Student Notes Prep Session Topic: Prticle Motion Number Line for AB Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be given s n eqution, grph or tble. There re lot of different things students my be sked to find. While prticles my not relly move in this wy, the question is verstile wy to test vriety of clculus concepts. These questions my give the position eqution, or the velocity eqution or the ccelertion eqution long with n initil condition. Students my be sked bout the motion of the prticle: its direction, when it chnges direction, its mximum position in one direction, etc. Speed, the bsolute vlue of velocity, is lso common topic. Wht you should know how to do: Move esily between the position, velocity nd ccelertion equtions by differentiting or integrting. If you re given the velocity nd n initil position, or given the ccelertion nd n initil velocity, you re looking t differentil eqution initil vlue problem. Be sure you know how to do initil vlue problems. If you re given the velocity nd n initil position, or given the ccelertion nd n initil velocity, you my often be ble to pproch the problem s n ccumultion problems using t1 either ( ) ( ) ( ) t1 = + or ( ) ( ) ( ) x t x t v t dt 1 0 the sitution s n initil vlue problem. t0 vt = vt + t dt. This is often the esier tht treting 1 0 Speed is the bsolute vlue of velocity (it is not vector quntity). When the velocity nd ccelertion hve the sme sign the speed is incresing; if the signs re different the speed is decresing. Be sure you understnd why this is true. This is common topic on the AP exms. Grphiclly, speed is the non directed distnce from the velocity grph to the t xis. If the distnce of the velocity is incresing the speed is incresing. Reflecting the prts of the velocity grph tht lie below the t xis, will give you the grph of the speed. The totl distnce trveled t velocity ( ) By Lin Mcmullin 2010 Ntionl Mth nd Science Inititive Pge 1 AP* is trdemrk of the College Bord. The College Bord ws not involved in the production of this product. t0 b vt from t = to t = b is given by vt () dt b net distnce (displcement) over the sme intervl is vtdt ().. The Don t be reluctnt to use your grphing clcultor for either of the computtions in the previous bullet nd to clculte t ( ) v ( t) =. 1
Student Notes Prticle Motion Contributed by Teres Trter, Bob Jones HS, Huntsville, AL Wht you need to know bout motion long the x-xis: When you see Think Initilly t = 0 At rest vt ( ) = 0 Prticle moving right (forwrd or up) vt ( ) > 0 Prticle moving left (bckwrd or down) vt ( ) < 0 Averge velocity on [,b] 1 b vtdt () xt () Instntneous velocity t time t = v( ) = x ( ) ( ) x( ) 1 1 x b = = b b b Accelertion t time t = c ( c) = v ( c) = x ( c) Velocity is incresing t ( ) = v ( ) > 0 Velocity is decresing t ( ) = v ( ) < 0 Speed v( t ) Speed is incresing vt ( ) nd t ( ) hve sme sign (both + or both -) Speed is decresing vt ( ) nd t ( ) hve different signs b Totl distnce trveled on [, b] vt () dtabsolute vlue is importnt! b Net distnce trveled vtdt () b Position of object t time t = b ( ) ( ) ( ) Prticle is frthest left (right) x b x v t dt = + Compre positions (x-vlues) t endpoints & t locl minim (mxim). b 1 Do NOT think ( ) + v( b) v 2 By Lin Mcmullin 2010 Ntionl Mth nd Science Inititive Pge 2 AP* is trdemrk of the College Bord. The College Bord ws not involved in the production of this product. 2
Student Notes Free response questions: 2002 AB3 Anlytic stem 2008 AB4/BC4 Grph stem 2005 AB5 Grph stem 2006 AB 4 Tble stem Multiple choice Selection: 2, 4, 5, 6, 7, 9, 10 No clcultor: 1 6; Grphing clcultor llowed: 7 12. Answers: 1 C, 2 B, 3 C, 4 D, 5 B, 6 E, 7 E, 8 B, 9 C, 10 D, 11 C, 12 B WATCH nd LISTEN to the multiple-choice questions being solved Go to http://tinyurl.com/nmsi-mth-9 Click on the Full Screen rrow. Then click nywhere on the pge to see nd her from tht point on. Click nywhere to go bck nytime. By Lin Mcmullin 2010 Ntionl Mth nd Science Inititive Pge 3 AP* is trdemrk of the College Bord. The College Bord ws not involved in the production of this product. 3
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AP CALCULUS AB 2006 SCORING GUIDELINES Question 4 t (seconds) vt () (feet per second) 0 10 20 30 40 50 60 70 80 5 14 22 29 35 40 44 47 49 Rocket A hs positive velocity vt () fter being lunched upwrd from n initil height of 0 feet t time t = 0 seconds. The velocity of the rocket is recorded for selected vlues of t over the intervl 0 t 80 seconds, s shown in the tble bove. () Find the verge ccelertion of rocket A over the time intervl 0 t 80 seconds. Indicte units of mesure. 70 (b) Using correct units, explin the mening of vt () dtin terms of the rocket s flight. Use midpoint Riemnn sum with 3 subintervls of equl length to pproximte vt () dt. 10 3 (c) Rocket B is lunched upwrd with n ccelertion of t () = feet per second per second. At time t + 1 t = 0 seconds, the initil height of the rocket is 0 feet, nd the initil velocity is 2 feet per second. Which of the two rockets is trveling fster t time t = 80 seconds? Explin your nswer. 70 10 () Averge ccelertion of rocket A is 1 : nswer v( 80) v( 0) 49 5 11 ft sec 2 = = 80 0 80 20 (b) Since the velocity is positive, vt () dtrepresents the distnce, in feet, trveled by rocket A from t = 10 seconds to t = 70 seconds. 70 10 1 : explntion 3 : 1 : uses v( 20 ), v( 40 ), v( 60) 1 : vlue A midpoint Riemnn sum is 20[ v( 20) + v( 40) + v( 60) ] = 20[ 22 + 35 + 44] = 2020 ft (c) Let vb () t be the velocity of rocket B t time t. 3 vb () t = dt = 6 t + 1 + C t + 1 2 = v ( 0) = 6 + C B vb () t = 6 t + 1 4 v ( 80) = 50 > 49 = v( 80) B 4 : 1 : 6 t + 1 1 : constnt of integrtion 1 : uses initil condition 1 : finds vb ( 80 ), compres to v( 80 ), nd drws conclusion Rocket B is trveling fster t time t = 80 seconds. Units of 2 ft sec in () nd ft in (b) 1 : units in () nd (b) 2006 The College Bord. All rights reserved. Visit pcentrl.collegebord.com (for AP professionls) nd www.collegebord.com/pstudents (for AP students nd prents). 5 8
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