Welcome Back o Physics 215! (General Physics I) Thurs. Jan 19 h, 2017 Lecure01-2 1 Las ime: Syllabus Unis and dimensional analysis Today: Displacemen, velociy, acceleraion graphs Nex ime: More acceleraion and free fall! 2 1
Ask a physicis. If you email me quesions, I m happy o answer hem here. 3 Undergraduae colloquium moved To Tuesday, January 31 s a 3:30pm 4 2
Kinemaics-- describing moion 1D Four basic ypes of moion Slide 1-19 3
The Paricle Model Ofen moion of he objec as a whole is no influenced by deails of he objec s size and shape. We only need o keep rack of a single poin on he objec. So we can rea he objec as if all is mass were concenraed ino a single poin. A mass a a single poin in space is called a paricle. Paricles have no size, no shape and no op, boom, fron or back. Below is a moion diagram of a car sopping, using he paricle model. Slide 1-27 The Paricle Model Moion Diagram in which he objec is represened as a paricle Moion diagram of a rocke launch Slide 1-26 4
SG 1-1.2 Three moion diagrams are shown. Which is a dus paricle seling o he floor a consan speed, which is a ball dropped from he roof of a building, and which is a descending rocke slowing o make a sof landing on Mars? A. (a) is dus, (b) is ball, (c) is rocke. B. (a) is ball, (b) is dus, (c) is rocke. C. (a) is rocke, (b) is dus, (c) is ball. D. (a) is rocke, (b) is ball, (c) is dus. E. (a) is ball, (b) is rocke, (c) is dus. Slide 1-29 Posiion and Displacemen Neglec shape of objec and represen by poin moving in space (1D) Posiion may be specified by giving disance o origin x coordinae Choice of origin arbirary! many choices o describe same physical siuaion. Hence x-coordinae no unique 5
Displacemen = change in posiion x 2 x 1 Q O P origin Displacemen (P Q) = x 2 - x 1 = D x Displacemen does NOT depend on origin! Displacemen Displacemen is disance plus direcion Displacemen D x is a vecor quaniy change in posiion (vecor) of objec In one dimension, his amouns o a sign Displacemen owards increasing x posiive Displacemen owards decreasing x negaive 6
SG 1-2.02.1 An an zig-zags back and forh on a picnic able as shown. The an s disance raveled and displacemen are A. 50 cm and 50 cm. B. 30 cm and 50 cm. C. 50 cm and 30 cm. D. 50 cm and 50 cm. E. 50 cm and 30 cm. Slide 2-29 Velociy Definiion: Average velociy in some ime inerval D is given by v av = (x 2 - x 1 )/( 2-1 ) = D x/d Displacemen D x can be posiive or negaive so can velociy i is a vecor, oo Average speed is no a vecor, jus (disance raveled)/d Example of average velociy: Driving from Ihaca o Syracuse 7
Insananeous velociy Bu here is anoher ype of velociy which is useful insananeous velociy Measures how fas my posiion (displacemen) is changing a some insan of ime Example -- nohing more han he reading on my car s speedomeer and my direcion Describing moion Average velociy (for a ime inerval): v average = Insananeous velociy (a an insan in ime) v insan = v = Insananeous speed v 8
Velociy from graph x Q V av = D x/d P D D x As D ges small, Q approaches P and v dx/d = slope of angen a P insananeous velociy Inerpreaion Slope of x() curve reveals v ins (= v) Seep slope = large velociy Upwards slope from lef o righ = posiive velociy Average velociy = insananeous velociy only for moions where velociy is consan x 9
When does v av = v ins? When x() curve is a sraigh line Tangen o curve is same a all poins in ime x We say ha such a moion is a consan velociy moion we ll see ha his occurs when no forces ac No fan. car demo Provide an iniial push, bu almos no forces ac while car is moving Wha do he posiion and velociy plos look like? 20 10
SG 1-2.1 Here is a posiion graph of an objec: A = 1.5 s, he objec s velociy is A. 40 m/s. B. 20 m/s. C. 10 m/s. D. 10 m/s. E. None of he above. QuickCheck 2.7 SG1-2.2 Here is a posiion graph of an objec: A = 3.0 s, he objec s velociy is A. 40 m/s. B. 20 m/s. C. 10 m/s. D. 10 m/s. E. None of he above. 11
Summary of erms Posiions: Displacemens: Insans of ime: Time inervals: Average velociy: Insananeous velociy: Insananeous speed: x iniial, x final D x = x final - x iniial iniial, final D = final - iniial v av = D x/d v = dx/d v = dx/d Acceleraion Someimes an objec s velociy is consan as i moves. More ofen, an objec s velociy changes as i moves. Acceleraion describes a change in velociy. Consider an objec whose velociy changes from o during he ime inerval. The quaniy is he change in velociy. The rae of change of velociy is called he average acceleraion: The Audi TT acceleraes from 0 o 60 mph in 6 s. 12
