Uniform Accelerated Motion

Transcription

1 5 h Year Applied Mahs Higher Level Kieran Mills Uniform Acceleraed Moion No par of his publicaion may be copied, reproduced or ransmied in any form or by any means, elecronic, mechanical, phoocopying, recording, or oherwise, wihou prior wrien permission from The Dublin School of Grinds. Ref: 5/appmahs/h/km/UAM

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4 Conens: Uniform Acceleraed Moion Secion Using he Formulae... Exercise...6 Exercise...8 Secion Successive Times and Disances...0 Exercise Secion 3 Cach-up Problems...6 Exercise Secion 4 Velociy-ime Graphs...8 Exercise Secion 5 Free-fall...4 Exercise Leaving Cer Quesions The Dublin School of Grinds Page Kieran Mills & Tony Kelly

5 The Dublin School of Grinds Page Kieran Mills & Tony Kelly Secion : Using he Formulae 0 m = 0 s a = 4 m s - Quaniy Symbol Unis Acceleraion a m s Iniial velociy u m s Final velociy v m s Time s Displacemen s m - Lef s = 300 m Equaions of Moion Ex. How long does i ake he bus o ravel from A o B? How far is i from A o B? Soluion a = 4 ms u = 0 ms v = 50 ms =? + Righ +a: acceleraion a: deceleraion A B u = 0 m s - v = 50 m s - v= u+ a 50 = = 4 = 0 s v= u+ a...( )[ No No s] s= ( v+ u )...( )[ a] s= u+ a...( 3)[ No v] v = u +as...( 4)[ No No ] s= v a...( 5)[ u] s= ( v+ u ) = ( )( 0) = ( 60)( 0) = 300 m

6 The Dublin School of Grinds Page 3 Kieran Mills & Tony Kelly Example u = 3 m s, a = 6 m s, = 4 s, s =? Example u = 6 m s, a = 0 m s, = 4 s, v =? Example 3 u = 8 m s, v = 5 m s, = 6 s, s =? Example 4 u = 4 m s, v = 9 m s, a = 0.5 m s, s =? Example 5 s = 0 m, v = 0 m s, = 3 s, a =? Example 6 s = 8 m, u = 6 m s, a = m s, =?

7 The Dublin School of Grinds Page 4 Kieran Mills & Tony Kelly Example u = 3 m s, a = 6 m s, = 4 s, s =? Soluion Mahemaical Calculaions u = 3 m s a = 6 m s = 4 s s =? No v presen. Use Equaion (3). s= u+ a = ()( 3 4) + ( 6)( 4) = + 48 = 60 m Example u = 6 m s, a = 0 m s, = 4 s, v =? Soluion Mahemaical Calculaions u = 6 m s a = 0 m s = 4 s v =? No s presen. Use Equaion (). v= u+ a = ( ) = = 46 ms Example 3 u = 8 m s, v = 5 m s, = 6 s, s =? Soluion Mahemaical Calculaions u = 8 m s v = 5 m s = 6 s s =? No a presen. Use Equaion (). s= ( v+ u ) = ( ) = 33 ( ) = 69 m

8 The Dublin School of Grinds Page 5 Kieran Mills & Tony Kelly Example 4 u = 4 m s, v = 9 m s, a = 0.5 m s, s =? Soluion Mahemaical Calculaions u = 4 m s v = 9 m s a = 0.5 m s s =? No presen. Use Equaion (4). v = u + as 9 = (. ) s 8 = 6 + s s = 65 m Example 5 s = 0 m, v = 0 m s, = 3 s, a =? Soluion Mahemaical Calculaions s = 0 m v = 0 m s = 3 s a =? No u presen. Use Equaion (5). s= v a 0 = 0() 3 a() = 30 a 40 = 60 9a 9a = 80 a = 0 ms Example 6 s = 8 m, u = 6 m s, a = m s, =? Soluion Mahemaical Calculaions s = 8 m u = 6 m s a = m s =? No v presen. Use Equaion (3). s= u+ a 8= ( 6) + ( ) 8= = 0 ( 4)( ) = 0 = s, 4s

9 Exercise. Using he formulae. u = 0 m s, a= 4 m s, = 6 s, s =?. u = 5 m s, a= 3 m s, = 3 s, v=? 3. u = 0 m s, a= 4 m s, = 4 s, s =? 4. u = 7 m s, v= 5 m s, = 0 s, s =? 5. u = 4 m s, v= 0 m s, s = 7m, a =? 6. v= 30 m s, a = 6 m s, = s, u =? 7. u = 6 m s, v= 3 m s, s = 5. m, a =? 8. s = 35 m, a= 6 m s, = s, u =? 9. s = u = 40 m, 5 m s, v= 5 m s, =? 0. u = 5 m s, v= 0 m s, a = 0. 5 m s, s =?. u = 48 m s, v= m s, = 9 s, a =?. s = 30 m, = s, a = 4 m s, v=? 3. s = 0 m, = 3 s, v= 0 m s, a=? 4. s = a= 4 m, m s, v= 3 m s, =? 5. s = v= 8 m, 7 m s, a = 3 m s, =? 6. s = u = 4 m, 5 m s, a= m s, =? 7. s = u = 8 m, 6 m s, a= m s, =? 8. v= 7 km h, a = 05. m s, = min, u =? 9. s = v= 8 m, 8 m s, a= 3m s, u =? 0. s = v= 6 m, 4 m s, a= 4 m s, =? The Dublin School of Grinds Page 6 Kieran Mills & Tony Kelly

10 Answers Exercise. 7 m. 4 m s 3. 3 m 4. 0 m. 4 m s. 9 m s 3. 0 m s 4. 4 s 5. 6 m s 6. 8 m s 7. 9 m s 8 5. s, s 7. s, 4s s 8..5 m s 9. 4 s m 8. 0 m s 9. 4 m s 0. 3 s The Dublin School of Grinds Page 7 Kieran Mills & Tony Kelly

