Motion along a Straight Line

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1 chaper 2 Moion along a Sraigh Line verage speed and average velociy (Secion 2.2) 1. Velociy versus speed Cone in he ebook: fer Eample 2. Insananeous velociy and insananeous acceleraion (Secions 2.3, 2.4) 2. cceleraion curve Moion wih consan acceleraion (Secion 2.5) 3. Displacemen curves 4. Collision ime wihou deceleraion The acceleraion of free fall (Secion 2.6) 5. cceleraion a he op 6. Maimum heigh 7. Rising ball 8. Timing a rising ball by heigh 9. ball passes by a window Supplemenary problems for his secion. Inegraion of equaions of moion (Secion 2.7) 10. verage velociy This is an eercise on he noion ha disance raveled is he area under velociy curve as implied by Eq. (2.35). Conens PhysiQuiz 9

2 The following figure shows he displacemen versus ime graph of a paricle moving along he ais: (m) 2 O 1 2 B (sec) Deermine he average velociy and he average speed (in unis of m/s) of he paricle along he whole pah shown from O o o B. B C D verage velociy verage speed Era: Deermine he insananeous velociy and he insananeous speed a B. Eplanaion: The average velociy is given by v displ acemen B O 0. ime B O The average speed is given by average speed dis ance O B ime B O nswer B. 2 m/s. Eplanaion era: B he insananeous velociy is given by he slope of he displacemen curve. From he plo we see ha he slope is given by v ( B )/( B ) (0 2)/(2 1) 2 m/s. The insananeous speed is he magniude of he velociy, which is 2 m/s. 10 PhysiQuiz 1. Velociy versus Speed

3 a P Q O The diagram describes he acceleraion versus ime behavior for a car moving in he direcion. he poin P he car is moving how? B C Wih an increasing velociy Wih a consan velociy Wih a decreasing velociy Era: Describe he change of velociy a Q. Eplanaion: a dv/d. s long as he acceleraion is posiive, he velociy is always increasing. nswer. Eplanaion era: Here again a 0, so he velociy coninues o increase. 2. cceleraion Curve PhysiQuiz 11

4 car saring from he origin is moving along he posiive direcion wih a consan velociy. Which graph correcly describes he coordinae versus ime for his moion? B C D E Era: Which graph represens he coordinae of a consan acceleraion saring from 0? Eplanaion: The moion along he ais wih a consan velociy v is described by v. nswer B. Eplanaion era: consan acceleraion saring from 0 is described by 1 2 a 2. nswer C. 12 PhysiQuiz 3. Displacemen Curves

5 Refer o Eample 8 of Secion 2.5 in he ebook. The iniial velociy of he minivan is v m/s; he iniial velociy of he ruck is v m/s; and he iniial disance beween he wo vehicles is 0 12 m. Find an epression for he collision ime if he minivan does no decelerae. Choose one of he following: B C D 0 v0 0 v 0 0 v0 v 0 0 v0 v 0 Era: Verify ha he collision ime obained here (wihou any deceleraion) is shorer han he collision ime when he brake is applied in Eample 8, which is c 1.1 s. Eplanaion: In Eample 8, if we se a 0, he collision condiion leads o he equaion v 0 0 v 0. Solving for gives 0 /(v 0 v 0 ). nswer C. Eplanaion era: Using he numbers given, 0 /(v 0 v 0 ) 12/(22 6.9) 0.8 s c 1.1 s, which shows ha when he brake is applied, he collision akes place a a laer ime. 4. Collision Time wihou Deceleraion PhysiQuiz 13

6 When a sone is hrown sraigh upward, is velociy a is highes poin is zero. Wha is is acceleraion a his poin? B C cceleraion 9.8 m/s m/s 2 a op down up Eplanaion: Near he surface of he Earh, for all pracical purposes he graviaional acceleraion is a consan 9.8 m/s 2 poining downward. nswer. To illusrae how his works, le us ake for eample an upward iniial velociy of 9.8 m/s. fer he firs second he velociy will be 0. fer he second second he velociy will be 9.8 m/s, bu heading down. This can be wrien as 9.8 m/s, where he minus sign indicaes he downward velociy. In oher words, in each second he velociy is decreased by 9.8 m/s. 14 PhysiQuiz 5. cceleraion a he Top

