he original posiion x0 600m a ime 0 0 and hen i moves along he x- axis in he posiive x- direcion. A ime in he same direcion aferwards. 40 s, i crosses he origin of coordinae axis and coninues o move Think abou various possibiliies for he graphs of posiion vs. ime. To undersand how fas his car is moving one can inroduce several quaniies. The firs will be he average velociy which is he raio of he displacemen x o he ime inerval for which his displacemen occurs

4 x x x v avg. (-.) As you can see he average velociy depends on he ime inerval for which i is found. Since he xbg-dependence is, generally speaking, a curve, no a sraigh line, he average velociy will be differen for every differen ime inerval. I is, in fac, he slope of he sraigh line connecing any wo poins of he curve. Differen poins provide differen sraigh lines wih differen slopes. As we can see from is definiion, he average velociy always has he same sign as displacemen x. So, i has cerain direcion and i is a vecor in a same way as displacemen. If displacemen is equal o zero hen he average velociy is zero oo. For a car, which afer a long rip has finally come back o he same poin where i sared, he average velociy is zero. I does no maer, how fas he car was moving during is rip. This does no provide enough informaion. So, in addiion o average velociy we shall also consider an average speed, which is defined as he raio of he oal disance raveled (does no maer in which direcion) for he cerain ime inerval o his ime inerval s avg oal disance. (-.3) Average speed, in conras o average velociy, does no have any sign or direcion, so i is a scalar, bu i sill depends on he ime inerval for which i is aken. 3. Insananeous Velociy and Speed Average velociy, as well as average speed, refer o some ime inerval. However, for he mos par, we wan o know how fas somehing is moving a a given insan. To see ha we have o shrink he ime inerval in definiion of average velociy o zero, which is x v lim 0, (-.4) called insananeous velociy. Graphically insananeous velociy represens he slope of he curve in your graph a any given momen in ime. Insananeous velociy is also a vecor, having boh magniude and direcion. A he same ime, he magniude of insananeous velociy is called insananeous speed. Insananeous speed is he quaniy shown by he car's speedomeer. Example -.. Suppose a car passes hrough 0 miles consrucion zone a 0 mph and hen ravels a 60 mph for anoher 0 miles. Wha is he average velociy of he car during ha ime? Work hrough his example before looking a he soluion.

5 In order o be able o use he definiion of he average velociy -., we firs have o find ime i akes for his car o pass ha disance. I passes firs 0 mi for ime x 0mi hr, v 0mph and second en miles for ime x 0mi hr. v 60mph 6 So he average velociy will be x x x vavg 0mi 0mi 0mi 30 mph. hr 6hr 4 6hr The common misake for his problem is o calculae he average of wo velociies 40 mph insead of average velociy 30mph. Exercise Prove ha he average velociy equals o he average of he wo velociies v bv v g only if he car moves he same ime wih each of hese wo velociies. 4. Acceleraion If you look again a he xbg-graph, you can see ha i has a differen slope a every poin. In fac, in he picure he slope ges larger wih ime, his means ha he car is no moving wih consan velociy, bu acceleraes all he ime. This is wha happens, when he driver is pressing gas pedal. To see how fas velociy of he car is changing during he cerain ime inerval, one can inroduce he average acceleraion in a same way as we have done for he average velociy: a avg v v v. (-.5) Average acceleraion depends on he ime inerval for which i is calculaed. I is a vecor. On he oher hand acceleraion and velociy are quie differen hings, since he laer represens how fas he objec is moving while he former represens how fas is velociy changes. Insananeous acceleraion is acceleraion a a given momen in ime v a lim 0. (-.6) Insananeous acceleraion can be seen as he slope of he curve, which depics dependence of velociy on ime, vbg. As i can be seen from he definiions, since he SI uni for velociy and speed is m/s, he SI uni for acceleraion is m s.