Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl moion we only need he sign. Since only consn ccelerion will be considered in his course, here is no need o differenie beween verge nd insnneous ccelerion. Unis for Accelerion Accelerion is he re of chnge of. Unis for ccelerion You cn derive he unis by subsiuing he correc unis ino he righ hnd side of hese equions. = Δv Δ m/s s = m/s A B C D displcemen disnce speed velociy Ms. Levine
The uni for velociy is: 3 The meric uni for ccelerion is: A m A m B m/s B m/s C m/s D f/s C m/s D f/s 4 A horse gllops wih consn ccelerion of 3m/s. Which semen below is rue? Solving Problems Afer you red he problem crefully: A The horse's velociy doesn' chnge.. Drw digrm (include coordine es). B The horse moves 3m every second.. Lis he given informion. C The horse's velociy increses 3m every second. 3. Idenify he unknown (wh is he quesion sking?) D The horse's velociy increses 3m/s every second. 4. Choose formul (or formuls o combine) 5. Rerrnge he equions o isole he unknown vrible. 6. Subsiue he vlues nd solve! 7. Check your work. (You cn do he sme operions o he unis o check your work... ry i!) 5 Your velociy chnges from 60 m/s o he righ o 00 m/s o he righ in 0 s; wh is your verge ccelerion? 6 Your velociy chnges from 60 m/s o he righ o 0 m/s o he righ in 0 s; wh is your verge ccelerion? Ms. Levine
7 Your velociy chnges from 50 m/s o he lef o 0 m/s o he righ in 5 s; wh is your verge ccelerion? 8 Your velociy chnges from 90 m/s o he righ o 0 m/s o he righ in 5.0 s; wh is your verge ccelerion? Kinemics Equion Moion Consn Accelerion = Δv Δ bu since "Δ" mens chnge Δv = v - vo nd = v - vo Δ = - o if we lwys le o = 0, Δ = = v - vo v - v o = Solving for "v" Reurn o Tble of Conens v = vo + This equion ells us how n objec's velociy chnges s funcion of ime. 9 Sring from res, you ccelere 4.0 m/s for 6.0s. Wh is your finl velociy? 0 Sring from res, you ccelere 8.0 m/s for 9.0s. Wh is your finl velociy? Ms. Levine 3
You hve n iniil velociy of 5.0 m/s. You hen eperience n ccelerion of -.5 m/s for 4.0s; wh is your finl velociy? You hve n iniil velociy of -3.0 m/s. You hen eperience n ccelerion of.5 m/s for 9.0s; wh is your finl velociy? 3 How much ime does i ke o ccelere from n iniil velociy of 0m/s o finl velociy of 00m/s if your ccelerion is.5 m/s? 4 How much ime does i ke o come o res if your iniil velociy is 5.0 m/s nd your ccelerion is -.0 m/s? 5 An objec cceleres re of 3 m/s for 6 s unil i reches velociy of 0 m/s. Wh ws is iniil velociy? 6 An objec cceleres re of.5 m/s for 4 s unil i reches velociy of 0 m/s. Wh ws is iniil velociy? Ms. Levine 4
Grphing Moion Consn Accelerion In physics here is noher pproch in ddiion o lgebric which is clled grphicl nlysis. The formul v = v0 + cn be inerpreed by he grph. We jus need o recll our memory from mh clsses where we lredy sw similr formul y = m + b. From hese wo formuls we cn some nlogies: Moion Consn Accelerion Below we cn find he geomeric eplnion o he ccelerion =Δv/Δ. If slope is equl o: m = Δy/Δ Then consider grph wih velociy on he y-is nd ime on he -is. Wh is he slope for he grph on he righ? v y (dependen vrible of ), v0 b (inersecion wih vericl is), (independen vrible), m ( slope of he grph- he rio beween rise nd run Δy/Δ). Moion Consn Accelerion The grph on he righ hs slope of Δv/Δ, which is equl o ccelerion. Therefore, he slope of velociy vs. ime grph is equl o ccelerion. 7 The velociy s funcion of ime is presened by he grph. Wh is he ccelerion? (slope) y =Δy/Δ (slope of velociy vs. ime) =Δv/Δ 8 The velociy s funcion of ime is presened by he grph. Find he ccelerion. Moion Consn Accelerion The ccelerion grph s funcion of ime cn be used o find he velociy of moving objec. When he ccelerion is consn he velociy is chnging by he sme moun ech second. This cn be shown on he grph s srigh horizonl line. In order o find he chnge in velociy for cerin limi of ime we need o clcule he re under he ccelerion line h is limied by he ime inervl. Ms. Levine 5
