Introduction to Mechnics Prctice using the Kinemtics Equtions Ln Sheridn De Anz College Jn 24, 2018
Lst time finished deriing the kinemtics equtions some problem soling prctice
Oeriew using kinemtics equtions more problem soling prctice free fll (?)
The Kinemtics Equtions For constnt ccelertion: = 0 + t = 0 t + 1 2 t2 = t 1 2 t2 = 0 + t 2 2 = 2 0 + 2 For zero ccelertion: = t
Using the Kinemtics Equtions to sole problems Process: 1 Identify which quntity we need to find nd which ones we re gien. 2 Is there quntity tht we re not gien nd re not sked for? 1 If so, use the eqution tht does not include tht quntity. 2 If there is not, more tht one kinemtics eqution my be required or there my be seerl equilent pproches. 3 Input known quntities nd sole.
Emple 4 A cr drier sees speed trp hed, while driing t 30.0 m/s ( 67 mph). Assume the cr brkes with constnt decelertion of 2.30 m/s 2. It slows to 20.0 m/s, wht ws the time tken to slow?
eed of the object increses. In this cse, the elocity nd ccelertion int in the sme direction. Emple 4 hen the elocity nd ccelertion of n object he opposite signs, the eed of the object decreses. In this cse, the elocity nd ccelertion int in opposite directions. A cr drier sees speed trp hed, while driing t 30.0 m/s ( 67 mph). Assume the cr brkes with constnt decelertion of 2.30 m/s 2. It slows to 20.0 m/s, wht ws the time tken to slow? o possibilities re illustrted in Figure 2 11. Notice tht when prticle s reses, it mens either tht its elocity becomes more positie, s in 11 (), or more negtie, s in Figure 2 11 (d). In either cse, it is the e of the elocity the speed tht increses. Sketch: () (b) (c) (d)
eed of the object increses. In this cse, the elocity nd ccelertion int in the sme direction. Emple 4 hen the elocity nd ccelertion of n object he opposite signs, the eed of the object decreses. In this cse, the elocity nd ccelertion int in opposite directions. A cr drier sees speed trp hed, while driing t 30.0 m/s ( 67 mph). Assume the cr brkes with constnt decelertion of 2.30 m/s 2. It slows to 20.0 m/s, wht ws the time tken to slow? o possibilities re illustrted in Figure 2 11. Notice tht when prticle s reses, it mens either tht its elocity becomes more positie, s in 11 (), or more negtie, s in Figure 2 11 (d). In either cse, it is the e of the elocity the speed tht increses. Sketch: () (b) Hypothesis: He s slowing by 10 m/s t bout 2 m/s 2, perhps round 10 2 = 5 s. (c) (d)
eed of the object increses. In this cse, the elocity nd ccelertion int in the sme direction. Emple 4 hen the elocity nd ccelertion of n object he opposite signs, the eed of the object decreses. In this cse, the elocity nd ccelertion int in opposite directions. A cr drier sees speed trp hed, while driing t 30.0 m/s ( 67 mph). Assume the cr brkes with constnt decelertion of 2.30 m/s 2. It slows to 20.0 m/s, wht ws the time tken to slow? o possibilities re illustrted in Figure 2 11. Notice tht when prticle s reses, it mens either tht its elocity becomes more positie, s in 11 (), or more negtie, s in Figure 2 11 (d). In either cse, it is the e of the elocity the speed tht increses. Sketch: () (b) Hypothesis: He s slowing by 10 m/s t bout 2 m/s 2, perhps (c) round 10 2 = 5 s. Gien: 0 = +30.0 m/s i, = +20.0 (d) m/s i, = 2.30 m/s 2 i Asked for: t
Emple 4 Strtegy: Use = 0 + t
Emple 4 Strtegy: Use = 0 + t Rerrnge: t = 0 20.0 m/s 30.0 m/s = 2.30 m/s 2 = 4.35 s Resonble / Agrees with hypothesis: yes, just bit smller. Tht mkes sense becuse the ccelertion hd mgnitude 2.30 m/s 2 rther thn 2 m/s 2. It is resonble mount of time to need to slow for speed trp.
