Mechanics Acceleration The Kinematics Equations

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1 Mechanics Acceleraion The Kinemaics Equaions Lana Sheridan De Anza College Sep 27, 2018

2 Las ime kinemaic quaniies graphs of kinemaic quaniies

3 Overview acceleraion he kinemaics equaions (consan acceleraion) applying he kinemaics equaions

4 Quesion: Average Velociy vs Average Speed Quick Quiz Under which of he following condiions is he magniude of he average velociy of a paricle moving in one dimension smaller han he average speed over some ime inerval? A A paricle moves in he +x direcion wihou reversing. B A paricle moves in he x direcion wihou reversing. C A paricle moves in he +x direcion and hen reverses he direcion of is moion. D There are no condiions for which his is rue. 1 Serway & Jewe, page 24.

5 Quesion: Average Velociy vs Average Speed Quick Quiz Under which of he following condiions is he magniude of he average velociy of a paricle moving in one dimension smaller han he average speed over some ime inerval? A A paricle moves in he +x direcion wihou reversing. B A paricle moves in he x direcion wihou reversing. C A paricle moves in he +x direcion and hen reverses he direcion of is moion. D There are no condiions for which his is rue. 1 Serway & Jewe, page 24.

6 Insananeous Velociy and Posiion-Time Graphs aper 2 Moion in One Dimension x (m) (s) The blue line beween posiions and approaches he green angen line as poin is moved closer o poin. a b 2 y graph of represens in he n he veriin he quanhorizonal Figure 2.3 (a) Graph represening he moion of he car in Figure 2.1. (b) An enlargemen of he upper-lef-hand corner of he graph. v = lim x( + ) x() = lim represens he velociy of he car a poin. Wha we have done is deermine he insananeous velociy a ha momen. In oher words, he insananeous velociy v x equals he limiing value of he raio Dx/D as D approaches zero: 1 Dx x = dx d

7 v 2 v v 2 v leraion Velociy vs. Time Graphs ding o he expresseconds. o s. The acceleraion a is equal o he slope of he green angen line a 2 s, which is 20 m/s 2. v x (m/s) (s) 2.9 (Example 2.6) ociy ime graph for a moving along he x axis ng o he expression v x (0) m/s v x (2.0) m/s

8 leraion Velociy vs. Time Graphs ding o he expresseconds. o s. The acceleraion a is equal o he slope of he green angen line a 2 s, which is 20 m/s 2. v x (m/s) (s) 2.9 (Example 2.6) ociy ime graph for a moving along he x axis ng o he expression v x (0) m/s The slope a any poin of he velociy-ime curve is he v x acceleraion 5 a40 ha 2 5(2.0) ime m/s v 2 v v 2 v

9 Acceleraion acceleraion average acceleraion a = dv d = d2 x d 2 a avg = v Acceleraion is also a vecor quaniy.

10 Acceleraion acceleraion average acceleraion a = dv d = d2 x d 2 a avg = v Acceleraion is also a vecor quaniy. If he acceleraion vecor is poined in he same direcion as he velociy vecor (ie. boh are posiive or boh negaive), he paricle s speed is increasing.

11 Acceleraion acceleraion average acceleraion a = dv d = d2 x d 2 a avg = v Acceleraion is also a vecor quaniy. If he acceleraion vecor is poined in he same direcion as he velociy vecor (ie. boh are posiive or boh negaive), he paricle s speed is increasing. If he acceleraion vecor is poined in he opposie direcion as he velociy vecor (ie. one is posiive he oher is negaive), he paricle s speed is decreasing. (I is deceleraing.)

12 Acceleraion and Velociy-Time Graphs Acceleraion is he slope of a velociy-ime curve. Unis: meers per second per second, m/s 2

13 Acceleraion and Velociy-Time Graphs Acceleraion is he slope of a velociy-ime curve. Unis: meers per second per second, m/s 2 In general, acceleraion can be a funcion of ime a().