11 Exercise. Simple problems using he formulae. A rain sars from res and acceleraes uniformly a.5 m s unil i reaches a speed of 5 m s. Find he disance moved and he ime aken for his moion.. A car can accelerae from res o 90 km h in 7.5 seconds. Find is acceleraion. 3. In ravelling 65 cm along he barrel of a rifle a bulle acceleraes from res o 30 m s. Find he accceleraion and he ime he bulle is in he barrel. 4. A car ravelling a 4 m s requires a minimum braking disance of 36 m. Wha is is deceleraion? How long does i ake o sop? 5. A car sars from res wih acceleraion 4 m s. How far does i go in (i) s, (ii) 3 s, (iii) he hird second. 6. A body moves in a sraigh line and increases is velociy from 3 m s o 5 m s uniformly in 6 s. Find he acceleraion and he disance ravelled. 7. A paricle sars wih a velociy of 3 m s and acceleraes uniformly a.5 m s. How far does i go in (i) s, (ii) 5 s, (iii) he fourh second. 8. A body is projeced from he origin wih a velociy of 8 m s and acceleraion m s. Find (i) he velociy when = 3 s, (ii) when i comes o insananeous res. 9. A paricle moves along a sraigh line beween wo poins P and Q wih consan acceleraion 0.8 m s. Is velociy a Q is. m s greaer han he velociy a P. If he disance PQ is 48 m, find he velociy a P. How long afer passing P does i ake he velociy o reach 48 m s. 0. A car is moving wih speed u m s. The brakes of he car can produce a consan deceleraion of 5 m s. I is known ha when he driver decides o sop, a period of 5 s elapses before he brakes are applied. As he car passes a poin O, he driver decides o sop. Find in erms of u he minimum disance of he car from O when he car comes o res. The driver is approaching raffic lighs and is 0 m away when he ligh changes from green o amber. The lighs remain amber for 3 s before changing o red. Show (a) when u < 30 he driver can sop before reaching he lighs, (b) when u > 34 he driver can pass he ligh before i urns red. The Dublin School of Grinds Page 8 Kieran Mills & Tony Kelly

12 Answers Exercise. 5 m, 0 s. 0 3 ms 3. 40, 69 m s, s 4. 8 m s, 3 s 5. 8 m, 8 m, 0 m 6. m s, 54 m m, m, 8.5 m 8. m s, 4 s m s, 0.75 s 0. 0 u + 5 u The Dublin School of Grinds Page 9 Kieran Mills & Tony Kelly

13 The Dublin School of Grinds Page 0 Kieran Mills & Tony Kelly Secion : Successive Times and Disances Example In wo successive seconds a uniformly acceleraing body ravels 5 m and 3 m. Find is acceleraion and is iniial velociy. Example A uniformly deceleraing body covers successive 00 m disances in 5 s and 0 s. Find is iniial speed, he deceleraion and he furher ime for he body o come o res.

14 The Dublin School of Grinds Page Kieran Mills & Tony Kelly Example In wo successive seconds a uniformly acceleraing body ravels 5 m and 3 m. Find is acceleraion and is iniial velociy. Soluion In hese ypes of problems always use he equaion shown. Take all values from he beginning poin. u 5= u() + a() 5 = u+ a 0 = u+ a...( ) = s a s= u+ a = s, s = 8 m = s s = 5 m s = 3 m 8 = u( ) + a( ) 8 = u+ a...( ) Mahemaical Calculaions 0 = u+ a...( )( ) 8 = u+ a...( ) 0 = u a 8 = u+ a 8 = a 0 = u+ a...( ) 0 = u + 8 = u u = Answers a= 8 ms, u = ms

15 Con... The Dublin School of Grinds Page Kieran Mills & Tony Kelly Example A uniformly deceleraing body covers successive 00 m disances in 5 s and 0 s. Find is iniial speed, he deceleraion and he furher ime for he body o come o res. Soluion u In hese ypes of problems always use he equaion shown. Take all values from he beginning poin. 00 = u() 5 + a() = 5u+ a 40 = u+ 5a...( ) = 5 s a s= u+ a s = 00 m s = 00 m = 5 s, s = 00 m 00 = u( 5) + a( 5) 5 00 = 5u+ a 80 = 6u+ 45a...( ) = 0 s =? 0 m s - Mahemaical Calculaions 40 = u+ 5a...( )( 3) 80 = 6u+ 45a...( ) 0 = 6u 5a 80 = 6u + 45a 40 = 30a 40 = u+ 5a...( ) 4 40 = u + 5( 3) 0 40 = u 3 0 = 6u 0 40 = 6u = u 40 6 u = ms 70 3

16 The Dublin School of Grinds Page 3 Kieran Mills & Tony Kelly Example A uniformly deceleraing body covers successive 00 m disances in 5 s and 0 s. Find is iniial speed, he deceleraion and he furher ime for he body o come o res. Soluion u = 5 s a s= u+ a s = 00 m s = 00 m = 5 s, s = 00 m Mahemaical Calculaions Furher ime for body o come o res: u = 70 3 a = 4 3 ms v = 0 ms =? ms v= u+ a = 3 + ( 3) = 3 4 = = = 7. 5 s = 0 s =? Answer: Furher ime = =.5 s 4 0 m s - Mahemaical Calculaions 40 = u+ 5a...( )( 3) 80 = 6u+ 45a...( ) 0 = 6u 5a 80 = 6u + 45a 40 = 30a 40 = u+ 5a...( ) 4 40 = u + 5( 3) 0 40 = u 3 0 = 6u 0 40 = 6u = u 40 6 u = ms 70 3

17 Exercise 3. Successive Times/Successive Disances. In wo successive seconds a uniformly acceleraing body ravels 4 m and 8 m. Find is acceleraion.. A uniformly acceleraing body ravels 5 m and m repecively in is firs wo seconds. How far does i ravel in he fourh second? 3. A uniformly deceleraing body covers successive 00 m disances in 5 s and 0 s. Find is iniial speed, he deceleraion and he furher ime for he body o come o res. 4. A paricle sars from res and moves in a sraigh line wih uniform acceleraion. I passes hree poins A, B and C where AB =05 m and BC = 63 m. If i akes 6 s o ravel from A o B and s from B o C find (i) is acceleraion, (ii) he disance of A from he saring posiion. 5. A spriner runs a race wih consan acceleraion hroughou. During he race he passes four poss A, B, C, D such ha AB = BC = CD = 36 m. If he spriner akes 3 s o run from A o B and s o run from B o C, how long does i akes o run from C o D? 6. A paricle moving in a sraigh line wih uniform acceleraion describes 3 m in he fifh second of is moion and 3 m in he sevenh second. Calculae is iniial velociy. 7. A body ravels in a sraigh line wih uniform acceleraion. The paricle passes hree poins A, B and C a = 0, = 3 s and = 6 s. If BC = 90 m and he speed of he paricle a B is m s, find he acceleraion of he body and is speed a A. 8. A, B, C are hree poins which lie in ha order on a sraigh road wih AB = 45 m and BC = 3 m. A car ravels along he road in he direcion ABC wih consan acceleraion f. The car passes A wih speed u and passes B five seconds laer and passes C wo seconds afer ha. Find u and f. 9. A car is moving along a a seady 0 m s when he driver suddenly sees a ree across he road 56 m ahead. He immediaely applies he brakes giving he car a consan deceleraion of 4 m s. How far in fron of he ree does he car come o res? If he driver had no reaced immediaely and he brakes were applied one second laer wih wha speed would he car have hi he ree? 0. A, B, C are hree poins on a sraigh line in ha order. A body is projeced from B owards A wih a speed of 5 m s. The body experiences an acceleraion of m s owards C. If BC = 4 m, find he ime o reach C, and he disance ravelled by he body from he insan of projecion unil i reaches C.. A bus.5 m long ravels wih consan acceleraion. The fron of he bus passes a poin P wih speed u and he rear passes P wih speed v. Find in erms of u and v (i) he ime aken for he bus o pass P, (ii) wha fracion of he bus passes P in half his ime. The Dublin School of Grinds Page 4 Kieran Mills & Tony Kelly

