Chapter The Big Three: Acceleratin, Distance, and Time In This Chapter Thinking abut displacement Checking ut speed Remembering acceleratin Being able t cnnect displacement, speed, and acceleratin is undamental t wrking with physics. These things cncern peple every day, and physics has made an rganized study them. Prblems that cnnect displacement, speed, and acceleratin are all abut understanding mvement, and that s the tpic this chapter putting numbers int the discussin. Yu ll ten ind physics prblems abut cars starting and stpping, hrses racing, and rcket ships zming back and rth. And ater yu inish this chapter, yu ll be a real pr at slving them. Frm Pint A t B: Displacement Displacement ccurs when smething mves rm here t there. Fr example, suppse that yu have a ball at the zer psitin, as in Figure -1A. A 4 3 1 0 1 3 4 meters Figure -1: A mving ball. B 4 3 1 0 1 3 4 meters Nw suppse that the ball rlls ver t a new pint, 3 meters t the right, as yu see in Figure -1B. The ball is at a new lcatin, s there s been displacement. In this case, the displacement is just 3 meters t the right. In physics terms, yu ll ten see displacement reerred t as the variable s. In this case, s = +3 meters.
6 Part I: Applying Physics Like any ther measurement in physics, displacement is always expressed in units, usually centimeters r meters, as in this example. O curse, yu als can use kilmeters, inches, eet, miles, r even light years (the distance light travels in ne year 5,865,696,000,000 miles). The llwing example questin cuses n displacement. Q. Yu ve taken the pineers advice t G West. Yu started in New Yrk City and went west 10 miles the irst day, 14 miles the next day, and then back east 9 miles n the third day. What is yur displacement rm New Yrk City ater three days? A. s = 15 miles west New Yrk City 1. Yu irst went west 10 miles, s at the end the irst day, yur displacement was 10 miles west.. Next, yu went west 14 days, putting yur displacement at 10 + 14 miles = 4 miles west New Yrk City. 3. Finally, yu traveled 9 miles east, leaving yu at 4 9 = 15 miles west New Yrk City. S s = 15 miles west New Yrk City. 1. Suppse that the ball in Figure -1 nw mves 1 meter t the right. What is its new displacement rm the rigin, 0?. Suppse that the ball in Figure -1, which started 4 meters t the right the rigin, mves 6 meters t the let. What is its new displacement rm the rigin in inches?
Reading That Speedmeter Chapter : The Big Three: Acceleratin, Distance, and Time In physics terms, what is speed? It s the same as the cnventinal idea speed: Speed is displacement divided by time. 7 Fr example, i yu went a displacement s in a time t, then yur speed, v, is determined as llws: v = s t Technically speaking, speed is the change in psitin divided by the change in time, s yu als can represent it like this i, r example, yu re mving alng the x axis: v x x- x = = t - t Q. Suppse that yu want t drive rm New Yrk City t Ls Angeles t visit yur uncle s amily, a distance abut 781 miles. The trip takes yu ur days. What was yur speed in miles per hur? A. v x = = =. x - x t - t 8 97 miles per hur 1. Start by iguring ut yur speed (the distance traveled divided by the time taken t travel that distance): 781 miles = 695. 5 4 days. Okay, the speed is 695.5, but 695.5 what? This slutin divides miles by days, s it s 695.5 miles per day nt exactly a standard unit measurement. S what is that in miles per hur? T determine that, yu cancel days ut this equatin and put in hurs. Because 4 hurs are in a day, yu can multiply as llws (nte that days cancel ut, leaving miles ver hurs, r miles per hur): 781 miles 1 day # = 8. 97 miles per hur 4 days 4 hurs S yur speed was 8.97 miles per hur. That s yur average speed, averaged ver bth day and night.
