Ground Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan

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1 Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure is going on No gossiping while he lecure is going on Raise your hand if you have quesion o ask Be on ime for lecure Be on ime o come back from he recess break o coninue he lecure Bring your lecurenoes o lecure Kinemaics Describes moion while ignoring he agens ha caused he moion In hese lecures we will consider moion in one dimension (along a sraigh line) We will use he paricle model A paricle is a poin-like objec, has mass bu infiniesimal size (i.e., he size is very very small) Posiion Posiion is defined in erms of a frame of reference One dimensional, so generally he -ais or y- ais The objec s posiion is is locaion wih respec o he frame of reference The frame can be saionary or moving 3 4

2 Posiion-Time Graph Displacemen The posiion-ime graph shows he moion of he paricle (car) The smooh curve is an approimaion as o wha happened beween he poins 5 Defined as he change in posiion during some ime inerval Represened as (pronounced as dela ) = f i (final posiion iniial posiion) SI unis are meers (m), can be posiive or negaive Displacemen may no be always be equal o disance. Disance refers o he lengh of a pah followed by a paricle. Give an eample of disance of 5 m bu he corresponding displacemen is 0. 6 Vecors and Scalars Average Velociy Vecor quaniies need boh magniude (size or numerical value) and direcion o compleely describe hem We use + and signs o indicae vecor direcions Scalar quaniies are compleely described by magniude only 7 The average velociy is he rae (w.r.. ime) a which he displacemen occurs v average f i The dimensions for velociy are lengh / ime, [L/T] The SI unis are m/s, or ms -1 When becomes a very small ime inerval, he average velociy is also he insananeous velociy and is magniude is he slope of he line (gradien of he angen line) in he posiion ime graph 8

3 Insananeous Velociy Average Speed The limi of he average velociy refers o ha insance when he ime inerval becomes infiniesimally shor (very very shor), or when he ime inerval approaches zero The insananeous velociy indicaes wha is happening a every poin of ime. The informaion includes he direcion of movemen and magniude of velociy. Speed is a scalar quaniy same unis as velociy oal disance / oal ime The average speed is no necessarily he magniude of he average velociy Give an eample where average speed is m/s and is corresponding velociy is Insananeous Velociy, equaions Insananeous Velociy, graph The general equaion for insananeous velociy is v lim0 d d (he value of he limi when Δ ends o 0) The insananeous velociy can be posiive, negaive, or zero The insananeous velociy is he slope of he line angen o he vs. curve This would be he green line The blue lines show ha as ges smaller, hey approach he green line Posiion Time 11 1

4 Some Tangen Values Average Acceleraion an (45 ) = y/ = 1 an (0 ) = 0 45 y y y Acceleraion is he rae of change of he velociy a Dimensions are = L/T SI unis are m/s² v v f L /T T v i an (90 ) = Insananeous Acceleraion Insananeous Acceleraion - graph The insananeous acceleraion is he limi of he average acceleraion as approaches 0, which is he value of he consan when ends o 0. a lim 0 d d v dv d Differeniae wih respec o wo imes. Please ake noe ha his has nohing o do wih square The slope of he angen in he velociy vs. ime graph is he acceleraion The green line represens he insananeous acceleraion The blue line is he average acceleraion such as 7 =

5 Acceleraion and Velociy, 1 When an objec s velociy and acceleraion are in he same direcion, he objec is speeding up When an objec s velociy and acceleraion are in he opposie direcion, he objec is slowing down Acceleraion and Velociy, The car is moving wih consan posiive velociy (shown by red arrows mainaining he same size) Acceleraion equals zero Acceleraion and Velociy, 3 Acceleraion and Velociy, 4 Velociy and acceleraion are in he same direcion Acceleraion is uniform (blue arrows mainain he same lengh) Velociy is increasing (red arrows are geing longer) This shows posiive acceleraion and posiive velociy 19 Acceleraion and velociy are in opposie direcions Acceleraion is uniform (blue arrows mainain he same lengh) Velociy is decreasing (red arrows are geing shorer) Posiive velociy and negaive acceleraion 0

6 Kinemaic Equaions - summary Similar se of equaions: 1 v = u + a s = s 0 + ½ (u + v) s = s 0 + u + ½ a v = u + a (s s 0 ) u : iniial velociy v : final velociy s 0 : iniial displacemen s : final displacemen a : acceleraion We will derive hem!! Le Linear Acceleraion Final velociy be v Iniial velociy be u Disance ravelled be s Acceleraion be a Time duraion be By definiion: v u a a v u v u a Le he small change in velociy be v v a v u v i.e. final velociy = iniial velociy + small change in velociy in he ime inerval Consider a consan velociy: In v vs. graph, v velociy V v s Area under he curve s v i.e. he disance ravelled is equal o he area under he curve of v vs. graph. ime

7 Now consider a linear change in velociy in he following graph: velociy v u v a ime Area under he curve = area of recangle + area of riangle 1 = u a 1 s u a If he saring disance is no 0, say s 0, we have 1 s s 0 u a h l 1 l The disance ravelled due o linear acceleraion can also be derived by he area of a rapezium: Area s l 1 l u v h Ne, from, we have Subsiue v u a v u 1 ino s s 0 u a a v u a We have: uv u av u s s0 a a uv u av vu u s s0 a a uv u v vu u s s0 a a v u s s0 a a s s v u v u 0 a s s 0 If he saring poin is a origin, i.e. s 0 = 0 v u as Kinemaic Equaions The kinemaic equaions may be used o solve any problem involving onedimensional moion wih a consan acceleraion Someime you may need o use wo of he equaions o solve one problem Many imes here is more han one way o solve a problem 8