18 . A body moving in a sraigh line wih consan acceleraion passes in succession hrough poins A, B, C and D where AB = x, BC = y and CD = z where he disances x, y and z are covered in equal inervals of ime. Show y = x + z. 3. A uniformly deceleraing rain of lengh 40 m eners a saion of lengh 80 m. The fron engine leaves he saion 5 s laer and he rear of he rain leaves he saion afer a furher 5 s. Find he deceleraion of he rain. 4. A uniformly acceleraing body sars wih a speed of u, in successive imes of ravels disances s and s. Prove ha is acceleraion is 4 u s. 5. A body sars moving in a sraigh line wih velociy u and acceleraion a. If when he velociy has increased o 5u he acceleraion is reversed in direcion is magniude being unalered prove ha when he paricle reurns o is saring poin is velociy will be 7u. Answers Exercise 3. 4 m s. 3 m ms, 3 ms, 5. s 4. (i) 3.5 m s, (ii) 7 m 5..6 s 6. 5 m s 7. 6 m s, 3 m s 8. 4 m s, m s 9. 6 m, 0.6 m s 0. 8 s, 36.5 m 5 3u+ v., v+ u 4( u+ v) 3..6 m s The Dublin School of Grinds Page 5 Kieran Mills & Tony Kelly

19 The Dublin School of Grinds Page 6 Kieran Mills & Tony Kelly Secion 3: Cach-up Problems Example Two bodies sar ogeher a he same ime a he same place and move along he same sraigh line. If one moves wih a consan speed of 6 m s while he oher sars from res and moves a a consan acceleraion of 4 m s. How long will i ake before hey are ogeher? Example Two bodies A and B ravel in he same direcion along he same line. Body A sars wih velociy 5 m s and acceleraion 3 m s. The oher body sars from he same place wih velociy m s and acceleraion 4 m s. Find when and where hey are ogeher again. Example 3 Two bodies move in he same direcion along parallel pahs. A sars from a poin O wih velociy 8 m s and acceleraion m s and B sars 8 m ahead of A and moves off wih velociy m s a acceleraion 4 m s. Find when hey will be ogeher and heir disances from O a hese imes. Wha are heir respecive speeds when hey are ogeher? Example 4 If A sars seconds before B find when and where hey are ogeher. Find he maximum disance ha A moves ahead of B in he subsequen moion. u = 0 m s a = m s A 64 m B u = m s a = 4 m s

20 The Dublin School of Grinds Page 7 Kieran Mills & Tony Kelly Example Two bodies sar ogeher a he same ime a he same place and move along he same sraigh line. If one moves wih a consan speed of 6 m s while he oher sars from res and moves a a consan acceleraion of 4 m s. How long will i ake before hey are ogeher? Soluion = 0 s A B u = 6 m s a = 0 m s u = 0 m s a = 4 m s s= u+ a s = s A B Separaion Equaion: sa sb s s = 6 A B sa = 6+ ( 0) = 6 s = ( 0) + ( 4) B = Mahemaical Calculaions Togeher (level): sa sb =0 6 = 0 6 = 0 8 = 0 ( 8) = 0 = 0 s, 8s Answers A and B are ogeher afer 0 s and afer 8 s.

21 The Dublin School of Grinds Page 8 Kieran Mills & Tony Kelly Example Two bodies A and B ravel in he same direcion along he same line. Body A sars wih velociy 5 m s and acceleraion 3 m s. The oher body sars from he same place wih velociy m s and acceleraion 4 m s. Find when and where hey are ogeher again. Soluion = 0 s A B u = 5 m s a = 3 m s - u = m s a = 4 m s s= u+ a s = s A B Separaion Equaion: sa sb 3 s s = 5+ A B = 3 s = () 5 + () 3 A 3 = 5+ s = ( ) + ( 4) B = + Mahemaical Calculaions Togeher (level): sa sb =0 3 = 0 6 = 0 ( 6) = 0 = 0 s, 6 s A and B are ogeher afer 0 s and afer 6 s. Disance A and B have ravelled afer 6 s: s = B + = 6 ( ) + 6 ( ) = 6 ( ) + 36 ( ) = + 7 = 84 m

22 The Dublin School of Grinds Page 9 Kieran Mills & Tony Kelly Example 3 Two bodies move in he same direcion along parallel pahs. A sars from a poin O wih velociy 8 m s and acceleraion m s and B sars 8 m ahead of A and moves off wih velociy m s a acceleraion 4 m s. Find when hey will be ogeher and heir disances from O a hese imes. Wha are heir respecive speeds when hey are ogeher? Soluion = 0 s o A u = 8 m s a = m s 8 m - s= u+ B sars 8 m ahead of A. Therefore, add a 8 o he disance equaion for B. - B s = s A B u = m s a = 4 m s - - Separaion Equaion: sa sb s s = 8+ 8 A B = s = () 8 + ( ) A = 8+ s = 8+ ( ) + ( 4) B = 8+ + Mahemaical Calculaions Togeher (level): sa sb = = = 0 ( )( 4) = 0 = s, 4 s Con...