8 Part I: Applying Physics 3. Suppse that yu used yur new SpeedPass t get yu thrugh the tllbths at bth ends yur trip, which was 90 miles n the turnpike and tk yu 1 hur and 15 minutes. On yur return hme, yu re surprised t ind a traic ticket r speeding in the mail. Hw ast did yu g, n average, between the tllbths? Was the turnpike authrity justiied in sending yu a ticket, given that the speed limit was 65 mph? 4. Suppse that yu and a riend are determined t ind ut whse car is aster. Yu bth start yur trips in Chicag. Driving nnstp, yu reach Ls Angeles a distance 018 miles in 1.9 days, and yur riend, als driving nnstp, reaches Miami a distance 1380 miles in 0.89 days. Whse car was aster? Putting Pedal t Metal: Acceleratin In physics terms, acceleratin is the amunt by which yur speed changes in a given amunt time. In terms equatins, it wrks like this: a = v Given initial and inal velcities, v and v, and initial and inal times ver which yur speed changed, t and t, yu can als write the equatin like this: a v v - v = = t - t T get the units acceleratin, yu divide speed by time as llws: v - v a = t - t distance/ time = = distance time time Distance ver time squared? Dn t let that thrw yu. Yu end up with time squared in the denminatr just because it s velcity divided by time that s smething yu get used t when slving physics prblems. In ther wrds, acceleratin is the rate at which yur speed changes because rates have time in the denminatr. S r acceleratin, yu can expect t see units meters per secnd, r centimeters per secnd, r miles per secnd, r eet per secnd, r even kilmeters per hur.
Chapter : The Big Three: Acceleratin, Distance, and Time 9 Q. Suppse that yu re driving at 75 miles an hur and suddenly see red lashing lights in the rearview mirrr. Great, yu think, and yu pull ver, taking 0 secnds t cme t a stp. Yu culd calculate hw quickly yu decelerated as yu were pulled ver (inrmatin abut yur law-abiding tendencies that, n dubt, wuld impress the icer). S just hw ast did yu decelerate, in cm/sec? A. / sec a v 3350 cm = = = 168 cm/ sec 0 secnds 1. First cnvert t miles per secnd: 75 miles 1 hur 1 minute - # # =. 008 =. 08 # 10 miles per secnd hur 60 minutes 60 secnds. Cnvert rm miles per secnd t inches per secnd: - 08. # 10 miles 580 eet # # 1 inches = 1318 in secnd 1 mile 1 t secnd 3. Yur speed was 1318 inches per secnd. What s that in centimeters per secnd? - 08. # 10 miles 580 eet 1 inches # # #. 54 centimeters = 3350 cm secnd 1 mile 1 t inch secnd 4. What was yur acceleratin? That calculatin lks like this: a v v - v = = t - t 0 3350 cm/ secnd = - =-168 cm/ sec 0 secnds In ther wrds, 168 cm/sec, nt +168 cm/sec. There s a big dierence between psitive and negative in terms slving physics prblems and in terms law enrcement. I yu accelerated at +168 cm/sec instead accelerating at 168 cm/sec, yu d end up ging 150 miles per hur at the end 0 secnds, nt 0 miles per hur. In ther wrds, the sign the acceleratin tells yu hw the speed is changing. A psitive acceleratin means that the speed is increasing in the psitive directin, and a negative acceleratin (als knwn as deceleratin) tells yu that the speed is increasing in the negative directin.
30 Part I: Applying Physics 5. A rcket ship is ging t land n the mn in exactly hurs. There s nly ne prblem: It s ging 17,000 miles an hur. What des its deceleratin need t be, in miles per secnd, in rder t land n the mn saely at 0 miles per hur? 6. Yu re stpped at a red light when yu see a mnster SUV careening tward yu. In a lightning calculatin, yu determine yu have 0.8 secnds bere it hits yu and that yu must be ging at least 1.0 miles an hur rward at that time t avid the SUV. What must yur acceleratin be, in miles per hur? Can yu avid the SUV? 7. A bullet cmes t rest in a blck wd in 1.0 10 secnds, with an acceleratin 8.0 10 4 meters per secnd. What was its riginal speed, in meters per secnd? 8. The light turns red, and yu cme t a screeching halt. Checking yur stpwatch, yu see that yu stpped in 4.5 secnds. Yur deceleratin was 1.3 10 3 miles per secnd. What was yur riginal speed in miles per hur?
