Chapter 4 Kinematics in Two Dimensions

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Esmond Bell
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1 D Kinemtic Quntities Position nd Velocit Acceletion Applictions Pojectile Motion Motion in Cicle Unifom Cicul Motion Chpte 4 Kinemtics in Two Dimensions

2 D Motion Pemble In this chpte, we ll tnsplnt the conceptul ppoch fom the 1D motion to motion in plne, o D cse (the concepts cn be etpolted to 3D motion): The ppoch will eintoduce the kinemtic quntities in the new contet: Define position Instntneous Velocit Acceletion Applictions: Pojectile Motion nd Displcement m R Cicul Motion t

3 D Kinemtics Position nd Displcement Phsicl contet: Conside pticle moing in plne, long D tjecto epesented on mp-like SC: In ee point one cn define ecto position iˆ ˆ j, Then, the displcement between positions 1, is Tjecto Mgnitude: Diection:, θ 1 1 ctn 1 1 1

4 Fo n point long D tjecto, the instntneous elocit is ecto tngent to the tjecto in tht point The components of the elocit e gien b the fist deities of the espectie position components: d d, dt dt d ˆ d i ˆj dt dt, d dt d 1 tn dt D Kinemtics Velocit θ Tjecto

5 D Kinemtics Acceletion Conside two points long D tjecto. The ege cceletion hs the diection of the ege elocit Δ: t t t 1 1 Een if the mgnitude of the espectie elocit is constnt long the tjecto, the cceletion is not zeo if the diection of elocit chnges Hence, since this is lws tue fo cued D tjecto, the ecto Δ does not point long the tjecto: it points towd the conce side of the cued pth So, n cued pth hs n cceletion diected inside (conce pt of) the cue The ecto instntneous cceletion chcteizes ee point on the pth, nd hs components gien b the fist deitie of the espectie elocit components diection of the cceletion gien b the diection of Δ 1 d d, dt dt d d ˆ i ˆj dt dt 1 Tjecto,

6 Poblem: 1. Two dimensionl kinemtic nlsis: A web pge designe cetes n nimtion in which dot on compute sceen hs position ) Find the mgnitude nd diection of the dot s ege elocit between t = s nd. s b) Find the mgnitude nd diection of the instntneous elocit t t = s, 1. s, nd. s c) Sketch the dot s tjecto fom t = s to t =. s, nd sketch the instntneous elocities clculted in pt (b) d) Find the mgnitude nd diection of the dot s ege cceletion between t = s nd. s e) Find the cceletion of the pticle t t =. s 4. cm.5 cm s t iˆ 5. cm s tj ˆ

7 D Kinemtics Pllel nd pependicul components of cceletion Fo n bit point on D tjecto, it is intuitiel useful to conside the components of cceletion pllel (o tngent t ) nd pependicul (o dil ) with espect to the diection of motion The pllel (o nti-pllel) component descibes how the elocit chnges in mgnitude, while the pependicul component descibes how the elocit chnges in diection E. 1: In line motion, tht is, the cceletion hs onl pllel component, so if the diection of motion sts the sme onl the mgnitude of the elocit ies Tjecto E. : In cicul motion. Since the diection of elocit lws chnges, must be non-zeo, but m be zeo if the speed doesn t chnge ( = const) inceses constnt deceses

8 Eecise Components of cceletion: A pticle moes s shown in the figue. Between points B nd D, the pth is stight line. Sketch the cceletion ectos t A, C nd E in the cses in which ) the pticle moes with stedil constnt speed b) the pticle moes with stedil incesing speed c) the pticle moes with stedil decesing speed

9 Pojectile motion Sstemtic Appoch Since onl git cts, the cceletion of the pojectile is onl the gittionl cceletion g pointing eticll downwd Time t -components: 1 t gt gt sin (t) Poblem setup:,, g g g t tjecto Gound t Pojectile Time t Initil elocit cos (t) Time t -components: t

Pojectile motion Summ In conclusion, the pojectile motion cn be split into independent eticl nd hoizontl components: one with constnt elocit nd one unifoml cceleted In ode to nlze the motion, the nd

10 Pojectile motion Summ In conclusion, the pojectile motion cn be split into independent eticl nd hoizontl components: one with constnt elocit nd one unifoml cceleted In ode to nlze the motion, the nd components cn be woked out independentl, nd then the D kinemtics cn be obtined t n moment bsed on the components t the espectie time Position: ˆ 1 t i t gt ˆj Velocit: Acceletion: ˆ i gt g ĵ E.: If two objects e elesed simultneousl one eticll down nd one with n initil hoizontl elocit the eticl motion is the sme s the eticl pojection (, ) of the pbolic motion (like the motion of shdow on eticl sceen) ˆj (t) (t)