9 The following grph shows ccelerion s funcion of ime of moving objec. Wh is he chnge in velociy during firs 0 seconds? Free Fll: Accelerion Due o Grviy Reurn o Tble of Conens Free Fll All unsuppored objecs fll owrds Erh wih he sme ccelerion. We cll his ccelerion he "ccelerion due o grviy" nd i is denoed by g. g = 9.8 m/s Keep in mind, ALL objecs ccelere owrds he erh he sme re. g is consn! I slows down. Wh hppens when i (negive ccelerion) goes up? g = -9.8 m/s I Wh sops hppens momenrily. he v op? = 0 g = -9.8 m/s I Wh speeds hppens up when i (negive goes down? ccelerion) g = -9.8 m/s Click here o wch Glileo's fmous eperimen performed on he moon An objec is hrown upwrd wih iniil velociy, v o I reurns Wh hppens wih is when i originl lnds? velociy. I sops momenrily. v = 0 g = -9.8 m/s On he wy up: v = 3 s On he wy down: v v0 = 0 s I slows down. (negive ccelerion) g = -9.8 m/s I speeds up. (negive ccelerion) g = -9.8 m/s v v v = s = s v v = s An objec is hrown upwrd wih iniil velociy, vo I reurns wih is originl velociy. v0 v = 0 s v v v = s = 3 s Ms. Levine 6
Lb Physics For ny objec hrown srigh up ino he ir, his is wh he velociy vs. ime grph looks like. v (m/s) I sops momenrily. v=0 g = -9.8 m/s A bll is dropped from res nd flls (do no consider ir resisnce). Which is rue bou is moion? A ccelerion is consn B velociy is consn C velociy is decresing D ccelerion is decresing I reurns wih is originl velociy bu in he opposie direcion. An corn flls from n ok ree. You noe h i kes.5 seconds o hi he ground. How fs ws i going when i hi he ground? A rock flls off cliff nd his he ground 5 seconds ler. Wh velociy did i hi he ground wih? 4 A bll is hrown down off bridge wih velociy of 5 m/s. Wh is is velociy seconds ler? An rrow is fired ino he ir nd i reches is highes poin 3 seconds ler. Wh ws is velociy when i ws fired? 3 0 An objec is hrown upwrd wih iniil velociy, vo Ms. Levine 7
5 A rocke is fired srigh up from he ground. I reurns o he ground 0 seconds ler. Wh ws is lunch speed? Moion Consn Accelerion If velociy is chnging consn re, he verge velociy is jus he verge of he iniil nd finl velociies. v = v + vo And we lerned erlier h v = Δ Some problems cn be solved mos esily by using hese wo equions ogeher. Δ = Δ = v + vo (v + vo) 6 Sring from res you ccelere o 0 m/s in 4.0s. Wh is your verge velociy? 7 Sring wih velociy of m/s you ccelere o 48 m/s in 6.0s. Wh is your verge velociy? 8 A cr cceleres from res o 30m/s while rveling disnce of 0m; wh ws is ccelerion? 9 An objec cceleres from res, wih consn ccelerion of 8.4 m/s, wh will is velociy be fer s? Ms. Levine 8
Velociy vs. Time Grphs Grphing Similrly, he sme pproch cn be used o cree velociy vs. ime grph. v (m/s) A velociy versus ime grph differs by hving he velociy on he vericl is. A velociy versus ime grph shows describes n objecs velociy, i's displcemen, nd ccelerion. Reurn o Tble of Conens Sring he posiion, 0 = 4 m, you rvel consn velociy of + m/s for 6s. b. Drw he Posiion versus Time for your rvel during his ime. Drg nd drop he d poins on he grph in order o consruc he v vs pern! 0 9 8 7 6 5 4 3 Sring he posiion, 0 = 4 m, you rvel consn velociy of + m/s for 6s. c. Drw he Velociy versus Time grph for your rip. Drg nd drop he d poins on he grph in order o consruc he v vs pern! v (m/s) 4 3 3 4 5 6 3 4 5 6 Drw line of bes fi o observe he pern. Drw line of bes fi o observe he pern. Sring he posiion, 0 = 0 m, you rvel consn velociy of -m/s for 6s. b. Drw he Posiion versus Time for your rvel during his ime. Drg nd drop he d poins on he grph in order o consruc he v vs pern! 0 9 8 7 6 5 4 3 Anlyzing Posiion vs Time Grphs Recll erlier in his uni h slope ws used o describe moion. The slope in posiion vs. ime grph is Δ/Δ, which is equl o velociy. Δ Δ v = Δ/Δ 3 4 5 6 Therefore, slope is equl o velociy on posiion vs. ime grph. Drw line of bes fi o observe he pern. Ms. Levine 9
Anlyzing Posiion vs Time Grphs A posiive slope is posiive velociy, negive slope is negive velociy, nd slope of zero mens zero velociy. posiive slope v > 0 negive slope v < 0 zero slope v = 0 The posiion versus ime grph, below, describes he moion of hree differen crs moving long he -is. b. Clcule he velociy of ech of he crs. Posiion A posiive velociy mens moving in he posiive direcion, negive velociy mens moving in he negive direcion, nd zero velociy mens no moving ll. The velociy vs ime grph, below, describes he moion 73 The velociy vs ime grph, below, describes he moion of n objec moving long he -is. of n objec moving long he -is. v 4 (m/s) 3 v 4 (m/s) 3 4 6 4 6 Describe in words wh is hppening o he speed during he following inervls. ) 0s o s b) s o 3s c) 3s o 4 sec d) 4s o 5s e) 5s o 6s Deermine he verge speed during he following inervls. ) 0s o s b) s o 3s c) 3s o 4 sec d) 4s o 5s e) 5s o 6s f) 4s o 6s Summry Kinemics is he descripion of how objecs move wih respec o defined reference frme. Displcemen is he chnge in posiion of n objec. Averge velociy is he displcemen divided by he ime. Averge ccelerion is he chnge in velociy divided by he ime. Ms. Levine 0