23 The Dublin School of Grinds Page 0 Kieran Mills & Tony Kelly Example 3 Two bodies move in he same direcion along parallel pahs. A sars from a poin O wih velociy 8 m s and acceleraion m s and B sars 8 m ahead of A and moves off wih velociy m s a acceleraion 4 m s. Find when hey will be ogeher and heir disances from O a hese imes. Wha are heir respecive speeds when hey are ogeher? Soluion o A u = 8 m s a = m s 8 m - - s= u+ a s = s = 0 m A B B u = m s a = 4 m s - - v A = m s - v B = 0 m s - v= u+ a = 0 s = s = 4 s s = + A () 8 ( ) = 8+ Mahemaical Calculaions Disances and speeds afer s: sa = 8+ = 8 ( ) + ( ) = = 0 m va = u+ a = () 8 + ( )( ) = 8+ 4 = ms v B = u+ a = ( ) + ( 4)( ) = + 8 = 0 ms Con...

24 The Dublin School of Grinds Page Kieran Mills & Tony Kelly Example 3 Two bodies move in he same direcion along parallel pahs. A sars from a poin O wih velociy 8 m s and acceleraion m s and B sars 8 m ahead of A and moves off wih velociy m s a acceleraion 4 m s. Find when hey will be ogeher and heir disances from O a hese imes. Wha are heir respecive speeds when hey are ogeher? Soluion s= u+ a v= u+ a = 0 s = s = 4 s s A = s B = 48 m o A u = 8 m s a = m s 8 m - - s = s = 0 m A B B u = m s a = 4 m s - - sa = () 8 + ( ) = 8+ - v A = m s - v B = 0 m s - v A = 6 m s - v B = 8 m s Mahemaical Calculaions Disances and speeds afer 4 s: sa = 8+ = 84 ( ) + ( 4) = = 48 m va = u+ a = () 8 + ( )( 4) = 8+ 8 = 6 ms vb = u+ a = ( ) + ( 4)( 4) = + 6 = 8 ms

25 The Dublin School of Grinds Page Kieran Mills & Tony Kelly Example 4 If A sars seconds before B find when and where hey are ogeher. Find he maximum disance ha A moves ahead of B in he subsequen moion. Soluion s= u+ a - u = m s - B a = 4 m s 64 m = 0 s If A sars s before B, ake A s ime as ( + ) s and B s ime as s. Mahemaical Calculaions = s s = A ( 0)( ) ( )( ) = 0 s A = sb = ( + ) = 0 - u = 0 m s - A a = m s + = ( 5)( 8) = 0 = = 5s, 8s Separaion Equaion: sa sb s s = A B = s = 64 + B () + ( 4) = Togeher (level): sa sb =0 A and B are ogeher (level) 5 s and 8 s afer B sars. Con...

26 The Dublin School of Grinds Page 3 Kieran Mills & Tony Kelly Example 4 If A sars seconds before B find when and where hey are ogeher. Find he maximum disance ha A moves ahead of B in he subsequen moion. Soluion A s= u+ a ( + ) = s ( + ) = 7 s ( + ) = 0 s s A = s B = 00 m u = 0 m s a = m s - - s = s = 9 m A B + - u = m s - B a = 4 m s 64 m = 0 s s A = s B = = 5 s = 8 s Separaion Equaion: s s = A B Mahemaical Calculaions Disance afer 5 s: sa = = () () + 4 = = 9 m Disance afer 8 s: sa = = () () + 4 = = 00 m Con...

27 The Dublin School of Grinds Page 4 Kieran Mills & Tony Kelly Example 4 If A sars seconds before B find when and where hey are ogeher. Find he maximum disance ha A moves ahead of B in he subsequen moion. Soluion A The maximum separaion occurs when he velociies of A and B are equal. s= u+ a ( + ) = s ( + ) = 7 s ( + ) = 0 s s A = s B = 00 m u = 0 m s a = m s - - s = s = 9 m A B + - u = m s - B a = 4 m s 64 m = 0 s s A = s B = = 5 s = 8 s Separaion Equaion: s s = A B Mahemaical Calculaions Maximum Separaion: va = vb va = u+ a = 0 + ( )( + ) = = + 4 vb = u+ a = + ( 4) = 4 + va = vb + 4 = 4+ 3 = = 65. s s s = A B = ( 65. ) + 3( 65.) 40 = 5. m

28 Exercise 4. Cach up. Two bodies sar ogeher a he same ime a he same place and move along he same sraigh line. If one moves wih a consan speed of 8 m s while he oher sars from res and moves a a consan acceleraion of m s. How long will i ake before hey are ogeher?. A car A passes a poin P on a sraigh road a a consan speed of 0 m s. A he same ime anoher car B sars from res a P wih uniform acceleraion.5 m s. (i) When and how far from P will B overake A. (ii) If B ceases o accelerae on overaking, wha ime elapses beween he wo cars passing a poin Q which is 3 km from P. 3. A boy runs a 4 m s away from a cyclis who sars a res and acceleraes a m s. If he boy has an iniial lead of 5 m, how long does he cyclis ake o cach him? 4. Two bodies A and B ravel in he same direcion along he same line. Body A sars wih velociy 3 m s and acceleraion m s. The oher body sars from he same place wih velociy m s and acceleraion 3 m s. Find when and where hey are ogeher again. 5. Two bodies move along parallel racks in he same direcion. Body A sars wih velociy m s and acceleraion 6 m s. Body B sars from he same place and he same ime wih velociy 5 m s and acceleraion m s. Find when and where hey are ogeher again. Find heir velociies when hey are ogeher for he second ime. 6. Two bodies move in he same direcion along parallel pahs. A sars from poin O wih velociy m s and acceleraion 4 m s. B sars 6 m ahead of A wih velociy 3 m s and acceleraion m s. Find when and where hey are ogeher and heir velociies a his insan. 7. Two bodies move in he same direcion along parallel pahs. A sars from a poin O wih velociy 8 m s and acceleraion m s and B sars 8 m ahead of A and moves off wih velociy m s a acceleraion 4 m s. Find when hey will be ogeher and heir disances from O a hese imes. 8. Two bodies A and B ravel in he same direcion along he same line from he same poin P a he same ime. A sars wih velociy 5 m s and acceleraion 3 m s. B sars wih velociy m s and acceleraion 4 m s. They are ogeher again a poin Q. Find he ime a which hey are ogeher and he disance PQ. Find heir maximum disance apar beween P and Q. 9. Two bodies move in he same direcion along parallel pahs. They sar a he same poin P a he same ime. A sars from P wih velociy 3 m s and acceleraion m s. B sars wih velociy m s and acceleraion 3 m s. They are ogeher again a poin Q. Find he ime a which hey are ogeher and he disance PQ.Find heir maximum disance apar beween P and Q. 0. Two bodies A and B move along parallel sraigh lines in he same direcion from he same poin P. A sars wih velociy 4 m s and acceleraion m s. B sars second afer A wih velociy m s and acceleraion 4 m s. Find when and where hey will be ogeher. The Dublin School of Grinds Page 5 Kieran Mills & Tony Kelly

