Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion under graviy is exended o objecs projeced a an angle. When you have compleed i, you should undersand displacemen, velociy and acceleraion as vecor quaniies be able o inerpre he moion as a combinaion of he efecs of he iniial velociy and of graviy know ha his implies he independence of horizonal and verical moion be able o use equaions of horizonal and verical moion in calculaions abou he rajecory of a projecile know and be able o obain general formulae for he greaes heigh, ime of fligh, range on horizonal ground and he equaion of he rajecory be able o use your knowledge of rigonomery in solving problems. in his web service Cambridge Universiy Press
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Cambridge Inernaional AS and A Level Mahemaics: Mechanics Any objec moving hrough he air will experience air resisance, and his is usually signiican for objecs moving a high speeds hrough large disances. The answers obained in his chaper, which assume ha air resisance is small and can be negleced, are herefore only approximae.. Velociy as a vecor When an objec is hrown verically upwards wih iniial velociy u, is displacemen s afer ime is given by he equaion s u g, where g is he acceleraion due o graviy. One way o inerpre his equaion is o look a he wo erms on he righ separaely. The irs erm, u, would be he displacemen if he objec moved wih consan velociy u, ha is if here were no graviy. To his is added a erm ( g ), which would be he displacemen of he objec in ime if i were released from res under graviy. You can look a he equaion v = u g in a similar way. Wihou graviy, he velociy would coninue o have he consan value u indeiniely. To his is added a erm ( g), which is he velociy ha he objec would acquire in ime if i were released from res. Now suppose ha he objec is hrown a an angle, so ha i follows a curved pah hrough he air. To describe his you can use he vecor noaion which you have already used (in M Chaper 0) for force. The symbol u wrien in bold sands for he velociy wih which he objec is hrown, ha is a speed of magniude u in a given direcion. If here were no graviy, hen in ime he objec would have a displacemen of magniude u in ha direcion. I is naural o denoe his by u, which is a vecor displacemen. To his is added a verical displacemen of magniude g verically downwards. In vecor noaion his can be wrien as g, where he symbol g sands for an acceleraion of magniude g in a direcion verically downwards. To make an equaion for his, le r denoe he displacemen of he objec from is iniial posiion a ime = 0. Then, assuming ha air resisance can be negleced, = u g +. In his equaion he symbol + sands for vecor addiion, which is carried ou by he riangle rule, he same rule ha you use o add forces. EXAMPLE.. A ball is hrown in he air wih speed m s a an angle of 70 o he horizonal. Draw a diagram o show where i is.5 seconds laer. If here were no graviy, in.5 seconds he ball would have a displacemen of magniude.5, ha is 8 m, a 70 o he horizonal. This is represened by he arrow OA in Fig.., on a scale of cm o 5 m. To his mus be added a in his web service Cambridge Universiy Press
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper : The moion of projeciles displacemen of magniude 0 5. m, ha is.5 m, verically downwards, represened by he arrow AB. The sum of hese is he displacemen OB. So afer.5 seconds he ball is a B. You could if you wish calculae he coordinaes of B, or he disance OB, bu in his example hese are no asked for. 8 m r 70.5 m Fig.. EXAMPLE.. A sone is hrown from he edge of a cliff wih speed 8 m s. Draw diagrams o show he pah of he sone in he nex 4 seconds if i is hrown a horizonally, b a 30 o he horizonal. 0.5.5.5 3 3.5 4 0.5.5.5 3 0.5 0.5.5.5.5.5 3 3.5 4 3 3.5 3 3.5 4 Fig.. Fig..3 4 These diagrams were produced by superimposing several diagrams like Fig... In Figs.. and.3 (for pars a and b respecively) his has been done a inervals of 0.5 s, ha is for = 0.5,,.5,, 4. The displacemens u in hese imes have magniudes 9 m, 8 m,, 7 m. The verical displacemens have magniudes.5 m, 5 m,.5 m,, 80 m. The poins corresponding o A and B a ime are denoed by A and B. You can now show he pahs by drawing smooh curves hrough he poins O, B 0.5, B,, B 4 for he wo iniial velociies. The word projecile is ofen used o describe an objec hrown in his way. The pah of a projecile is called is rajecory. A vecor riangle can also be used o ind he velociy of a projecile a a given ime. If here were no graviy he velociy would have he consan value u indeiniely. The effec of graviy is o add o his a velociy of magniude g verically downwards, which can be wrien as he vecor g. This gives he equaion v = u + g, assuming ha air resisance can be negleced. in his web service Cambridge Universiy Press
