2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance

Transcription

1 Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion Free-Fall Acceleraion Graphical Inegraion in Moion Analysis.: Wha is physics? In his chaper we sudy he moion of objecs: How far hey move in a given ime inerval; displacemen and disance How fas hey move; velociy and speed Time rae for velociy or speed change; acceleraion.: Moion We always find moving objecs all around us. The sudy of moion called Kinemaics. Examples on moion: Earh orbi around he sun Car moving on a sree People walking in he marke Ec. In his chaper we will sudy moion ha akes place in a sraigh line Moving objecs will be considered as a paricle ha represen all he objec (all pars of rigid exended objecs moves in he same fashion) Indeed, forces are responsible for objec speeding up, slowing down, or remaining a he same rae.3: Posiion, Displacemen, Disance Posiion (x): Locaion wih respec o a reference poin (origin of coordinae sysem) Displacemen (): Change in posiion from iniial posiion x o final posiion x in a ime inerval (vecor) x x SI uni is ( m) (righ) (lef) Displacemen is posiive (+ve) if moion is in +ve direcion (righ) and negaive ( ve) if moion is in ve direcion (lef)

2 Disance (d): Toal lengh of a pah followed in a ime inerval (scalar always posiive). * Example: a paricle moves from x 5m o x m, hen i reurns back o x 7m. Find a) he firs displacemen, b) he second displacemen, c) he oal displacemen, and d) he oal disance. a) b) c) d).3: Posiion, Displacemen, Disance The firs displacemen x x 5 7m Thesecond displacemen x3 x 7 5m The oaldisplacemen x3 x 7 5 m The oal disance d m.3: Posiion, Displacemen, Disance We can describe he posiion wih ime is o draw he posiion x as a funcion of ime ; o draw he funcion x(). Consider a car moves a from A o B o C o D and o E as shown. The ime beween any wo successive poins is minues Time (min) Posiion x x from A D - - Disance d form A A B A B - C 4 B C D 6 - c d E 8 3 D E 4 8 C A B E x 3 x 3 E B A C - D - -3 ime (min Overall displacemen x E -x A 3 - km, overall disance 8 km Posiion, Displacemen, Disance Consider a car moves from A o B o C o D and o E as shown. The ime inerval beween wo successive poins is min D C A B E Time (min) Posiion - - x f x i x 3 from A Disance d form A A B A B - C 4 B C D 6 - c d E 8 3 D E 4 8 Overall displacemen x E -x A 3 - km, x 3 E B A C - D - ime (min overall disance 8 km.4: Average Velociy and Average Speed Average velociy: rae of change in posiion x x vecor quaniy v avg oal disance oal ime ( m s) Average velociy is he slope of he sraigh line beween he wo posiions x and x on he x- graph Average speed s avg ( m s) scalar quaniy

3 Ex: From he figure below, find he displacemen, average velociy and average speed beween he wo posiions A and F x x x x 5 3 8m 8 v avg.6 m / s 5 s avg.4: Average Velociy and Average Speed.5: Insananeous Velociy and Speed oal disance oal ime +.4 m / s 5 F A Average velociy is he slope of he sraigh line beween wo poins on x graph of ime inerval difference. If he ime inerval decease poin B will move o lef approaching A, unil hey are he same poin a (insan of ime) Velociy a a given insan is he slope of angen line a ha insan of ime (green line on he figure) Speed a a given insan is he magniude of he insananeous velociy. dx v lim d speed v ( m s).6: Acceleraion Acceleraion occurs when he velociy of an objec changes over ime. Consan v a V a consan V and a same direcions speeding up.6: Acceleraion Average acceleraion Slope of he line beween he wo seleced poins on velociy ime graph (blue line) v v v a avg ( m s ) V a consan V and a differen dv d dx d x direcions slowing a down (deceleraion) d Insananeous acceleraion Slope of of he angen line a given insan of ime (green line) on velociy-ime graph or derivaive of velociy wih respec o ime v dv a lim d d d d ( m s )

4 .6: Acceleraion Ex: The displacemen of a paricle wih ime is given by: x 4 + Find he displacemen of he paricle from during firs 3 seconds. Calculae he average velociy in his ime inerval. Find he insananeous velociy a.5 s Find he average speed during firs 3 seconds Average acceleraion during firs 3 seconds Insananeous acceleraion a s The ime a which he objec change direcion.8: Graphical Relaionships and calculus consideraion f x x v d x x i Area under v-graph v v v ad Area under a-graph a() x() Slope of x - graph. dx v ( ) d Slope of v - graph dv d x a ( ) d d Ex: mach acceleraion wih velociy graphs parabola Example velociy of a paricle wih ime is given by: Find he displacemen of he paricle from during firs 4 seconds. Calculae he average velociy in his ime inerval. Find he insananeous velociy a.5 s Find he average speed during firs 3 seconds Average acceleraion during firs 3 seconds Insananeous acceleraion a s The ime a which he objec change direcion Draw he acceleraion graph

5 .7: Consan acceleraion (aconsan): A special case Assuming a ime we have xx and vv v v + a Velociy as funcion of ime x x + ( v + v) x x + v + a x v + ( v + v ) v v + a a Posiion as funcion of ime and velociy Posiion as funcion of ime Velociy as funcion of posiion.7: Consan acceleraion: Ex. A je plane lands wih a speed of 5 m/s and can accelerae a a maximum rae of -5. m/s as i comes o res. a) From he insan he plane ouches he runway, wha is he minimum ime inerval needed before i can come o res? v v 5 m/s, a -5m/s², v v v v 5 v v + a s a 5 b) Can his plane land on a small island airpor where he runway is.8 km long? x v a 5() ( 5)() m is he disance o sop i canno land on he island airpor.9: Free fall acceleraion (Freely Falling Objecs).9: Free fall acceleraion (Freely Falling Objecs) Free falling is an example of moion in consan acceleraion Only he influence of graviy is assumed Downward acceleraion (-ve) regardless of iniial velociy Free fall acceleraion : a - g m / s +y For he equaions of moion in y-direcion for free falling objecs, we replace a by g and x by y Equaions of moion becomes v y y y + v g ( v + v) y y + v g y v ( v + v) g v v g y

6 .9: Free fall acceleraion : Ex: In he figure shown, a picher osses a baseball up along a y axis, wih an iniial speed of m/s. (a) How long does he ball ake o reach is maximum heigh? v m/s and a maximum high v v v v v g. g 9.8 s (b) Wha is he ball s maximum heigh above is release poin? y v g ()(.) (9.8)(.) 7. 3m (c) How long does he ball ake o reach a poin 5. m above is release poin? Posiion y v g reference poin Solve for.53 s and.9 s Example Iniial velociy m/s (poin A) Building heigh 5 m - Time o reach maximum heigh (poin B) - The maximum heigh 3- Time o reurn o he heigh i was hrown a (poin C) 4- Velociy a his ime 5- Velociy and posiion a 5 s. (D) Review Average and insananeous (displacemen, velociy, and acceleraion) v is derivaive (slope) of x, a is derivaive of v and second derivaive of x. For consan acceleraion, x v + v v + a v + v) ( v a v + a Free falling acceleraion is g where g 9.8 m/s² For free falling equaions, replace x by y, and replace a by -g