29 . Two bodies A and B move along parallel sraigh lines in he same direcion from he same poin P. A sars from poin P wih velociy 5 m s and acceleraion 4 m s. B sars second before A wih velociy 6 m s and acceleraion 3 m s from a poin a disance of.5 m o he righ of P. Find when and where hey are ogeher.. Find when and where hey are ogeher. Find heir maximum separaion beween he wo imes when hey are ogeher. A u = 0 m s a = m s B u = m s a = 4 m s 8m 3. If A sars seconds before B find when and where hey are ogeher. Find heir maximum separaion beween he wo posiions. A u = 0 m s a = m s B u = m s a = 4 m s 4. A car A sars from a poin P wih iniial velociy 8 m s and hen ravels wih uniform acceleraion 4 m s. Two seconds laer a second car B sars from P wih an iniial velociy of 30 m s and hen moves wih a uniform acceleraion of 3 m s. Show ha afer passing A, B will never be ahead by more han 74 m. 5. Bodies A and B sar ogeher and move along he same sraigh line. A sars wih a speed of 0 m s and moves wih a consan deceleraion, while B sars a 5 m s and acceleraes a 4 m s. Find he deceleraion of A if hey mee when he velociy of B is wice ha of A. 6. The driver of a car ravelling a 0 m s sees a second car 0 m in fron ravelling in he same direcion a a uniform speed of 8 m s. (a) Wha is he leas uniform reardaion ha mus be applied o he faser car o avoid collision? (b) If he acual reardaion is m s find (i) he ime inerval in seconds for he faser car o reach a poin 66 m behind he slower car, (ii) he shores disance beween he cars. 64 m The Dublin School of Grinds Page 6 Kieran Mills & Tony Kelly

30 Answers Exercise 4. 8 s. (i) 8 s, 80 m (ii) 46 s 3. 5 s 4. 4 s, 8 m from saring poin 5..5 s, 9.75 m, m s, 8 m s 6. 3 s, 4 m, 4 m s, 9 m s 7. s, 4 s, 0 m, 48 m 8. 6 s, 84 m, 4.5 m 9. 4 s, 8 m, m 0. 5 s afer B sars, 60 m. 0 s afer A sars, 50 m from P. s, 8 s,.5 m 3. 5 s and 8 s afer B sars, 9 m from A and 00 m from A,.5 m 5. 4 m s 6. (a) 0.6 m s (b) (i) 6 s, 8 s; (ii) 48 m The Dublin School of Grinds Page 7 Kieran Mills & Tony Kelly

31 The Dublin School of Grinds Page 8 Kieran Mills & Tony Kelly Secion 4: Velociy-ime Graphs Example A car saring from res acceleraes uniformly over 8 s o a velociy of 6 m s. I hen mainains a consan velociy for he nex 0 s. I finally deceleraes uniformly o res for 4 s. Draw a velociy-ime curve o represen he moion of he car. Use he graph o find (i) he acceleraion and disance ravelled by he car in he firs 8 s, (ii) he disance ravelled by he car over he nex 0 s, (iii) he deceleraion and disance ravelled by he car over he las 4 s, (iv) he average velociy of he car for is enire journey. Example A car ravels from A o B. I sars from res a A and acceleraes a m s unil i reaches a speed of 30 m s. I hen ravels a his speed for 600 m and hen deceleraes a.5 m s o come o res a B. Find (i) he oal ime for he journey, (ii) he disance from A o B, (iii) he average speed for he journey. Example 3 A paricle P wih speed 40 m s begins o decelerae uniformly a a cerain insan while anoher paricle Q sars from res 6 s laer and acceleraes uniformly. When he second paricle Q has ravelled 5 m, boh paricles have a speed of 5 m s. (i) Show he moion of boh on he same speed-ime curve. (ii) How many seconds afer he commencemen of deceleraion does he firs paricle P come o res?

32 The Dublin School of Grinds Page 9 Kieran Mills & Tony Kelly Example A car saring from res acceleraes uniformly over 8 s o a velociy of 6 m s. I hen mainains a consan velociy for he nex 0 s. I finally deceleraes uniformly o res for 4 s. Draw a velociy-ime curve o represen he moion of he car. Use he graph o find (i) he acceleraion and disance ravelled by he car in he firs 8 s, (ii) he disance ravelled by he car over he nex 0 s, (iii) he deceleraion and disance ravelled by he car over he las 4 s, (iv) he average velociy of he car for is enire journey. Soluion - v (m s ) (s) Velociy Time Curves Acceleraion = Slope of curve Disance = Area under curve Con...

33 The Dublin School of Grinds Page 30 Kieran Mills & Tony Kelly Example A car saring from res acceleraes uniformly over 8 s o a velociy of 6 m s. I hen mainains a consan velociy for he nex 0 s. I finally deceleraes uniformly o res for 4 s. Draw a velociy-ime curve o represen he moion of he car. Use he graph o find (i) he acceleraion and disance ravelled by he car in he firs 8 s, (ii) he disance ravelled by he car over he nex 0 s, (iii) he deceleraion and disance ravelled by he car over he las 4 s, (iv) he average velociy of he car for is enire journey. Soluion - v (m s ) a = m s - 64 m Run = 8 s Base b = 8 Area of a riangle = bh Rise = 6 m s - Heigh h = 6 (s) Velociy Time Curves Acceleraion a = Slope of curve Disance s = Area under curve Mahemaical Calculaions (i) Rise 6 ms a = = = ms Run 8 s s= bh = ()( 8 6 ) = 64 m Con...