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Cambridge Inernaional AS and A Level Mahemaics: Mechanics EXAMPLE..3 For he ball in Example.., ind he velociy afer.5 seconds. The vecor u has magniude m s a 70 o he horizonal. The vecor g has magniude 0.5 m s, ha is 5 m s, direced verically downwards. To draw a vecor riangle you need o choose a scale in which velociies are represened by displacemens. Fig..4 is drawn on a scale of cm o 5 m s. You can verify by measuremen ha he magniude of v is abou 5.5 m s, and i is direced a abou 4 below he horizonal. ms 5ms v Fig..4 Fig..5 combines he resuls of Examples.. and..3, showing boh he posiion of he ball afer.5 seconds and he direcion in which i is moving. 4 r v Fig..5 As a reminder of wha was said a he very sar of his chaper, any objec moving hrough he air will experience air resisance. How signiican his is will depend upon a number of facors, including he naure of he objec. For example, a feaher is far more affeced by air resisance han a solid meal ball. Also, experimens show ha air resisance increases wih speed. Air resisance is usually signiican for objecs moving a high speeds hrough large disances. In his chaper, i is assumed ha air resisance is small enough ha i can be ignored, and so all of he answers are only approximae. Some people hink ha anoher limiaion of his model for projeciles is he assumpion ha he acceleraion is consan hroughou he moion. They are aware ha graviy reduces wih heigh. However, while his las poin is rue, he acceleraion change is miniscule: i only reduces by abou 0.003% for every 00 meres above he ground. This is far less signiican han any measuremen error or air resisance effecs for he ype of projeciles we are considering, and so canno be considered a signiican limiaion. The change in graviy does need o be aken ino accoun when designing spacecraf launchers, bu ha is beyond he scope of his course. Exercise A A sone is hrown horizonally wih speed 5 m s from he op of a cliff 30 meres high. Consruc a diagram showing he posiions of he paricle a 0.5 second inervals. Esimae he disance of he sone from he hrower when i is level wih he foo of he cliff, and he ime ha i akes o fall. in his web service Cambridge Universiy Press
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper : The moion of projeciles A pipe discharges waer from he roof of a building, a a heigh of 60 meres above he ground. Iniially he waer moves wih speed m s, horizonally a righ angles o he wall. Consruc a diagram using inervals of 0.5 seconds o ind he disance from he wall a which he waer srikes he ground. 3 A paricle is projeced wih speed 0 m s a an angle of elevaion of 40. Consruc a diagram showing he posiion of he paricle a inervals of 0.5 seconds for he irs.5 seconds of is moion. Hence esimae he period of ime for which he paricle is higher han he poin of projecion. 4 A ball is hrown wih speed 4 m s a 35 above he horizonal. Draw diagrams o ind he posiion and velociy of he ball 3 seconds laer. 5 A paricle is projeced wih speed 9 m s a 40 o he horizonal. Calculae he ime he paricle akes o reach is maximum heigh, and ind is speed a ha insan. 6 A cannon ires a sho a 38 above he horizonal. The iniial speed of he cannonball is 70 m s. Calculae he disance beween he cannon and he poin where he cannonball lands, given ha he wo posiions are a he same horizonal level. 7 A paricle projeced a 40 o he horizonal reaches is greaes heigh afer 3 seconds. Calculae he speed of projecion. 8 A ball hrown wih speed 8 m s is again a is iniial heigh.7 seconds afer projecion. Calculae he angle beween he horizonal and he iniial direcion of moion of he ball. 5 9 A paricle reaches is greaes heigh seconds afer projecion, when i is ravelling wih speed 7 m s. Calculae he iniial velociy of he paricle. When i is again a he same level as he poin of projecion, how far has i ravelled horizonally? 0 Two paricles A and B are simulaneously projeced from he same poin on a horizonal plane. The iniial velociy of A is 5 m s a 5 o he horizonal, and he iniial velociy of B is 5 m s a 65 o he horizonal. a Consruc a diagram showing he pahs of boh paricles unil hey srike he horizonal plane. b From your diagram esimae he ime ha each paricle is in he air. c Calculae hese imes, correc o 3 signiican igures.. Coordinae mehods For he purposes of calculaion i ofen helps o use coordinaes, wih column vecors represening displacemens, velociies and acceleraions, jus as was done for forces in M Chaper 0. I is usual o ake he x-axis horizonal and he y-axis verical. For insance, in Example..(a), he iniial velociy u of he sone was 8 m s 8 horizonally, which could be represened by he column vecor 0. Since he unis 0 are meres and seconds, g is 0 m s verically downwards, represened by 0. Denoing he displacemen r by, he equaion becomes r = u + g becomes in his web service Cambridge Universiy Press