Emple 5 A cr drier sees n obstcle in the rod nd pplies the brkes. It tkes him 4.33 s to stop the cr oer distnce of 55.0 m. Assuming the cr brkes with constnt ccelertion, wht ws the cr s decelertion?
int in the sme direction. hen the elocity nd ccelertion of n object he opposite signs, the eed of the object decreses. In this cse, the elocity nd ccelertion Emple 5 int in opposite directions. A cr drier sees n obstcle in the rod nd pplies the brkes. It tkes him 4.33 s to stop the cr oer distnce of 55.0 m. Assuming the cr brkes with constnt ccelertion, wht ws the cr s decelertion? o possibilities re illustrted in Figure 2 11. Notice tht when prticle s reses, it mens either tht its elocity becomes more positie, s in 11 (), or more negtie, s in Figure 2 11 (d). In either cse, it is the e of the elocity the speed tht increses. Sketch: () (b) (c) (d)
int in the sme direction. hen the elocity nd ccelertion of n object he opposite signs, the eed of the object decreses. In this cse, the elocity nd ccelertion Emple 5 int in opposite directions. A cr drier sees n obstcle in the rod nd pplies the brkes. It tkes him 4.33 s to stop the cr oer distnce of 55.0 m. Assuming the cr brkes with constnt ccelertion, wht ws the cr s decelertion? o possibilities re illustrted in Figure 2 11. Notice tht when prticle s reses, it mens either tht its elocity becomes more positie, s in 11 (), or more negtie, s in Figure 2 11 (d). In either cse, it is the e of the elocity the speed tht increses. Sketch: () (b) Hypothesis: 55 m is not ery fr nd4.33 s is not long time: the decelertion must be high-ish. I would guess mybe 5 m/s 2 which is bout hlf of g. (c) (d)
int in the sme direction. hen the elocity nd ccelertion of n object he opposite signs, the eed of the object decreses. In this cse, the elocity nd ccelertion Emple 5 int in opposite directions. A cr drier sees n obstcle in the rod nd pplies the brkes. It tkes him 4.33 s to stop the cr oer distnce of 55.0 m. Assuming the cr brkes with constnt ccelertion, wht ws the cr s decelertion? o possibilities re illustrted in Figure 2 11. Notice tht when prticle s reses, it mens either tht its elocity becomes more positie, s in 11 (), or more negtie, s in Figure 2 11 (d). In either cse, it is the e of the elocity the speed tht increses. Sketch: () (c) (b) Hypothesis: 55 m is not ery fr nd4.33 s is not long time: the decelertion must be high-ish. I would guess mybe 5 m/s 2 which is bout hlf of g. Gien: t = 4.33 s, = 55.0 m, = 0 m/s Asked for: (d)
int in the sme direction. hen the elocity nd ccelertion of n object he opposite signs, the eed of the object decreses. In this cse, the elocity nd ccelertion Emple 5 int in opposite directions. A cr drier sees n obstcle in the rod nd pplies the brkes. It tkes him 4.33 s to stop the cr oer distnce of 55.0 m. Assuming the cr brkes with constnt ccelertion, wht ws the cr s decelertion? o possibilities re illustrted in Figure 2 11. Notice tht when prticle s reses, it mens either tht its elocity becomes more positie, s in 11 (), or more negtie, s in Figure 2 11 (d). In either cse, it is the e of the elocity the speed tht increses. Sketch: () (c) (b) Hypothesis: 55 m is not ery fr nd4.33 s is not long time: the decelertion must be high-ish. I would guess mybe 5 m/s 2 which is bout hlf of g. Gien: t = 4.33 s, = 55.0 m, = 0 m/s Asked for: Strtegy: use (d) = t 1 2 t2
Emple 5 A cr drier sees n obstcle in the rod nd pplies the brkes. It tkes him 4.33 s to stop the cr oer distnce of 55.0 m. Assuming the cr brkes with constnt ccelertion, wht ws the cr s decelertion? = t 1 2 t2 1 2 t2 = t = 2(t ) t 2
Emple 5 A cr drier sees n obstcle in the rod nd pplies the brkes. It tkes him 4.33 s to stop the cr oer distnce of 55.0 m. Assuming the cr brkes with constnt ccelertion, wht ws the cr s decelertion? = t 1 2 t2 1 2 t2 = t = = 2(t ) t 2 2(0 55.0 mi) (4.33 s) 2 = 5.87 m/s 2 i Or, the cr s ccelertion is 5.87 m/s 2, opposite the direction of the cr s trel.
