# 3 Motion with constant acceleration: Linear and projectile motion

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1 3 Moion wih consn ccelerion: Liner nd projecile moion cons, In he precedin Lecure we he considered moion wih consn ccelerion lon he is: Noe h,, cn be posiie nd neie h leds o rie of behiors. Clerl similr equions cn be wrien for he moion lon oher es: cons, cons, All hese moions cn hppen he sme ime independenl from ech oher. We cn wrie our equions in more enerl form wih shifed ime Below we will consider differen problems s illusrions ec. This mens h = he eloci is/ws nd he coordine is/ws. PHY66 Sprin 8

2 Liner moion wih consn ccelerion An emple is free fll of n objec under he influence of ri if he ir resisnce is neleced. If we use he is direced upwrd, hen he ccelerion lon he is is Problem:, 9.8 m/s A sone is dropped from he leel () = m, (b) = m boe he round wih ero iniil eloci. Wh is he fll ime? Soluion: Use fll nd find fll Anlicl resul (Frme our finl nlicl resul!) Now plu numbers ino our finl resul: ( ) : ( b) : fll fll s.4 s Numericl resuls

3 Problem: A sone is hrown upwrd from he round ( = ) wih he iniil eloci () = 5 m/s; (b) = m/s. Wh is he miml heih h he sone reches? Wh ime does i ke for he sone o fll on he round? Soluion: The miml heih corresponds o he ero eloci. Thus we use This ields -m -m Now he miml heih cn be found s m m m m (Frme our finl resul!) Now plu numbers ino he finl resul: ( ) : ( b) : m m To find he fll ime, use 5.8 m m 9.8 fll fll fll The wo soluions of his equion re fll = nd fll m of which he firs is riil nd second is relen. fll Plu numbers 3

4 4 Problem: Two sones re dropped from he sme poin wih ero iniil eloci, one = nd noher = >. How he disnce beween he wo sones depends on ime? Wh do ou epec before ou sr solin he problem? Soluion: Use subscrip for he firs sone nd for he second sone. The hihs of he wo sones ime re ien b, The disnce beween he wo sones is hus d d The disnce linerl increses wih ime! Morl: Alws mke ood sr..

5 Projecile moion Projecile moion is he moion of objecs in he riionl field of he erh if he do no o fr w from he Erh s surfce nd he ir resisnce is neleced. This is he moion wih consn ccelerion direced downwrd nd hin he lue. There is no ccelerion in he direcions prllel o he Erh s surfce, hus projecions of he eloci on hese direcions remins consn. Formuls for he projecile moion cn be pu ino he form wih he pproprie choice of he es (so h = = ). The rjecor is prbol, if. Le us show i in he priculr cse = =. In his cse = / nd subsiuin ino he formul for ields h is, is qudric funcion of. 5

6 Problem: A bll rolls off shelf wih horionl eloci of m/s. A wh horionl disnce from he shelf does he bll lnd if he shelf is m boe he floor? Soluion: Use equions, wih = h = m nd = m/s. From he firs equion follows he fll ime fll Pluin i ino he second equion ies d fll This is our finl nlicl resul. Now plu numbers o find numericl resul: d.64 m 9.8 6

7 Problem: A missile ws lunched from he surfce leel wih he iniil eloci he nle q wih he horion. Wh horionl disnce d will i rel before hiin he round? For which lue of q is his disnce miml? Soluion: The componens of he iniil eloci re ien b = cosq, = sinq. The horionl disnce is ien b d = fll, where fll is he fll ime, he ime from he sho unil rechin he re. Here i is imporn h = is consn. The fll ime cn be found considerin moion lon : = (/) = ( /). The condiion = resuls in he fll ime fll = /. This ields d = / = sinqcosq/ = sin(q/. The disnce hs mimum q = 45º. q d 7

8 Relie eloci, ddiion of elociies In some cses he bod (for insnce, person) is moin wih respec o lrer bod or medi (for insnce rin or rier) h is, in urn, moin wih respec o he min frme of reference such s he Erh. We cll he min frme of reference lboror frme nd noher frme (rin, rier) moin frme. If - eloci of he bod wih respec o he moin frme (relie eloci) u - eloci of he moin frme wih respec o he lboror frme hen = + u is he eloci of he bod wih respec o he lboror frme (bsolue eloci). This simpl mens h elociies dd up. The ler holds in he non-reliisic mechnics where ll speeds re much smller hn he speed of lih c = 3 km/s = 3 8 m/s. 8

9 Emple: A person wlks in rin moin u= km/h. The speed of he wlker is =5 km/h. This is he relie eloci (speed). The lue of he ol eloci is = ± +u, dependen on he direcion of he wlker, h is, 5 or 95 km/h. Problem: A swimmer swims wih speed 5 km/h cross rier flowin wih speed of 3 km/h. A wh nle wih respec he srih crossin line should he swimmer swim o cross he rier perpendiculrl? Wh will be speed wih which he swimmer is crossin he rier? Mhemicl formulion: = 5 km/h, u = 3 km/h; q? =? Soluion: Use eloci-ddin formul = + u. mus be perpendiculr o u. Choose coordine ssem s shown. Projec ono is o find he nle: u ' u 'sinq u q rcsin ' Projec ono is o find : ' u 'cosq ' ' ( u / ') sin ' u 4 km/h Anle soluion: Soluion eiss onl for > u, s epeced. q q u 9

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