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Cambridge Inernaional AS and A Level Mahemaics: Mechanics = 8 0 0, or more simply 0 = 8 + 0 = 8 0 5. 5 You can hen read off along each line o ge he pair of equaions x = 8 and y = 5. From hese you can calculae he coordinaes of he sone afer any ime. You can make he subjec of he irs equaion as x and hen subsiue his in 8 he second equaion o ge y 5( 8 x), or (approximaely) y = 0.05x. This is he equaion of he rajecory. You will recognise his as a parabola wih is verex a O. You can do he same hing wih he velociy equaion v = u + g, which becomes v = 8 + 0 = 8 + 0 = 8 0 0 0 0 0. This shows ha he velociy has componens 8 and 0 in he x- and y-direcions respecively. Noice ha 8 is he derivaive of 8 wih respec o, and 0 is he derivaive of 5. This is a special case of a general rule. 6 If he displacemen of a projecile is dx, is velociy is d. dy d This is a generalisaion of he resul given in M Secion. for moion in a sraigh line. Here is a good place o use he shorhand noaion (do noaion) inroduced in M Secion.5, using x o sand for d x and y for d y. You can hen wrie he velociy x d d vecor as. Now consider he general case, when he projecile sars wih an iniial speed u a an angle θ o he horizonal. Is iniial velociy u can be described eiher in erms of u and θ, or in erms of is horizonal and verical componens p and q. These are conneced by p = u cos θ and q = u sin θ. The noaion is illusraed in Figs..6 and.7. Fig..6 Fig..7 θ in his web service Cambridge Universiy Press
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper : The moion of projeciles The acceleraion g is represened by becomes r = u g + p = q + 0 g or 0, so he equaion g u = u i + 0 g. By reading along each line in urn, he separae equaions for he coordinaes are and x p y q g or x ucos θ, or y u si n g. In a similar way, v = u + g becomes p = q + 0 g or u = cosθ u sinθ + 0. g So and x p y q g or or x ucos θ, y u sinθ g Since g, p, q, u and θ are all consan, you can see again ha x and y are he derivaives of x and y wih respec o. Now he equaions x = p and x p are jus he same as hose you would use for a paricle moving in a sraigh line wih consan velociy p. And he equaions y q g and y = q g are he same as hose for a paricle moving in a verical line wih iniial velociy q and acceleraion g. This esablishes he independence of horizonal and verical moion. 7 If a projecile is launched from O wih an iniial velociy having horizonal and verical componens p and q, under he acion of he force of graviy alone and neglecing air resisance, and if is coordinaes a a laer ime are (x,y), hen he value of x is he same as for a paricle moving in a horizonal line wih consan velociy p; he value of y is he same as for a paricle moving in a verical line wih iniial velociy q and acceleraion g. EXAMPLE.. A golf ball is driven wih a speed of 45 m s a 37 o he horizonal across horizonal ground. How high above he ground does i rise, and how far away from he saring poin does i irs land? To a good enough approximaion cos 37 = 0.8 and sin 37 = 0.6, so he horizonal and verical componens of he iniial velociy are p = 45 0.8 m s = 36 m s and q = 45 0.6 m s = 7 m s. The approximae value of g is 0 m. in his web service Cambridge Universiy Press
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Cambridge Inernaional AS and A Level Mahemaics: Mechanics 8 To ind he heigh you only need o consider he y-coordinae. To adap he equaion v = u + as wih he noaion of Fig..6, you have o inser he numerical values u (ha is q) = 7 and a = 0, and replace s by y and v by y. This gives y = 7 0 y = 79 0y When he ball is a is greaes heigh, y = 0, so 79 0y = 0. This gives 79 y = = 0 36. 