Cnnecting Acceleratin, Time, and Displacement Chapter : The Big Three: Acceleratin, Distance, and Time Yu knw that yu can relate speed with displacement and time. And yu knw that yu can relate speed and time t get acceleratin. Yu als can relate displacement with acceleratin and time: s= 1 a_ t-ti 31 I yu dn t start at zer speed, yu use this equatin: s= v t t 1 _ - i + a_ t-ti Q. Yu climb int yur drag racer, waving nnchalantly at the cheering crwd. Yu lk dwn the quarter-mile track, and suddenly the lag ges dwn. Yu re, getting a tremendus kick rm behind as the car accelerates quickly. A brie 5.5 secnds later, yu pass the end the curse and pp the chute. Yu knw the distance yu went: 0.5 miles, r abut 40 meters. And yu knw the time it tk: 5.5 secnds. S just hw hard was the kick yu gt the acceleratin when yu blasted dwn the track? A. 6.6 meters/secnd 1. Yu knw that 1 s= a t Yu can rearrange this equatin with a little algebra (just divide bth sides by t and multiply by ) t get a = s t. Plugging in the numbers, yu get a s ^40 metersh = = = 6. 6 meters/ secnd t ^55. secndsh What s 6.6 meters/secnd in mre understandable terms? The acceleratin due t gravity, g, is 9.8 meters/secnd, s this is abut.7 g. And that s quite a kick.
3 Part I: Applying Physics 9. The light turns green, and yu accelerate at 10 meters per secnd. Ater 5 secnds, hw ar have yu traveled? 10. A stne drps under the inluence gravity, 9.8 meters per secnd. Hw ar des it drp in 1 secnds? 11. A car is ging 60 miles per hur and accelerating at 10 miles per hur. Hw ar des it g in 1 hur? 1. A mtrcycle is ging 60 miles per hur, and decelerating at 60 miles per hur. Hw ar des it g in 1 hur?
Chapter : The Big Three: Acceleratin, Distance, and Time 33 13. An eagle starts at a speed 50 meters per secnd and, decelerating at 10 meters per secnd, cmes t rest n a peak 5 secnds later. Hw ar is the peak rm the eagle s riginal psitin? 14. A trailer breaks lse rm its truck n a steep incline. I the truck was mving uphill at 0 meters per secnd when the trailer brke lse, and the trailer accelerates dwn the hill at 10.0 meters per secnd, hw ar dwnhill des the trailer g ater 10 secnds? 15. A blck wd is shting dwn a track at 10 meters per secnd and is slwing dwn because rictin. I it cmes t rest in 0 secnds and 100 meters, what is its deceleratin, in meters per secnd? 16. A minivan puts n the brakes and cmes t a stp in 1 secnds. I it tk 00 meters t stp, and decelerates at 10 meters per secnd, hw ast was it riginally ging, in meters per secnd?
34 Part I: Applying Physics Cnnecting Speed, Acceleratin, and Displacement Suppse yu have a drag racer whse acceleratin is 6.6 meters/secnd, and its inal speed was 146.3 meters per secnd. What is the ttal distance traveled? This scenari sets yu up t use ne the imprtant equatins mtin: v v = as = a (x x ) This is the equatin yu use t relate speed, acceleratin, and distance. Q. A drag racer s acceleratin is 6.6 meters/secnd, and at the end the race, its inal speed is 146.3 meters per secnd. What is the ttal distance the drag racer traveled? A. s 1. a v 1 = 146 3 409 66. meters = ^ h = ^ h 1. T slve this prblem, yu need t relate speed, acceleratin, and distance, s yu start with this equatin: v v = as = a (x x ). In this scenari, v is 0, which makes this equatin simpler: 3. Slve r s: 4. Plug in the numbers: s = v = as 1 a v s 1. a v 1 = 146 3 409 66. meters = ^ h = ^ h S the answer is 409 meters, abut a quarter a mile standard r a drag racing track.
Chapter : The Big Three: Acceleratin, Distance, and Time 35 17. A bullet is accelerated ver a meter-lng rile barrel at an acceleratin 400,000 meters per secnd. What is its inal speed? 18. A car starts rm rest and is accelerated at 5.0 meters per secnd. What is its speed 500 meters later? 19. A rcket is launched at an acceleratin 100 meters per secnd. Ater 100 kilmeters, what is its speed in meters per secnd? 0. A mtrcycle is ging 40 meters per secnd and is accelerated at 6 meters per secnd. What is its speed ater 00 meters?