34 The Dublin School of Grinds Page 3 Kieran Mills & Tony Kelly Example A car saring from res acceleraes uniformly over 8 s o a velociy of 6 m s. I hen mainains a consan velociy for he nex 0 s. I finally deceleraes uniformly o res for 4 s. Draw a velociy-ime curve o represen he moion of he car. Use he graph o find (i) he acceleraion and disance ravelled by he car in he firs 8 s, (ii) he disance ravelled by he car over he nex 0 s, (iii) he deceleraion and disance ravelled by he car over he las 4 s, (iv) he average velociy of he car for is enire journey. Soluion - v (m s ) a = m s - Breadh b = 6 64 m 30 m (s) Lengh l = 0 Area of a recangle = l b Velociy Time Curves Acceleraion a = Slope of curve Disance s = Area under curve Mahemaical Calculaions (i) Rise 6 ms a = = = ms Run 8 s s= bh = ()( 8 6 ) = 64 m (ii) s= l b = ( 0)( 6) = 30 m Con...

35 The Dublin School of Grinds Page 3 Kieran Mills & Tony Kelly Example A car saring from res acceleraes uniformly over 8 s o a velociy of 6 m s. I hen mainains a consan velociy for he nex 0 s. I finally deceleraes uniformly o res for 4 s. Draw a velociy-ime curve o represen he moion of he car. Use he graph o find (i) he acceleraion and disance ravelled by he car in he firs 8 s, (ii) he disance ravelled by he car over he nex 0 s, (iii) he deceleraion and disance ravelled by he car over he las 4 s, (iv) he average velociy of he car for is enire journey. Soluion - v (m s ) a = m s - Rise = 6 m s - Heigh h = 6 a = -4 m s - 64 m 30 m 3 m (s) Run = 4 s Base b = 4 Area of a riangle = bh Velociy Time Curves Acceleraion a = Slope of curve Disance s = Area under curve Mahemaical Calculaions (i) Rise 6 ms a = = = ms Run 8 s s= bh = ()( 8 6 ) = 64 m (ii) s= l b = ( 0)( 6) = 30 m Rise 6 ms (iii) a = = = 4 ms Run 4 s s= bh = ( 4 )( 6 ) = 3 m Con...

36 The Dublin School of Grinds Page 33 Kieran Mills & Tony Kelly Example A car saring from res acceleraes uniformly over 8 s o a velociy of 6 m s. I hen mainains a consan velociy for he nex 0 s. I finally deceleraes uniformly o res for 4 s. Draw a velociy-ime curve o represen he moion of he car. Use he graph o find (i) he acceleraion and disance ravelled by he car in he firs 8 s, (ii) he disance ravelled by he car over he nex 0 s, (iii) he deceleraion and disance ravelled by he car over he las 4 s, (iv) he average velociy of he car for is enire journey. Soluion - v (m s ) a = m s m + 30 m + 3 m = 46 m a = -4 m s - (s) Mahemaical Calculaions (iv) Average Velociy = Toal Disance Toal Time 46 m Average velociy = = 3 ms 3 s Con...

37 The Dublin School of Grinds Page 34 Kieran Mills & Tony Kelly Example A car saring from res acceleraes uniformly over 8 s o a velociy of 6 m s. I hen mainains a consan velociy for he nex 0 s. I finally deceleraes uniformly o res for 4 s. Draw a velociy-ime curve o represen he moion of he car. Use he graph o find (i) he acceleraion and disance ravelled by he car in he firs 8 s, (ii) he disance ravelled by he car over he nex 0 s, (iii) he deceleraion and disance ravelled by he car over he las 4 s, (iv) he average velociy of he car for is enire journey. Soluion - v (m s ) a = m s m 3 a = -4 m s - (s) Finding he area under he curve in one go: A rapezium is a four sided shape where wo of he sides are parallel. The area of a rapezium is half he sum of he parallel sides by he perpendicular disance beween hem. Area = ( x + y) h Mahemaical Calculaions s = ( 0 + 3)( 6) = ( 5)( 6) = ( 5)() 8 = 46 m

38 The Dublin School of Grinds Page 35 Kieran Mills & Tony Kelly Example A car ravels from A o B. I sars from res a A and acceleraes a m s unil i reaches a speed of 30 m s. I hen ravels a his speed for 600 m and hen deceleraes a.5 m s o come o res a B. Find (i) he oal ime for he journey, (ii) he disance from A o B, (iii) he average speed for he journey. Soluion Toal ime T = 5 s + 0 s + s = 47 s Velociy Time Curves Acceleraion a = Slope of curve Disance s = Area under curve Mahemaical Calculaions V 30 (i) I : a = = 30 = = 5 s s s = 600 m v = V = 30 ms =? s 600 II :V = 30 = 600 = = 0 s 30 V 30 III : a = 5. = = = 5. s Con...

39 . 4 ms The Dublin School of Grinds Page 36 Kieran Mills & Tony Kelly Example A car ravels from A o B. I sars from res a A and acceleraes a m s unil i reaches a speed of 30 m s. I hen ravels a his speed for 600 m and hen deceleraes a.5 m s o come o res a B. Find (i) he oal ime for he journey, (ii) he disance from A o B, (iii) he average speed for he journey. Soluion Toal ime T = 5 s + 0 s + s = 47 s Mahemaical Calculaions (ii) Area = ( x + y) h Average Velociy = S = ( ) 30 = 005 m Toal Disance Toal Time 005 m (iii) Average velociy = = 47 s

40 The Dublin School of Grinds Page 37 Kieran Mills & Tony Kelly Example 3 A paricle P wih speed 40 m s begins o decelerae uniformly a a cerain insan while anoher paricle Q sars from res 6 s laer and acceleraes uniformly. When he second paricle Q has ravelled 5 m, boh paricles have a speed of 5 m s. (i) Show he moion of boh on he same speed-ime curve. (ii) How many seconds afer he commencemen of deceleraion does he firs paricle P come o res? Soluion Velociy Time Curves Acceleraion a = Slope of curve Disance s = Area under curve Mahemaical Calculaions (ii) Paricle Q: s= bh 5 = ( 5) 5 = = 0 s s = 5 m h = 5 ms b=

41 The Dublin School of Grinds Page 38 Kieran Mills & Tony Kelly Example 3 A paricle P wih speed 40 m s begins o decelerae uniformly a a cerain insan while anoher paricle Q sars from res 6 s laer and acceleraes uniformly. When he second paricle Q has ravelled 5 m, boh paricles have a speed of 5 m s. (i) Show he moion of boh on he same speed-ime curve. (ii) How many seconds afer he commencemen of deceleraion does he firs paricle P come o res? Soluion Velociy Time Curves Acceleraion a = Slope of curve Disance s = Area under curve Mahemaical Calculaions (ii) Paricle P: Rise 5 a = = = 78. ms Run 6 u = 40 ms a = 78. ms v = 0 ms =? v= u+ a 0 = 40 + ( 78. ) 78. = = = s