45. To ind how far away he ball lands you need o use boh coordinaes, and he link beween hese is he ime. So use he y-equaion o ind how long he ball is in he air, and hen use he x-equaion o ind how far i goes horizonally in ha ime. Adaping he equaion s = u a for he verical moion, y = 7 5. When he ball his he ground y = 0, so ha 7 = = 54.. A paricle moving 5 horizonally wih consan speed 36 m s would go 36 5.4 m, ha is 94.4 m, in his ime. So, according o he graviy model, he ball would rise o a heigh of abou 36 meres, and irs land abou 94 meres from he saring poin. In pracice, hese answers would need o be modiied o ake accoun of air resisance and he aerodynamic lif on he ball. EXAMPLE.. In a game of ennis a player serves he ball horizonally from a heigh of meres. I has o saisfy wo condiions. i I mus pass over he ne, which is 0.9 meres high a a disance of meres from he server. ii I mus hi he ground less han 8 meres from he server. A wha speeds can i be hi? I is simples o ake he origin a ground level, raher han a he poin from which he ball is served, so add o he y-coordinae given by he general formula. If he iniial speed of he ball is p m s, x = p and y = 5. Boh condiions involve boh he x- and y-coordinaes, and he ime is used as he link. i The ball passes over he ne when = p, ha is =. The value p of y is hen 5 70 =, and his mus be more han 0.9. So p p 70 > 09.. p in his web service Cambridge Universiy Press
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper : The moion of projeciles This gives 70 70 <., which is p > 5.. 6 p. ii The ball lands when y = 0, ha is when 5 = 0, or. I has hen 5 gone a horizonal disance of p meres, and o saisfy he second 5 condiion you need p < 8. This gives p < 8 5 8. 5. 5 So he ball can be hi wih any speed beween abou 5.6 m s and 8.5 m s. EXAMPLE..3 A crickeer scores a six by hiing he ball a an angle of 30 o he horizonal. The ball passes over he boundary 90 meres away a a heigh of 5 meres above he ground. Neglecing air resisance, ind he speed wih which he ball was hi. If he iniial speed was u m s, he equaions of horizonal and verical moion are x = u cos 30 and y = u sin 30 5. You know ha, when he ball passes over he boundary, x = 90 and y = 5. Using he values cos30 3 and sin30 =, 90 = u 3 3u and 5 5 = u 5 for he same value of. 80 From he irs equaion, u = = 60 3. Subsiuing his in he second 3 equaion gives 5 30 3 5, which gives = 6 3 = 3. 06. 9 I follows ha u = 60 3 = 60 3 306 33.. 9 The iniial speed of he ball was abou 34 m s. EXAMPLE..4 A boy uses a caapul o send a small ball hrough his friend s open window. The window is 8 meres up a wall meres away from he boy. The ball eners he window descending a an angle of 45 o he horizonal. Find he iniial velociy of he ball. One of he modelling assumpions we are making here is ha he window is jus a poin. In he real world, he window has a signiican heigh, so in a more sophisicaed model, we could ake his ino accoun o obain a range of possible velociies. Denoe he horizonal and verical componens of he iniial velociy by p m s and q m s. If he ball eners he window afer seconds, = p and 8 q 5. Also, as he ball eners he window, is velociy has componens x p and y q 0. Since his is a an angle of 45 below he horizonal, y x, so q 0 = p, or p + q = 0. in his web service Cambridge Universiy Press
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Cambridge Inernaional AS and A Level Mahemaics: Mechanics You now have hree equaions involving p, q and. From he irs wo equaions, p = and q = 8+ 5. Subsiuing hese expressions in he hird equaion gives 8 5 + = 0, ha is + (8 + 5 ) = 0, which simpliies o 5 = 0. So =, from which you ge p = = 6 8+ 5 and q = = 4. Fig..8 shows how hese componens are combined by he riangle rule o give he iniial velociy of he ball. This has magniude 6 4 ms 5ms a an angle an 66. 8 ο o he horizonal. 6 4 u 4ms 0 6ms Fig..8 The ball is projeced a jus over 5 m s a 67 o he horizonal. Exercise B Assume ha all moion akes place above a horizonal plane. A paricle is projeced horizonally wih speed 3 m s, from a poin high above a horizonal plane. Find he horizonal and verical componens of he velociy of he paricle afer seconds. The ime of ligh of an arrow ired wih iniial speed 30 m s horizonally from he op of a ower was.4 seconds. Calculae he horizonal disance from he ower o he arrow s landing poin. Calculae also he heigh of he ower. 3 Show ha he arrow in Quesion eners he ground wih a speed of abou 38 m s a an angle of abou 39 o he horizonal. 5 4 A sone is hrown from he poin O on op of a cliff wih velociy 0 ms. Find he posiion vecor of he sone afer seconds. 5 A paricle is projeced wih speed 35 m s a an angle of 40 above he horizonal. Calculae he horizonal and verical componens of he displacemen of he paricle afer 3 seconds. Calculae also he horizonal and verical componens of he velociy of he paricle a his insan. 6 A famine relief aircraf, lying over horizonal ground a a heigh of 45 meres, drops a sack of food. a Calculae he ime ha he sack akes o fall. b Calculae he verical componen of he velociy wih which he sack his he ground. in his web service Cambridge Universiy Press