11 Tjecto Shpe, Totl time t tot, Rnge R, Mimum height m m t gt g 1 sin tn cos cos t Poblem t tot sin sin g R R g depends on like, so the shpe of the tjecto is pbolic. Tjecto chcteistics: Conside the tjecto of pojectile fied with initil elocit (,θ ), lnding t the sme ltitude s the fiing loction, s depicted boe. Poe tht the totl time t tot spent b the pojectile in the i, the hoizontl nge R nd the mimum height m e gien b the following eltions m sin g

12 Eecise: Bd Phsics in Pel Hbo. When it comes to Phsics, if ignonce is bliss, Hollwood is the gden of Eden. Thei moies e bottomless fountin of efeences to how subcultue seems to offe moe espect to the ules of bsebll thn to the lws of ntue Tke fo instnce the 1 w dm Pel Hbo which is med not onl b histoicl inccucies, but lso b completel bogus iplne mneues nd bomb tjectoies. Tke fo instnce the tjecto of bomb tht hits USS Aizon ppentl fte being dopped fom n iplne lmost eticll boe the ill-fted deck. Let s nlze it: Knowing tht Aizon ws bombed fom bout 3 m ltitude nd the seice speed of B5N Kte bombe is bout 3 km/h, estimte the hoizontl distnce befoe the bttleship whee the bomb must he been dopped in ode to hit it (neglecting i dg). h d

13 Poblem 4. Intesection of pojectile tjectoies: Two pojectiles e lunched simultneousl fom two diffeent loctions septed b distnce d, s indicted in the figue. The initil elocit of the fist pojectile hs mgnitude nd mkes n ngle α boe the hoizontl. The second pojectile is lunched with initil elocit u. ) Wite out the position of the fist pojectile t n moment in time, in the -sstem of coodintes epesented in the figue. b) Wite out the position of the second pojectile t n moment in time c) Use the equtions of motion to clculte the initil elocit necess fo the second pojectile to hit the fist one in gien time T fom the lunch. u α θ d

14 Motion in Cicle Unifom motion D motion long cicul pth; conside fist Unifom Cicul Motion (tht is the mgnitude of the elocit is constnt) let s chcteize this kinemtics 1 Δθ s s Δθ 1 Using the simil tingles Aege cceletion is: t t 1 1 Then, dil o centipetl cceletion descibing the chnge in elocit diection is: s lim lim t t t t Since the motion is peiodic, we define peiod T: time necess to tel one ccle Hence, we cn obtin n ltentie epession fo the centipetl cceletion 4 T T

15 Motion in Cicle Non-unifom motion If the mgnitude of the elocit ies, we del with Non-unifom Cicul Motion: 1 t 1 1 t These epessions e lid fo n pticle moing long cued tjecto: if the cutue of the pth cn be fitted locll b cicle of dius nd the instntneous speed is, the epession boe gies the dil component of the cceletion in the espectie point In this cse, the cceletion hs two components: Rdil o centipetl Tngent (to the pth): t d dt descibes the chnge in diection descibes the chnge in mgnitude Net cceletion: t t tjecto

16 Poblem 5. Unifom cicul motion: A ock tied to ope moes in the -plne. The eqution of motion of the pticle is, cos, sin R t R t whee R nd ω e constnts of motion. Let s nlze this motion nd show tht we e deling with the unifom cicul motion. ) Show tht the ock s distnce fom the oigin is constnt nd equl to R tht tht its pth is indeed cicle of dius R. b) Show tht t ee point the ock s elocit is pependicul on its position ecto nd hs mgnitude ωr. c) Show tht the ocks cceletion is lws opposite in diection to its position ecto nd hs mgnitude ω R tht is, it is centipetl cceletion d) Combine the esults of pts (b) nd (c) to show tht the ock s cceletion hs constnt mgnitude /R tht is, the eltionship tht we deied elie

Chapter 4 Two-Dimensional Motion

Chapter 4 Two-Dimensional Motion D Kinemtic Quntities Position nd Velocit Acceletion Applictions Pojectile Motion Motion in Cicle Unifom Cicul Motion Chpte 4 Two-Dimensionl Motion D Motion Pemble In this chpte, we ll tnsplnt the conceptul