42 Exercise 5. Velociy Time Curves. A car is ravelling a 7 km h when he brakes are applied producing a reardaion of 4 m s. How long does i ake o sop?. An elecric rain sars from a saion and reaches a speed of 4 m s in 5 s wih uniform acceleraion. Skech he velociy-ime graph, and find how far i has gone by he ime i reaches his speed. 3. An aircraf can ake off when i reaches a speed of 80 km h. If i aains his speed in 30 s wih uniform acceleraion wha disance does i require for aking off? 4. An express rain is ravelling a 44 km h when is brakes are applied. If hese produce a reardaion of m s how long will i ake o sop and wha disance will i cover in doing so? 5. A rain sars from res and aains a speed of 50 km h in 4 minues wih uniform acceleraion. I runs a ha speed for 5 minues and hen slows down uniformly o res in minues. Draw he velociy-ime graph and find he oal disance ravelled. 6. Find from he velociy-ime graph shown (i) he acceleraion during he firs 4 s, (ii) he reardaion during he las s, (iii) he oal disance ravelled. v (m/s) (s) 7. A cyclis rides along a sraigh road from A o B. He sars from res a A and acceleraes uniformly o reach a speed of 0 m s in 8 s. He mainains his speed for 30 s and hen uniformly deceleraes o res a B. If he oal ime is 48 s, draw a velociy-ime curve and from i find (i) he acceleraion, (ii) he deceleraion, (iii) he oal disance ravelled. 8. A car ravels from A o B. I sars from res a A and acceleraes a.5 m s unil i reaches a speed of 30 m s. I hen ravels a his speed for km and hen deceleraes a m s o come o res a B. Find (i) he oal ime for he journey, (ii) he disance from A o B, (iii) he average speed for he journey. 9. A and B are wo poins on a sraigh road. A car ravelling along he road passes A when = 0 and mainains a consan speed unil = 0 s, and in his ime covers four-fifhs of he disance from A o B. The car hen deceleraes uniformly o res a B. Draw a velociy-ime curve and find he ime from A o B. The Dublin School of Grinds Page 39 Kieran Mills & Tony Kelly

43 0. A ram ravels along a sraigh rack and sars from res. I acceleraes uniformly for 0 s and during his ime i ravels 60 m. I mainains a consan speed for a furher 50 s and deceleraes o res in 8 s. Calculae (i) he acceleraion, (ii) he deceleraion, (iii) he oal ime, (iv) he oal disance.. A rain sars from res and ravels 8 km in minues ending a res. The acceleraion is half he reardaion, boh are uniform, and here is a period when he rain runs a is maximum speed of 50 km h. Find he ime aken o reach full speed.. A 00 m spriner sars wih a speed of 6 m s - and acceleraes uniformly o 0 m s and finishes he race a his speed. If his oal ime is 0.4 s, find his uniform acceleraion and afer wha disance he is going a full speed. 3. A car akes minues o ravel beween wo ses of raffic lighs 45 m apar. I has uniform acceleraion for 30 s, hen uniform velociy, and hen uniform reardaion for he las 5 s. Find he maximum velociy and he acceleraion. 4. A rain ravels 5 km beween wo saions a an average speed of 50 km h. Is acceleraion is half he reardaion and boh are uniform. If he maximum speed is 7 km h find he acceleraion in m s. Skech he velociy-ime curve. 5. A car acceleraes a m s in boom gear,.5 m s in second gear and m s in op gear. Each gear change akes.5 s during which ime he car ravels a consan speed. If a mooris changes gear when his speeds are 3 m s and 9 m s find how long he will ake o reach 5 m s from res. 6. A rain moving in a sraigh line sars from A wih uniform acceleraion of 0. m s. Afer i has aained full speed i moves uniformly for 0 minues. I is brough o res a B by he brakes, which apply a consan reardaion of 0.8 m s for 0 s. Draw a rough velociy-ime graph and from i find he ime of he journey and he disance from A o B. 7. A rain has a maximum speed of 7 km h which i can achieve a an acceleraion of 0.5 m s. Wih is brakes fully applied he rain has a deceleraion of 0.5 m s. Wha is he shores ime ha he rain can ravel beween saions 8 km apar if i sops a boh saions? 8. A paricle wih speed 50 m s begins o decelerae uniformly a a cerain insan while anoher paricle sars from res 8 s laer and acceleraes uniformly. When he second paricle has ravelled 35 m, boh paricles have a speed of 30 m s. (i) Show he moion of boh on he same speed-ime curve. (ii) How many seconds afer he commencemen of deceleraion does he firs paricle come o res? The Dublin School of Grinds Page 40 Kieran Mills & Tony Kelly

44 9. A body sars from res a P ravelling in a sraigh line and hen comes o res a Q which is 696 m from P. The ime aken is 66 s. For he firs 0 s i has uniform acceleraion a. I hen ravels a consan speed and is finally brough o res by a uniform deceleraion b acing for 6 s. Find a and b. If he journey from res a P o res a Q had been ravelled wih no inerval of consan speed bu a acceleraion of a for a ime immediaely followed by deceleraion b for a ime, show ha he ime for he journey is 8 9s. 0. An ahlee runs 00 m in s. Saring from res he acceleraes uniformly o a speed of 0 m s and hen coninues a ha speed. Calculae he acceleraion.. A cyclis has a maximum acceleraion of m s, a maximum speed of 5 m s and a maximum deceleraion of 4 m s. If he ravels from res o res in he shores possible ime show ha he covers a disance of m. Find he ime o ravel (i) 05 m, (ii) 54 m. Answers Exercise 5. 5 s. 75 m km 4. 0 s, 400 m km 6. (i) m s (ii) 4 m s (iii) 56 m 7. (i).5 m s (ii) m s (iii) 390 m 8. (i) s (ii) 55 m (iii) 4.83 m s 4. ms s 6. 3 minues,.04 km s 8. (ii).5 s 9.. m s, m s 0..5 m s. (i) 5 8 s (ii) 9 s s 0. (i) 0.8 m s (ii) m s (iii) 78 s (iv) 04 m. 3. minues. m s, 6 m 3. m s, 0.73 m s The Dublin School of Grinds Page 4 Kieran Mills & Tony Kelly

45 The Dublin School of Grinds Page 4 Kieran Mills & Tony Kelly Secion 5: Free Fall Example A body is hrown verically up from he ground a 4 m s -. Find he maximum heigh i reaches and he ime o reach his heigh. From is highes poin find he ime for he body o hi he ground and is speed when i his he ground. Example A ball is hrown verically up a 0 m s - from a poin 3 m above he ground. Find he speed i has when i his he ground and he ime i akes he ball o hi he ground. Example 3 A ho-air balloon ravels from he ground verically up a a consan speed of m s -. Find is heigh above he ground afer 5 s. Afer 5 s a ball is dropped from he baloon. How long does i ake o reach he ground?

