1.1 Speed and Velocity in One and Two Dimensions


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1 1.1 Speed and Velocity in One and Two Dienion The tudy of otion i called kineatic. Phyic Tool box Scalar quantity ha agnitude but no direction,. Vector ha both agnitude and direction,. Aerage peed i total ditance traelled diided by total tie of trael. d a t Intantaneou peed i the agnitude of intantaneou elocity, Poition, d i the ditance and direction of an object fro a reference point Diplaceent d i the change i poition (final poition initial poition) of an object in a gien direction, d d2 d1, d d1 d2... dn Aerage elocity, ae i the change in poition diided by tie interal for d that change, ae. t Intantaneou elocity, d i the elocity at a particular intant, li, t 0 t dd. dt in A in B in C Sin Law: a b c c 2 a 2 b 2 2abco C Coine Law: Slope of line on a poition tie graph repreent the elocity The area under the line on a elocity tie graph repreent Unierity Note: The diplaceent ariable in one dienion i uually the ariable that indicate the axi, ie x. d The diplaceent ariable in two or three dienion i r. Exaple At a racetrack, the ditance for one lap i.2 k. If Mr Burn coer a ingle lap in 2.1 inute. Deterine a) The aerage peed in k/h b) The aerage peed in / c) The aerage elocity in k/h d) The aerage elocity in /
2 Solution: a) ae d t.2k 60 in 2.1in 1hr k hr Since all data ha two ignificant digit. Therefore the aerage peed i about 1 k/h. b) Fro aboe: c) ae d t.2k 1in in 60 1k 25.7 Therefore the aerage i about 25/. ae d t 0k 2.1in k 0 hr The aerage elocity around a loop i zero d) Fro c) aboe, the aerage elocity around a loop i zero in any unit. Exaple You drie your car 2.00 k down a dragtrip, then 2.50 k in the oppoite direction, copleting your trip in 180 a) Find the aerage elocity b) Find the aerage peed c) Explain why the aerage peed of the car i not equal to the agnitude of the aerage elocity.
3 Solution: a) ae r t 0.5k k r 2k 2.5k 0.5k When in doubt, ue SI unit The aerage elocity i 2.78 / to the left (if you choe right to be poitie) b) ae d t 4.5k k 25.0 The aerage peed i 25.0 / c) The anwer are different becaue the total ditance traelled i not equal to the agnitude of the diplaceent. Exaple You obere the flight path of a trange ehicle. You record the 2 ditance and direction ector: d kn E, d kw S The craft coe to a top after the third ector in 55 econd, deterine: a) the total ditance traelled b) aerage peed c) total diplaceent d) aerage elocity Solution: Draw a ector picture to repreent the ituation k k O
4 a) The total ditance traelled i a calar quantity d 0.50k 0.0k 0.80k b) ae d t 0.80k k The aerage peed about 15 / c) Since we hae 2 ector, we can ue the coine law d d d 2 d d co k 0.0k 20.50k0.0kco k d k 0.41k Becaue thi i a 2D ector, we need the angle (direction) Fro the Sine Law in in 55 O k 0.41k 0.0k in O O Therefore the angle i The total diplaceent i 0.41k at An alternatie ethod for c) W7 N Rather than drawing a triangle, place each ector at the origin and calculate the x diplaceent and the ydiplaceent and ue thee alue to draw the reultant ector and angle.
5 r in 70 0.in co co 70 The reultant ector r i copoed of: r x 0.5kco 70 0.kco k r 0.5kin 70 0.kin k y So, r r r x y k k r k 0.41k 2 2 The angle i found by tan k k k k The total diplaceent i r 0.41 k [ W7 N] 1 tan 7 d) ae d t 0.41k W 7N k 7.45 W7 N The aerage elocity i 7 /
6 DiplaceentTie and VelocityTie Graph Graphing proide a ueful way of tudying otion, in fact graph are one of the tool of trade for a phyicit. The hape of the cure, and oetie the area underneath the cure can proide u a deeper undertanding of the ituation. Three Step to Graphical Analyi DiplaceentTie (diplaceent Tie) Reading of alue fro the graph Calculating lope of the cure Calculating the area between the cure and the taxi of the graph Analyzing Diplaceent Tie Graph locate Intantaneou poition d t Graph Slope / Velocity Area 1 point tangent line? ye Intantaneou elocity no Aerage elocity Reading a alue proide you with an ordered pair. Thi point gie you the intantaneou alue of the diplaceent If the line i traight, then the lope of the line gie you the elocity of the graph. Note the unit are ditance/tie. If lope poitie then poitie elocity. If lope negatie then negatie elocity. Since line i traight then intantaneou elocity=aerage elocity If the line i cured, then if cured up the elocity i increaing (particle ha poitie acceleration). If cured down, the elocity i decreaing (particle ha negatie acceleration). If you ue two point to draw ecant line then lope i aerage elocity. If you ue only one point (calculu) to draw tangent line then lope i intantaneou elocity.