8 Graphical Look a Moion Posiion ime curve (graph) The slope of he curve is he velociy The curved line indicaes ha he velociy is changing Therefore, here is an acceleraion Wha if he curve in he Posiion-Time graph is a horizonal sraigh line? Posiion Time The objec is no moving, i.e., velociy = 0. 9 Wha if he sraigh line is no horizonal? The objec is moving wih consan velociy, i.e., no acceleraion. Posiion Time 30 Graphical Look a Moion velociy ime curve Graphical Look a Moion acceleraion ime curve The slope gives he acceleraion The sraigh line in velociy ime graph indicaes a consan acceleraion, which can be 0 if i is an horizonal line. Velociy Time The zero slope in acceleraion ime graph indicaes a consan acceleraion Acceleraion Time 31 3

9 Eample: Speedy Sue, driving a 30.0 m/s, eners a one-lane unnel. She hen observes a slow-moving van 155 m ahead raveling a 5.00 m/s. Sue applies her brakes bu can accelerae only a.00 m/s because he road is we. Will here be a collision? If yes, deermine how far ino he unnel and a wha ime he collision occurs. If no, deermine he disance of closes approach beween Sue's car and he van. - ms - 30 m/s 5 m/s 30 m/s 5 m/s - ms m 155 m Freely Falling Objecs A freely falling objec is any objec moving freely under he influence of graviy alone. I does no depend upon he iniial moion of he objec, which can be: Dropped released from res Thrown downward Thrown upward Acceleraion of Freely Falling Objec The acceleraion of an objec in free fall is direced downward, regardless of he iniial moion The magniude of free fall acceleraion is g = 9.80 m/s g varies slighly a differen geographical locaions 9.80 m/s is he average a he Earh s surface 35 36

10 Acceleraion of Free Fall, con. We will neglec air resisance Free fall moion is consanly acceleraed moion in one dimension Le upward be posiive, so downward is negaive Use he kinemaic equaions wih a y = g = m/s Free Fall Eample Iniial velociy a A is upward (+) and acceleraion is g (-9.8 m/s ) A B, he velociy is 0 and he acceleraion is g (-9.8 m/s ) A C, he velociy has he same magniude as a A, bu is in he opposie direcion A E, he displacemen is 50.0 m (i ends up 50.0 m below is saring poin) Moion Equaions from Calculus Displacemen equals he area under he velociy ime curve f lim v v ( ) d n 0 n n n i The limi of he sum is a definie inegral Kinemaic Equaions General Calculus Form dv a d v v a d f i 0 d v d v d f i

11 Kinemaic Equaions Calculus Form wih Consan Acceleraion The inegraion form of v f v i gives v v a f i The inegraion form of f i gives 1 f i vi a v = u + a, or v u = a s = s 0 + u + ½ a, or s - s 0 = u + ½ a Eample: A baseball is hi so ha i ravels sraigh upward afer being sruck by he ba. A fan observes ha i akes 3.00 s for he ball o reach is maimum heigh. Find (a) is iniial velociy and (b) he heigh i reaches. y f y i v f = 0 v i 41 4 Eample: Two sones are released from res a a cerain heigh, one afer he oher a an inerval of c seconds. (a) Will he difference in heir speeds increase, decrease, or say he same? Le v1 be he speed of firs sone, and v he speed of second sone. We have v1 = 0 - g v = 0 g (-c); v1 v = -g + g (-c) = -gc consan w.r.. ime 43 (b) Will heir separaion disance increase, decrease or say he same? Le s1 be he displacemen of firs sone, and s he displacemen of second sone. We have: s1 = 0 - ½g for > 0 s = 0 - ½g ( c) for > c s1 s = - ½g + ½g ( c) = - ½g + ½g ( c + c ) Posiion = ½g ( c + c ) absolue difference is increasing w.r.. ime Time 44

12 (c) Will he ime inerval beween he insans a which hey hi he ground be smaller han, equal o, or larger han he ime inerval beween he insans of heir release? Le 1 be he ime for he firs sone o hi he ground, and he ime for he second sone o hi he ground. Le he heigh be h. We have: -h = 0 - ½g1 1 = 1 = h g h g As he disance ravelled by he second sone is he same. We have = c + h g Therefore 1 = c (same as he difference of he insans of heir release.) 45 Eample: A small piece of ligh paper and a meal coin are dropped from he same heigh in he air a he same ime. Which iem will reach he floor firs? The meal coin will reach he ground firs. Is he acceleraion due o graviy eperienced by he small piece of paper is smaller (as i akes a longer ime o reach he ground) han ha eperienced by he meal coin? Why do he wo objecs reach he floor a differen ime? The acceleraion due o graviy is he same for boh iems. Bu he effec of uphrus is more significan on he paper due o is ligh weigh and wide area. The ne acceleraion is herefore no he same on boh objecs hus heir differen arrival imes. 46 Eample: Wha if he paper is on op of a meal plae and all he 3 iems are released a he same ime? How o make he paper and coin hi he floor a he same ime wihou he help of meal plae, conainer and air pump? The meal plae has removed he air volume below he paper during he free fall so he paper and he coin will reach he ground a he same ime. There is almos no uphrus due o he air on he paper. 47 There is nohing you can do wih he weigh of he paper. Bu you can do somehing abou is area. How? 48