Emple 5 Resonble / Agrees with hypothesis: This is lrge decelertion, but cr cn mnge it. It is resonble considering he needs to stop before the obstcle nd is breking hrd.
ow will tht chnge the nswer to prt (B)? Emple 6 stop frther wy from its strting point, so the nswer to n 2.15 Atht cr if treling i is lrger, t constnt then f will speed be of lrger. 45.0 m/s psses trooper on motorcycle hidden behind billbord. One second fter the speeding cr psses the billbord, the trooper sets out from the billbord to ctch the cr, ccelerting t constnt rte of 3.00 m/s 2. How long does it tke the trooper to oertke the cr? - AM Sketch: t 1.00 s t 0 Figure 2.13 (Emple 2.8) A speeding cr psses hidt? + - 1 Serwy & Jewett, Physics for Scientists nd Engineers, pg 39.
Emple 6 Hypothesis: The trooper ccelertes t 3.00 m/s 2. It will tke 15 s for him to rech speed of 45.0 m/s, but then he still needs to ctch up to the cr. I would guess it will tke him bout nother 15 s to ctch up. Guess: bout 30 s or so.
Emple 6 Hypothesis: The trooper ccelertes t 3.00 m/s 2. It will tke 15 s for him to rech speed of 45.0 m/s, but then he still needs to ctch up to the cr. I would guess it will tke him bout nother 15 s to ctch up. Guess: bout 30 s or so. Gien: cr = 45.0 m/s, ( cr = 0), tr = 3.00 m/s 2, tr,0 = 0 m/s, nd the cr hs 1 s hed strt Asked for: t when they re t the sme positition.
Emple 6 Hypothesis: The trooper ccelertes t 3.00 m/s 2. It will tke 15 s for him to rech speed of 45.0 m/s, but then he still needs to ctch up to the cr. I would guess it will tke him bout nother 15 s to ctch up. Guess: bout 30 s or so. Gien: cr = 45.0 m/s, ( cr = 0), tr = 3.00 m/s 2, tr,0 = 0 m/s, nd the cr hs 1 s hed strt Asked for: t when they re t the sme positition. Strtegy: Use the ide tht when the trooper hs cught up cr = tr We cn isulize this on grph, or just do bit of lgebr.
Emple 6 A cr treling t constnt speed of 45.0 m/s psses trooper on motorcycle hidden behind billbord. One second fter the speeding cr psses the billbord, the trooper sets out from the billbord to ctch the cr, ccelerting t constnt rte of 3.00 m/s 2. How long does it tke the trooper to oertke the cr? cr = tr cr (t + 1) i = ( 0 ( tr,0 )t + 1 2 trt 2) i This is qudrtic epression in t. 0 = 1 2 trt 2 cr t cr
Emple 6 A cr treling t constnt speed of 45.0 m/s psses trooper on motorcycle hidden behind billbord. One second fter the speeding cr psses the billbord, the trooper sets out from the billbord to ctch the cr, ccelerting t constnt rte of 3.00 m/s 2. How long does it tke the trooper to oertke the cr? 0 = 1 2 trt 2 cr t cr This is qudrtic epression in t. 0 = 1 2 trt 2 cr t cr t = cr ± 2 cr + 2 tr cr tr t = 31.0 s
Emple 6 Answer: t = 31.0 s Anlysis: This nswer is quite close to the hypothesis! The cr is going 45.0 m/s, or 100 mi/h, which is ery fst, but the trooper lso hs high ccelertion of 3.00 m/s 2. Motorcycles cn he ccelertions een higher thn tht, so the numbers in this question re resonble.