46 The Dublin School of Grinds Page 43 Kieran Mills & Tony Kelly Example A body is hrown verically up from he ground a 4 m s -. Find he maximum heigh i reaches and he ime o reach his heigh. From is highes poin find is speed when i his he ground and he ime for he body o hi he ground. Soluion Up is posiive Down is negaive s =? GOING UP v = 0 m s Going up: u = 4 m s - v = 0 m s - a = -9.8 m s - s =? =? u = 4 m s The acceleraion due o graviy is denoed by g. This value is 9.8 m s -. In free fall alway le a = -g = -9.8 m s -. v = u + as 0 = 4 + ( 98. ) s 9. 6s = s = = m v= u+ a 0= = = = = 43. s Conclusions: Time up = Time Down s = 0 m GOING DOWN u = 0 m s Going down: u = 0 m s - a = -9.8 m s - s = -0 m =? v =? v =? Up is posiive Down is negaive v = u + as v = 0 + ( 98. )( 0) v = 96 v = 96 = 4 ms v= u+ a 4 = = = = 43. s Velociy going up = Velociy a same poin on way down bu in he opposie direcion

47 The Dublin School of Grinds Page 44 Kieran Mills & Tony Kelly Example A ball is hrown verically up a 0 m s - from a poin 3 m above he ground. Find he speed i has when i his he ground and he ime i akes he ball o hi he ground. Soluion u = 0 m s s = 3 m v =? u = 0 m s - s = -3 m a = -9.8 m s - v =? =? The acceleraion due o graviy is denoed by g. This value is 9.8 m s -. In free fall alway le a = -g = -9.8 m s -. Up is posiive Down is negaive v = u + as v = 0 + ( 98. )( 3) v = 0 + ( 9. 8)( 3) =. 6 ms The velociy is negaive as i is moving down. v= u+ a. 6= = = = 3. s 98.

48 The Dublin School of Grinds Page 45 Kieran Mills & Tony Kelly Example 3 A ho-air balloon ravels from he ground verically up a a consan speed of m s -. Find is heigh above he ground afer 5 s. Afer 5 s a ball is dropped from he balloon. How long does i ake o reach he ground? Give your answer o one place of decimal. Soluion s =? = 5 s v = m s v = m s Ball is dropped =? u = m s s = 60 m v = s v = ms = 5 s s v = s= v = 5= 60 m Ball is dropped from balloon: u = m s - s = -60 m a = -9.8 m s - =? s= u+ a 60 = + ( 98. ) 60 = = 0 ± ( ) 449 (. )( 60) = 49 (. ) = 49. s

49 Exercise 6. Uniformly g acceleraed moion [In all problems g = 9.8 m s ]. A vase falls from a shelf 40 cm above he floor. Find he speed wih which i srikes he floor.. A sone is dropped from a poin 49 m above he ground. Find he ime for i o reach he ground. 3. A sone is hrown down a 5 m s. If is speed on hiing he ground is 9 m s from wha heigh was i hrown. How long does i ake? 4. A sone is dropped from he op of a ower and falls o he ground. If i srikes he ground a 4 m s, how high is he ower? 5. A ball is hrown verically downwards from he op of a ower wih an iniial speed of m s. If i his he ground 3 s laer find (i) he heigh of he ower, (ii) he speed wih which i his he ground. 6. A sone is hrown upwards wih a speed of m s. Find is heigh (i) s afer projecion, (ii) s afer projecion, (iii) 3 s afer projecion. 7. A ball is hrown up a 4 m s from a poin m above he ground. Find (i) he speed when i reurns o he level of projecion, (ii) he speed on he ground. 8. A ball is hrown verically up a 8 m s. Find (i) he maximum heigh, (ii) he ime o reach he maximum heigh, (iii) he velociy of reurn, (iv) he oal ime for he journey. 9. A balloon is rising a a seady speed of 3 m s. How high is i above he ground afer 0 s? A his insan a man releases a sone. Wha is he iniial velociy of he sone? How long does i ake o reach he ground? How high is he balloon above he ground when he sone srikes he ground? 0. A sone is hrown up a 49 m s from he ground. Find he imes a which he paricle is 78.4 m above he ground. Find he ime inerval for which he paricle is above 78.4 m.. A ball is hrown up a 4 m s. Find he imes a which he paricle is 9. m above he ground. The Dublin School of Grinds Page 46 Kieran Mills & Tony Kelly

50 . A ball is hrown up a 49 m s. How long does i ake o reach is maximum heigh? If an oher ball was hrown up s afer he firs one, how high is i above he ground when he firs ball has reached is maximum heigh if i has he same iniial velociy? 3. A jumper can jump m on he Earh. Wha is his ake-off speed? How high can he jump on he moon? (Acceleraion due o graviy of moon g =.6 m s ) 4. A paricle is hrown verically upwards under graviy wih a speed of 6 m s. One second laer anoher paricle is fired upwards from he same poin. Find he iniial speed of his paricle in order ha he wo paricles will collide when he firs paricle has reached is highes poin. 5. An objec falls verically pas a window m high in s. Find he heigh above he window from which he objec was dropped. 6. A sone is dropped from a balloon rising a 0 m s and reaches he ground in 8 s. How high was he balloon above he ground when he sone was dropped? 7. A body falls from he op of a ower and during he las second i falls 9 5 of he oal disance. Find he heigh of he ower. 8. A paricle falls freely from res from a poin O passing hree poins A, B and C, he disances AB and BC being equal. If he paricle akes 3 s o pass from A o B and s from B o C, calculae AB. 9. A body falls freely from res from a poin O passing hree poins A, B and C, he disances AB and BC being equal. The ime aken o go from A o B is s and from B o C is s. Find AB. 0. A paricle falls freely under graviy from res a a poin P. Afer i has fallen for s anoher paricle is projeced verically downwards from P wih speed 4.7 m s. Find he ime and disance from P a which hey collide. The Dublin School of Grinds Page 47 Kieran Mills & Tony Kelly