Chapter 2 - Motion along a straight line

Transcription

1 MECHANICS Kinemaics Chaper - Moion along a sraigh line I. Posiion and displacemen II. Velociy III. Acceleraion IV. Moion in one dimension wih consan acceleraion V. Free fall Paricle: poin-like objec ha has a mass bu infiniesimal size. I. Posiion and displacemen Posiion: Defined in erms of a frame of reference: or y ais in D. - The objec s posiion is is locaion wih respec o he frame of reference. Posiion-Time graph: shows he moion of he paricle (car). The smooh curve is a guess as o wha happened beween he daa poins.

2 I. Posiion and displacemen Displacemen: Change from posiion o - (.) during a ime inerval. - Vecor quaniy: Magniude (absolue value) and direcion (sign). - Coordinae (posiion) Displacemen Coordinae sysem > Only he iniial and final coordinaes influence he displacemen many differen moions beween and give he same displacemen. Disance: lengh of a pah followed by a paricle. - Scalar quaniy Displacemen Disance Eample: round rip house-work-house disance raveled km displacemen Review: - Vecor quaniies need boh magniude (size or numerical value) and direcion o compleely describe hem. - We will use + and signs o indicae vecor direcions. - Scalar quaniies are compleely described by magniude only.

3 II. Velociy Average velociy: Raio of he displacemen ha occurs during a paricular ime inerval o ha inerval. v (.) Moion along -ais -Vecor quaniy indicaes no jus how fas an objec is moving bu also in which direcion i is moving. - SI Unis: m/s - Dimensions: Lengh/Time [L]/[T] - The slope of a sraigh line connecing poins on an -versus- plo is equal o he average velociy during ha ime inerval. Average speed: Toal disance covered in a ime inerval. Toal disance S (.3) S magniude V S always > Scalar quaniy Same unis as velociy Eample: A person drives mi a 3 mi/h and mi and 5 mi/h Is he average speed >,<, mi/h? < mi/h mi/(3 mi/h).3 h ; mi/(5 mi/h).8 h o.3 h S 8 mi/.3h 37.5mi/h 3

4 Insananeous velociy: How fas a paricle is moving a a given insan. v lim d d (.) - Vecor quaniy - The limi of he average velociy as he ime inerval becomes infiniesimally shor, or as he ime inerval approaches zero. - The insananeous velociy indicaes wha is happening a every poin of ime. - Can be posiive, negaive, or zero. () - The insananeous velociy is he slope of he line angen o he vs. curve (green line). Insananeous velociy: Posiion Slope of he paricle s posiion-ime curve a a given insan of ime. V is angen o () when Time When he velociy is consan, he average velociy over any ime inerval is equal o he insananeous velociy a any ime. Insananeous speed: Magniude of he insananeous velociy. Eample: car speedomeer. - Scalar quaniy Average velociy (or average acceleraion) always refers o an specific ime inerval. Insananeous velociy (acceleraion) refers o an specific insan of ime.

5 III. Acceleraion Average acceleraion: Raio of a change in velociy v o he ime inerval in which he change occurs. a v v v (.5) - Vecor quaniy - Dimensions [L]/[T], Unis: m/s - The average acceleraion in a v- plo is he slope of a sraigh line connecing poins corresponding o wo differen imes. V Insananeous acceleraion: Limi of he average acceleraion as approaches zero. - Vecor quaniy v dv d a lim d d (.6) - The insananeous acceleraion is he slope of he angen line (v- plo) a a paricular ime. (green line in B) - Average acceleraion: blue line. - When an objec s velociy and acceleraion are in he same direcion (same sign), he objec is speeding up. - When an objec s velociy and acceleraion are in he opposie direcion, he objec is slowing down. 5

6 - Posiive acceleraion does no necessarily imply speeding up, and negaive acceleraion slowing down. Eample (): v -5m/s ; v m/s in 5s paricle slows down, a 5m/s - An objec can have simulaneously v and a Eample (): ()A v()a a()a ; A s, v() bu a()a Eample (3): - The car is moving wih consan posiive velociy (red arrows mainaining same size) Acceleraion equals zero. Eample (): + acceleraion + velociy - Velociy and acceleraion are in he same direcion, a is uniform (blue arrows of same lengh) Velociy is increasing (red arrows are geing longer). Eample (5): - acceleraion + velociy - Acceleraion and velociy are in opposie direcions. - Acceleraion is uniform (blue arrows same lengh). - Velociy is decreasing (red arrows are geing shorer). 6

7 IV. Moion in one dimension wih consan acceleraion - Average acceleraion and insananeous acceleraion are equal. a a v v - Equaions for moion wih consan acceleraion: v v + a v v + v v + v and (.7) v (.8), (.9) (.7), (. ) (.7) a v a a v + v v (.8) a v + (. ) v + a + a( v ) v (.9) + a + a( v + a( ) (.) missing a ) PROBLEMS - Chaper P. A red car and a green car move oward each oher in adjacen lanes and parallel o The -ais. A ime, he red car is a and he green car a m. If he red car has a consan velociy of km/h, he cars pass each oher a.5 m, and if i has a consan velociy of km/h, hey pass each oher a 76.6m. Wha are (a) he iniial velociy, and (b) he acceleraion of he green car? v r km/h v r km/h O X r 76.6m 3 km h m.m h 36s km X r.5 m d m X g m r g + v r r + v + a g g () ().5m v 8s r r 5.55m 76.6m v 6.9s r r.m r v g v r g.5 g.5 g a a g g 76.6 v.5 v g g (6.9s).5 (6.9s) a g (8s).5 (8s) a g The car moves o he lef (-) in my reference sysem a<, v< a g. m/s v g 3.55 m/sc 7