Free-Flling Objects One common scenrio of interest where ccelertion is constnt is objects freely flling. When we refer to free fll, we men objects moing under the influence of grity, nd where we re ignoring resistie forces, eg. ir resistnce.
Glileo nd the Lening Tower of Pis Aristotle, n erly Greek nturl philosopher, sid tht heier objects fll fster thn lighter ones. Glileo tested this ide nd found it ws wrong. Any two mssie objects ccelerte t the sme rte. Glileo studied the motion of objects by eperiment, s well s by bstrct resoning.
Glileo nd Inerti He considered blls rolling on inclined surfces nd deeloped the notion of inerti.
Glileo nd Inerti He considered blls rolling on inclined surfces nd deeloped the notion of inerti. Inerti is the tendency of objects to sty doing whteer they re lredy doing, unless they re interfered with. Glileo s ide of inerti: A body moing on leel surfce will continue in the sme direction t constnt speed unless disturbed.
Accelertion nd Free-Fll Glileo resoned bout the ccelertion due to grity by thinking more bout inclined surfces. h θ O The steeper the incline the lrger the ccelertion. The finl elocity of the bll is lso lrger. Strting from rest: = t
Free-Fll When the bll drops stright downwrd, it gins pproimtely 10 m/s of speed in ech second. Time of fll (s) Velocity cquired (m/s) 0 0 1 10 2 20 3 30. This is constnt ccelertion! We cll this ccelertion g.. g = 9.8 m s 2 10 m s 2
Free-Fll When the bll drops stright downwrd, it gins pproimtely 10 m/s of speed in ech second. Time of fll (s) Velocity cquired (m/s) 0 0 1 10 2 20 3 30. This is constnt ccelertion! We cll this ccelertion g.. g = 9.8 m s 2 10 m s 2 After 6.5 s, (roughly) wht is the bll s speed?
Free-Fll When the bll drops stright downwrd, it gins pproimtely 10 m/s of speed in ech second. Time of fll (s) Velocity cquired (m/s) 0 0 1 10 2 20 3 30. This is constnt ccelertion! We cll this ccelertion g.. g = 9.8 m s 2 10 m s 2 After 6.5 s, (roughly) wht is the bll s speed? 65 m/s
Free-Flling Objects The importnt point is tht t the surfce of the Erth, ll objects eperience this sme ccelertion due to grity: g = 9.8 m s 2. 1 Figure from Wlker, Physics, pge 39.
Free-Flling Objects The importnt point is tht t the surfce of the Erth, ll objects eperience this sme ccelertion due to grity: g = 9.8 m s 2. In the bsence of ir resistnce, the ccelertion does not depend IONAL KINEMATICS on n object s mss! ir nd rubpping bll. () (b) The fct tht ccelertion due to grity is independent of mss cn be seen in irless enironments... moe with constnt ccelertion. His conclusions were bsed on eperiments done by rolling blls down inclines of rious steepness. By using n incline, 1 Figure Glileo fromws Wlker, ble to Physics, reduce the pge ccelertion 39. of the blls, thus producing motion
Free-Flling Objects Objects ner the Erth s surfce he constnt ccelertion of g = 9.8 ms 2. (Or, bout 10 ms 2 ) The kinemtics equtions for constnt ccelertion ll pply.
Summry prctice using the kinemtics equtions free fll (?) Quiz Mondy, strt of clss. First Test net Thursdy, Feb 1. Homework Plese bring ruler to clss tomorrow! Wlker Physics: Ch 2, onwrd from pge 14. Problems: 61, 63