7 Analyzing Cure on Diplaceent Tie Graph I the graph a traight line? ye I the line horizontal? ye Then topped no no Accelerating Then contant elocity I the graph curing up? ye Speeding Up no Slowing down In a diplaceenttie graph lope equal elocity. the "y" intercept equal the initial diplaceent. when two cure coincide, the two object hae the ae diplaceent at that tie. traight line iply contant elocity. cured line iply acceleration. an object undergoing contant acceleration trace a portion of a parabola. aerage elocity i the lope of the traight line connecting the endpoint of a cure. intantaneou elocity i the lope of the line tangent to a cure at any point. poitie lope iplie otion in the poitie direction. negatie lope iplie otion in the negatie direction. zero lope iplie a tate of ret. The area under the cure i eaningle
8 Velocity Tie Graph Analyzing Velocity Tie Graph locate / Intantaneou Velocity t Graph Slope 2 Acceleration Area Diplaceent The total diplaceent of a elocitytie graph i the net area between the tie axi and the cure. Each area i calculated eparately and then added together (keeping the poitie and negatie ign). The total ditance traelled i repreented by the odulu (abolute alue) of each area between the tie axi and the cure. If the cure on the elocitytie graph conit of traight egent, then you can eaily calculate the area between the cure and the tie axi by creating triangle and rectangle and adding up their appropriate area. If the cure on a elocitytie graph conit of cure, then it take integral calculu to calculate the area between the cure and the tie axi. On a elocitytie graph lope equal acceleration. The "y" intercept equal the initial elocity. when two cure coincide, the two object hae the ae elocity at that tie. traight line iply unifor acceleration. cured line iply nonunifor acceleration. an object undergoing contant acceleration trace a traight line. aerage acceleration i the lope of the traight line connecting the endpoint of a cure. Intantaneou acceleration i the lope of the line tangent to a cure at any point. poitie lope iplie an increae in elocity in the poitie direction. negatie lope iplie an increae in elocity in the negatie direction. zero lope iplie otion with contant elocity. the area under the cure equal the change in diplaceent.
9 During which extended period of tie wa he traeling in a poitie direction? none of thee During which extended period of tie wa he traeling in a negatie direction? none of thee During which extended period of tie wa he at ret? none of thee During which extended period of tie wa he traeling at a contant elocity? During what tie interal did he trael the greatet ditance? During what tie interal did he trael the leat nonzero ditance? During which tie interal() did he experience a negatie acceleration?
10 During which tie interal() did he experience a poitie acceleration? During which tie interal did he experience an acceleration with the greatet agnitude? none of thee What total ditance did he trael in the firt 8 econd? area = ½()(15) + (2.5)(15) + (1)(5) + ½(1)(10) + ½(2.5)(5) area = 7.75 eter What total ditance did he trael in the lat 8.5 econd? area = ½(1)(1) + (4)(1) + ½(.5)(1) area = eter What wa hi aerage peed in the firt 8 econd? aerage peed = total ditance / total tie aerage peed = 7.75 eter / 8 econd aerage peed =.2 /ec What wa hi aerage peed in the lat 8.5 econd? aerage peed = total ditance / total tie aerage peed = eter / 8.5 econd aerage peed =.6 /ec What wa hi aerage peed for the entire 16.5 econd? aerage peed = total ditance / total tie aerage peed = (7.75 eter eter) / 16.5 econd aerage peed = 155 eter / 16.5 econd aerage peed =. /ec What wa hi net diplaceent during the entire 16.5 econd? net diplaceent: = eter What wa hi aerage elocity during the entire 16.5 econd? aerage elocity: 7.5 eter / 16.5 econd = /ec
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