7 B h υ 0 O Find he maimum heigh h (in m) of a ball hrown verically upward wih an iniial velociy v 0 30 m/s (assume g 10 m/s 2 : B C h Hin: Wih he graviaional acceleraion a g, is velociy and is displacemen a locaions O and B are Locaion Velociy Displacemen O v 0 O B 0 h Rewrie Eq. (2.25) of Secion 2.5 as v 2 v 0 2 2ay. Era: Compare he acceleraion of he ball a he poin O (jus afer being released from he hrower) o ha a he poin B. Eplanaion: B, y h. his poin v 2 v 0 2 2ah gives 0 v 0 2 2( g)h. So h v 0 2 /(2g) 30 2 /(2 10) 45 m. nswer. Eplanaion era: The acceleraion hroughou he moion is a g, including boh he iniial poin O and B a he op. 6. Maimum Heigh PhysiQuiz 15

8 rise, h υ 0 O rise /4 Consider a rising ball problem. The iniial velociy of he ball is v 0. The ime i akes for i o rise o is maimum heigh h is rise v 0 /g. Deermine he velociy v a some inermediae poin, where O rise /4: B C D v v 0 /4 v 0 /2 3v 0 /4 v 0 Era: How high is he poin? Epress O in erms of h. Eplanaion:, v v 0 g( rise /4), where rise v 0 /g. This implies ha v v 0 gv 0 /(4g) 3v 0 /4. nswer C. Eplanaion era: To find O we use he relaion v 2 v 0 2 2ay (see Eq in he e for he presen case, he one dimensional coordinae variable is replaced by y, and he relaion 0 is used.). his relaion gives (3v 0 /4) 2 v 0 2 2g(O). Solving for O gives O (1 9/16)v 0 2 /(2g) 7h/16. In he las sep, h v 0 2 /(2g) was used. 16 PhysiQuiz 7. Rising Ball

9 y B h 2/3 h ball is hrown verically upward a 0 from he ground a poin O. I reaches a maimum heigh h a poin. The heigh a poin B is 2h/3. Find he raio of B o O (ha is, he raio of he ime o go from poin B o o he ime o go from poin O o ): B C D B / O 1/3 1/ 3 1/2 (2/3) Era: Show ha v B (2/3)g h. O Eplanaion: Consider he fall of he ball saring a res a. Eq. (2.22) implies ha B 1 2 g B 2, or B 2 /g, B and O 2 /g. O Thus B / O B/ D (k/3)/ h 1/ 3. Eq (2.25) ogeher wih Eq (2.23) implies ha a any heigh he upward speed is he same as he downward speed. In oher words, a any poin he ime aken for he ball o reach he op is he same as he ime aken for he ball o fall from he op o ha poin. In oher words, B / O B / O 1/ 3. nswer B. Eplanaion era: v 2 B 2g(B) 2g(h/3), or v B (2/3)g h. 8. Timing a Rising Ball by Heigh PhysiQuiz 17

10 C B d O Consider a ball hrown upward from he ground a poin O. I passes by a window wih heigh B d in he ime inerval B T. The choice of he correc pair among he following four equaions will le us solve for he wo unknowns v and v B. The equaions are (1) v v B 2d/T (2) v v B gt (3) v v B d/t (4) v B v gt B C D Correc pair (1) and (2) (2) and (3) (3) and (4) (1) and (4) Era: Given d 2 m and T 1 s, find v. ssume g 10 m/s 2. Eplanaion: The average velociy is given by 1 2 (v v B ) d/t. So (1) is correc and (3) is incorrec. Because v v B, (2) is correc and (4) is incorrec. nswer. Eplanaion era: [Eq. (1) Eq. (2)]/2 v 0.5(2d/T gt ) 0.5(2 2/ ) 4.5 m/s. 18 PhysiQuiz 9. Ball Passes by a Window

11 Consider he hree velociy curves beween poins and B shown here: Choose he correc relaionship among quaniies v 1, v 2, v 3, and (v v B )/2, where v 1, v 2, and v 3 represen he average velociies over he respecive pahs: υ υ B υ B 0 B v 1 v 2 (v v B )/2 v 3 B v 1 v 2 (v v B )/2 v 3 C v 1 v 2 (v v B )/2 v 3 Eplanaion: v displ acemen area under v versus curve ime B By inspecion area 1 area 2 area 3 ; in urn v1 v2 v3. nswer B. 10. verage Velociy PhysiQuiz 19

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