14 Acceleraion Graphs x i x Slope v xf x Slope v xi Slope v xf x i aslope v xi x i vslope x v a xi Slope a x v a x v xi Slope a x v x v xi Slope a x b v xi ab x a x b a x va xi x v xi v xf Slope 0 Slope 0 Slope 0 a x v xi a x a x a x v xf v xf 2.6 Analysis Mod If he acceleraion of a pari Under o analyze. 2.6 Analysis Consan A very common Mod Ac ha in which he accelera aunder x,avg over any Consan ime inerval A If he acceleraion of a paricl o analyze. A very common a a any insan wihin he in ha If he in acceleraion which he acceleraio ou he moion. This of a siua paric ao x,avg analyze. over any A ime very common inerval isa model: ha any in insan he paricle which wihin he acceleraio he under iner ou generae a x,avg he over moion. several any ime This equaions inerval siuaioi model: If we a any insan he replace paricle a wihin x,avg under by a he ine co x i generae, we find ou he moion. several ha equaions This siuaio h model: If we he replace paricle a x,avg under by a x inc, generae we find several ha equaions h or If we replace a x,avg by a x in, we find ha v or This powerful expression v e xf or if we know he objec s This velociy ime powerful expression graph for hi en v The if we graph know is he a sraigh objec s line xf velociy ime This slope powerful is consisen graph expression wih for his a x en 5c The ive, if we graph which know is indicaes a he sraigh objec s a line, pos in velociy ime slope is of consisen he line graph in wih for Figure a his x 5 ive, The san, which graph he indicaes graph is a sraigh of a acceler posii line,

15 Reurning o Velociy vs Time Graphs PROBLEMS 49 The he area moorcycle under a velociy-ime during each graph of he has following a special segmens inerpreaion: of he i is moion: he displacemen (a) A, (b) B, and of he (c) objec C. over he ime inerval considered. 15 Velociy, v (m/s) 10 5 A B C O Time, (s) FIGURE 2 31 Problem 32 x = v avg 33. A person on horseback moves according o he velociy-

16 Reurning o Velociy vs Time Graphs PROBLEMS 49 The he area moorcycle under a velociy-ime during each graph of he has following a special segmens inerpreaion: of he i is moion: he displacemen (a) A, (b) B, and of he (c) objec C. over he ime inerval considered. 15 Velociy, v (m/s) 10 5 A B C O Time, (s) FIGURE 2 31 Problem 32 x = (25 m m + 75 m)i = 200 m i 33. A person on horseback moves according o he velociy-

17 Area under Velociy vs. Time Graphs CHAPTER 2 KINEMATICS 65 v- and x- graphs for he same objec: Area under v- graph = x. Slope of x- curve = v.

18 n Velociy One Dimensionvs. Time Graphs v x The area of he shaded recangle is equal o he displacemen in he ime inerval n. v xn,avg i f n der he curve in he velociy ime x = lim graph. v n Therefore, = v d in he limi n S `, or D 0 n displacemen is n i where x represens he change in posiion (displacemen) in he Dx 5 lim Dn S 0 a v xn,avg D n (2 ime inerval i o f. n f

19 Velociy vs. Time Graphs n One Dimension v x The area of he shaded recangle is equal o he displacemen in he ime inerval n. v xn,avg i f Or we can wrie n der he curve in he velociy ime x() graph. = vtherefore, d in he limi n S `, or D n displacemen is i if he objec sars a posiion x = 0 when = i. Dx 5 lim D S 0 a v xn,avg D n (2

20 Quesion Wha does he area under an acceleraion-ime graph represen?

21 bes describes he moion. Maching Velociy o Acceleraion Graphs hs v x v x v x s e d), a b c a x a x a x d e f

22 Summary acceleraion Homework - CHANGED! Read Ch 2. Ch 2, Quesions: 1, 2, 4, 5; Problems: 19, 21, 90 (will be se on Monday: Ch 2, Problems: 23, 25, 31, 35, 41, 69, 73)

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