Student Session Topic: Particle Motion

Student Session Topic: Prticle Motion 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 mny 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 ccumultion problems using either t1 1 0 () x t x t v t dt or t0 n initil vlue problem. t1 1 0 () v t v t t dt. This is often esier thn treting the sitution s t0 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. b The totl distnce trveled t velocity v(t ) from t = to t = b is given by v() t dt b (displcement) over the sme intervl is v() t dt.. The net distnce Don t be reluctnt to use your grphing clcultor for either of these computtions to clculte (t ) = v (t ). Pge 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 v (t ) = 0 Prticle moving right (forwrd or up) v (t ) > 0 Prticle moving left (bckwrd or down) v (t ) < 0 1 1 ( ) ( ) Averge velocity on [,b] 1 b x b x v( t) dt x( t) b b b Instntneous velocity t time t == c v(c) = x (c) b Accelertion t time t = c (c) = v (c) = x (c) Velocity is incresing (t) = v ' (t) > 0 Velocity is decresing (t) = v '(t) < 0 Speed (v )t Speed is incresing (speeding up) v (t ) nd (t) hve sme sign (both + or both -) Speed is decresing (slowing down) v (t ) nd (t) hve different signs b Totl distnce trveled on [, b] v() t dt Absolute vlue is importnt! b Net distnce trveled (displcement) v() t dt b Position of object t time () Prticle is frthest left (right) x b x v t dt Compre positions (x-vlues) t endpoints & t locl minim (mxim). 1 Do NOT think v( ) v( b) 2 Pge 2

Multiple Choice Non-Clcultor. Select the best nswer Questions 1 nd 2 refer to the digrm nd sitution below. A bug begins to crwl up verticl wire t time t = 0. The velocity v of the bug t time t, 0t 8, is given by the function whose grph is shown bove. 1. At wht vlue of t does the bug chnge direction? A. 2 B. 4 C. 6 D. 7 E. 8 2. Wht is the totl distnce the bug trveled from t = 0 to t = 8? A. 14 B. 13 C. 11 D. 8 E. 6 3. A prticle moves long the x-xis so tht its position t time t is given by vlue of t is the velocity of the prticle zero? 2 x( t) t 6t 5. For wht A. 1 B. 2 C. 3 D. 4 E. 5 4. The mximum ccelertion ttined on the intervl 0t 3 by the prticle whose velocity is given 3 2 by v( t) t 3t 12t 4 is A. 9 B. 12 C. 14 D. 21 E. 40 5. A prticle moves long the x-xis so tht its ccelertion t ny time t is ( t) 2t 7. If the initil velocity of the prticle is 6, t wht time t during the intervl 0t 4 is the prticle frthest to the right? A. 0 B. 1 C. 2 D. 3 E. 4 Pge 5

6. A prticle moves long the x-xis so tht t ny time t 0, its position is given by 3 2 x( t) 2t 21t 72t 53. At wht time t is the prticle t rest? A. t = 1 only B. t = 3 only C. D. 7 t 3 nd t E. t 3 nd t 4 2 7 t only 2 Clcultor. Select the best nswer. 7. At time t 0, the ccelertion of prticle moving on the x-xis is ( t) t sin t. At t = 0, the velocity of the prticle is -2. For wht vlue of t will the velocity of the prticle be zero? A. 1.02 B. 1.48 C. 1.85 D. 2.81 E. 3.14 8. A prticle moves long the x-xis so tht t ny time t 0, its velocity is given by v( t) 3 4.1cos 0.9t. Wht is the ccelertion of the prticle t time t = 4? A. -2.016 B. -0.677 C. 1.633 D. 1.814 E. 2.978 1 1, where t is 6 4 time in seconds. In the first 4 seconds, how mny times is the velocity of the object equl to 0? 9. The position of n object ttched to spring is given by y( t) cos 5t sin 5t A. Zero B. Three C. Five D. Six E. Seven 10. A prticle moves long the x-xis so tht t ny time 0 2 t, its velocity is given by v( t) cos 2 t. The position of the prticle is 3 t time t = 0. Wht is the position of the prticle when its velocity is first equl to 0? A. 0.411 B. 1.310 C. 2.816 D. 0.091 E. 3.411 2 2 11. The height, h, in meters of n object t time t is given by h( t) 24t 24t 16t. Wht is the height of the object t the instnt when it reches its mximum upwrd velocity? A. 2.545 meters B. 10.263 meters C. 34.125 meters D. 54.889 meters E. 89.005 meters 3 Pge 6

x 0 1 2 3 4 vt () -1 2 3 0-4 12. The tble gives selected vlues of velocity, v(t), of prticle moving long the x-xis. At time t = 0, the prticle is t the origin. Which of the following could be the grph of the position, x(t), of the prticle for 0t 4? Pge 7