Speeding Up or Slowing Down? speeding up: acceleraion and velociy vecors poin in he same direcion. slowing down: acceleraion and velociy vecors poin in opposie direcions. consan velociy = acceleraion is zero. Posiive acceleraions do no always mean speeding up! Do his by yourself: 1-2.3 A cyclis riding a 20 mph sees a sop sign and acually comes o a complee sop in 4 s. He hen, in 6 s, reurns o a speed of 15 mph. Which is his moion diagram? 13
SG 1-2.4 A oy car on a sraigh 1D rack is measured o have a negaive acceleraion, if we define he x-axis o poin o he righ. Wha else mus be rue of he acceleraion? 1. The car is slowing down. 2. The car is speeding up. 3. The car is moving o he lef. 4. The acceleraion vecor poins o he lef. x 27 Acceleraion Average acceleraion -- keep ime inerval D non-zero a av = D v/d = (v F - v I )/D Insananeous acceleraion a ins = lim D à 0 D v/d = dv/d 28 14
Sample problem A car s velociy as a funcion of ime is given by v() = (3.00 m/s) + (0.100 m/s 3 ) 2. Calculae he avg. accel. for he ime inerval = 0 o = 5.00 s. Calculae he insananeous acceleraion for i) = 0; ii) = 5.00 s. 29 1-2 SG 5-7: Acceleraion from graph of v() v P Q R T Wha is a av for 5. PQ? 6. QR? 7. RT? Slope measures acceleraion Posiive a means v is increasing Negaive a means v decreasing A. a avg > 0 B. a avg < 0 C. a avg = 0 30 15
SG 1-2.8: You are hrowing a ball up in he air. A is highes poin, he ball s A. Velociy v and acceleraion a are zero B. v is non-zero bu a is zero C.Acceleraion is non-zero bu v is zero D.v and a are boh non-zero 31 Fan car demo Aach fan o car - provides a consan force (we ll see laer ha his implies consan acceleraion) Depending on orienaion, force acs o speed up or slow down iniial moion Skech graphs of posiion, velociy, and acceleraion for car ha speeds up 32 16
Fan car demo Skech graphs of posiion, velociy, and acceleraion for car ha speeds up 33 SG 1-2.9 x a b The graph shows 2 rains running on parallel racks. Which is rue: A. A ime T boh rains have same v B. Boh rains speed up all ime C. Boh rains have same v for some <T D. Somewhere, boh rains have same a T 34 17
Assignmens due his week Due his Friday Pre-assessmen (on blackboard) Problem se 1 SAGE assignmen (should ake <5 mins) Inerpreing x() and v() graphs Slope a any insan in x() graph gives insananeous velociy x Slope a any insan in v() graph gives insananeous acceleraion Wha else can we learn from an x() graph? v 18
Acceleraion from x() Rae of change of slope in x() plo equivalen o curvaure of x() plo Mahemaically, we can wrie his as a = Negaive curvaure a < 0 Posiive curvaure a > 0 No curvaure a = 0 x 37 Reading assignmen Kinemaics, consan acceleraion 2.4 2.7 in exbook 38 19
Displacemen from velociy curve? Suppose we know v() (say as graph), can we learn anyhing abou x()? v Consider a small ime inerval D v = D x/d à D x = vd So, oal displacemen is he sum of all hese small displacemens D x x = S D x = lim D à 0 S v()d = Displacemen inegral of velociy lim D à 0 S D v() = area under v() curve noe: `area can be posiive or negaive *Consider v() curve for car in differen siuaions v *Ne displacemen? 20
Graphical inerpreaion v v() T 1 T 2 D Displacemen beween T 1 and T 2 is area under v() curve Sample problem An objec s posiion as a funcion of ime is given by x() = (3.00 m) - (2.00 m/s) + (3.00 m/s 2 ) 2. Calculae he avg. accel. beween = 2.00s and = 3.00 s. Calculae he insananeous accel. a i) = 2.00 s; ii) = 3.00 s. 42 21
Velociy from acceleraion curve Similarly, change in velociy in some ime inerval is jus area enclosed beween curve a() and -axis in ha inerval. a() a T 1 T 2 D Summary velociy v = dx/d = slope of x() curve NOT x/!! displacemen D x is v()d = area under v() curve NOT v!! accel. a = dv/d = slope of v() curve NOT v/!! change in vel. D v is a()d = area under a() curve NOT a!! 22
Discussion Average velociy is ha quaniy which when muliplied by a ime inerval yields he ne displacemen For example, driving from Syracuse Ihaca Car on incline demo Raise one end of rack so ha graviy provides consan acceleraion down incline (we ll sudy his in much more deail soon) Give car iniial velociy direced up he incline Skech graphs of posiion, velociy, and acceleraion for car 46 23