8 P: A he insan he raffic ligh urns green, an auomobile sars wih a consan acceleraion a of. m/s. A he same insan, a ruck, raveling wih consan speed of 9.5 m/s, overakes and passes he auomobile. (a) How far beyond he raffic signal will he auomobile overake he ruck? (b) How fas will he auomobile be raveling a ha insan? a c. m/s, v c m/s s Car (m) s Truck v 9.5 m/s d v 9.5 T T () Truck d v + a d +.5 (.m ). c C c () (m) d? ( a) s d (9.5m )(8.63s) 8m Car ( b) v v + a d (.m ) (8m) v 9m f c f P3: A proon moves along he -ais according o he equaion: 5+, where is in meers and is in seconds. Calculae (a) he average velociy of he proon during he firs 3s of is moion. ( 3) ( ) ( 5)( 3) + ( )( 3) v 8 m. 3 (b) Insananeous velociy of he proon a 3s. d v ( ) 5 + v(3s) m d dv (c) Insananeous acceleraion of he proon a 3s. a ( ) m a(3s) d (d) Graph versus and indicae how he answer o (a) (average velociy) can be obained from he plo. (e) Indicae he answer o (b) (insananeous velociy) on he graph. (f) Plo v versus and indicae on i he answer o (c). 5 + v 5 + P. An elecron moving along he -ais has a posiion given by: 6 ep(-) m, where is in seconds. How far is he elecron from he origin when i momenarily sops? () when v()?? d v 6e 6e 6e ( ) d v ( ) ; ( e > ) s ( ) 6/ e 5. 9m 8

9 P5. When a high speed passenger rain raveling a 6 km/h rounds a bend, he engineer is shocked o see ha a locomoive has improperly enered ino he rack from a siding and is a disance D 676 m ahead. The locomoive is moving a 9 km/h. The engineer of he high speed rain immediaely applies he brakes. (a) Wha mus be he magniude of he resulan deceleraion if a collision is o be avoided? (b) Assume ha he engineer is a when a he firs spos he locomoive. Skech () curves represening he locomoive and high speed rain for he siuaion in which a collision is jus avoided and is no quie avoided. s Train Locomoive d L (m) > s D D+d L (m) v T 6km/h.7 m/s v T D movemen wih a<ce v L 9 km/h 8.5 m/s is consan d d v 8.5 L () Locomoive L L 8.5 d + D v + a d a () L T T L T Train P5. v Tf v + T.7m (.7m )(8.5m ) 36m a a ( eq. ) T T dl dl (.7m ) v Tf v T + a ( D+ d a T L) T (676m+ dl ) (3) () d 38.3m L () (3) from () 7.s 8.5 () + (3) a T d L 36m 38.3m.97m Locomoive Collision can be avoided L T a T - Collision can be avoided: Train Collision can no be avoided Slope of () vs. locomoive a 7. s (he poin were boh Lines mee) v insananeous locom > Slope of () vs. rain - Collision canno be avoided: Slope of () vs. locomoive a 7. s < Slope of () vs. rain 9

10 - The moion equaions can also be obained by indefinie inegraion: dv a d dv a d v a + C; v v d v d d v d d ( v + a) d d v d + a d v + a a v() + a() + C' C' + v + a a v ( a)() + C v C v v + a + C'; V. Free fall Moion direcion along y-ais ( y > upwards) Free fall acceleraion: (near Earh s surface) a -g -9.8 m/s (in mov. eqs. wih consan acceleraion) Due o graviy downward on y, direced oward Earh s cener Approimaions: - Locally, Earh s surface essenially fla free fall a has same direcion a slighly differen poins. - All objecs a he same place have same free fall a (neglecing air influence). VI. Graphical inegraion in moion analysis From a() versus graph inegraion area beween acceleraion curve and ime ais, from o v() v v a d Similarly, from v() versus graph inegraion area under curve from o () v d

11 P6: A rocke is launched verically from he ground wih an iniial velociy of 8m/s. I ascends wih a consan acceleraion of m/s o an aliude of km. Is moors hen fail, and he rocke coninues upward as a free fall paricle and hen falls back down. (a) Wha is he oal ime elapsed from akeoff unil he rocke srikes he ground? (b) Wha is he maimum aliude reached? (c) Wha is he velociy jus before hiing ground? ) Ascen a m/s y y y ma y km a -g a m/s v, 3 +v, a -g y y v +.5 a s v v a v (m ) (53.8s) + 8m/ s 9m ) Ascen a -9.8 m/s v m s a g 9 / 9. 96s 9.8m Toal ime ascen s+9.96 s 83. s v 8m/s v 3 3) Descen a -9.8 m/s y v +.5 a 9.9. s oal s s +. s37.6 s h ma y y - m v -.9 (9 m/s)(9.96s)-(.9m/s )(9.96s) m h ma. km v3 ( v ) a g v g v 53.35m 3