HW 2 Help cm sin 60 =12.5 cm 10.8 cm cm sin 30 =12.5 cm 6.25 cm 2. We can check the answer by using the Pythagorean Theorem:

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1 HW Help 8.OGANIZE AND PLAN opposite funtion sin hpotenuse Gien the hpotenuse and the anles we will use the trionometri to sole for eah of the unknown sides. We note that the 60 is the anle opposite the lon side L beause it is a wider anle and 30 is the anle opposite the short side. We are interested in the unknown sides so we will use a little alebra to find that opposite = hpotenuse sin. We should also reall the sin of the ommon anles used in this problem (table 3.1). sin 30 = 1/ and sin 60 3 /. OLVE Applin our trionometri relation for riht trianles we find: and EFLECT (1.5 m). L L m sin 60 =1.5 m 10.8 m m sin 30 =1.5 m 6.5 m We an hek the answer b usin the Pthaorean Theorem: 3. OGANIZE AND PLAN We need to find the riht trianle lurkin within the problem. From where ou are the peak of the mountain is aboe the horizon. The map puts ou a horizontal distane from the peak. o ou hae an anle and a side adjaent to the anle. We need to find the side opposite the anle to find the heiht, aboe the present eleation. The eleation will then be the eleation of the ar at the point of measurement, plus the lenth of the side opposite the anle. The trionometri funtion that relates the two thins we know to the side we don t is the tanent funtion: tan =. Isolatin for the unknown alue ields: = tan OLVE The heiht aboe the urrent eleation is: = tan = 5.0 km tab 3.1 = 1.35 km The peak s eleation is then 1350 m m = 930 m EFLECT This eleation is a respetabl sized peak, thouh one of the smaller ones in the ok Mountains.

2 36. OGANIZE AND PLAN We are ien a displaement etor in ardinal diretions, the time it takes to traerse the displaement and asked to find the displaement and the aerae eloit. The first part is just onertin the ardinal diretions to the Cartesian form. We reall that North, outh, East and West orrespond to the unit etors i ˆ, i ˆ, ĵ and ˆ, j respetiel. The aerae eloit (whih we remember is a etor) will be determined b diidin the displaement etor b the time of traersal. We are asked to find the aerae eloit in units of m/s so we will also need to onert minutes to seonds. OLVE Mappin the ardinal diretions to the Cartesian unit etors ields displaement r in Cartesian form: Conertin 0 min t s we hae The aerae eloit is then: r 150 miˆ900 mˆj 60 s 1 1 min 0 min so 60 s min 100 s r 1 (150 miˆ900 m ˆj) 1.04 m/siˆ0.75 m/sˆj T 100 s or, in ardinal diretions 1.04 m/s east and 0.75 m/s south. EFLECT This problem required two tpes of onersions: minutes to seonds and ardinal diretions to Cartesian form. Aquirin skill in unit onersions will be somethin ou thank ourself for the rest of our life. ubtratin etors is ahieed b ombinin like terms where the unit etors are treated as ariables. The differene between etors 1 and is a etor ien b 1 ˆ ˆ ( 1 ) i ( 1 ) j 39.OGANIZE AND PLAN OLVE r r ( r r ) i ˆ ( r r ) ˆ j ( ) mi ( ) m j 5.95 mi 6.05 m j r r EFLECT in is important when subtratin omponents and sometimes an be trik. You an alwas hek the answer b drawin the etors tail to tail and obtainin r 1 r b drawin a etor startin at r and endin at r 1. Unlike addition, r 1 r r r so order 1 matters.

3 48. OGANIZE AND PLAN There is a distintion between distane traeled and displaement. Distane traeled is the readin on the odometer of the ar (assumin it started at zero at the beinnin of the rae). Displaement is the displaement etor beinnin at the start and endin at the loation of interest. The distane traelled will just be the ar of a irle whih is ien b frations of a irumferene. The irumferene of a irle is C =. One-quarter lap is orresponds C C to, one-half lap and one whole lap C. 4 The displaement is a etor and requires both manitude and diretion. The displaement etor d an be obtained b alulatin the differene between the final position etor and the initial position etor, d rf ri. One we hae the position etors at the start, at one-quarter lap, one-half lab and one full lap it is a straiht forward matter to alulate the etor differene. OLVE Distane traeled: C quarter lap: 393 m half lap: 4 C 785 m full lap: C 1570 m Position etors in omponent form (r, r ) at: start: (50 m, 0) quarter lap: (0, 50 m) half lap: ( 50 m, 0) full lap: (50 m, 0) Displaement etors from start to quarter lap: (0,50 m) (50 m,0) ( 50 m,50 m) 50 mi ˆ 50 mˆj half lap: ( 50 m,0) (50 m,0) ( 500 m,0)=500 mˆ i full lap: (50 m,0) (50 m,0) ( 0,0)= 0 EFLECT This problem reinfores the differene between distane traeled (a salar quantit) and displaement. The stark differene is seen between the distane traeled to o all the wa around, one irumferene; and the displaement assoiated with endin upwhere ou started, zero.

4 The aerae aeleration oer an duration of time requires the differene between the eloities at the start and end of the duration as well as the duration itself. The eloities are obtained from the speed and diretion at the beinnin of the rae and 1/4 of the wa around the irle. If the rae starts with ars fain due north (whih will all the ĵ diretion) then 1/4 of the wa around the irle the ar will be fain due west (whih we will all the î diretion) (see fiure below). 58.OGANIZE AND PLAN! f = ( ı) N! i = N ae trak The time is determined b usin rate time = distane. We an use the eometr of the problem to determine the total distane traeled and we are ien the rate of trael oer this distane. The aeleration is then just obtained b diidin the hane in eloit b the time. We also note that the units are km/h. To obtain an answer that we an ompare to eperiene we want to onert eerthin to meters and seonds. OLVE The differene in eloities is: ˆ ˆ 90 km / h 1000 m 1 hr 90 km / h( i j) ( iˆ ˆj) 5 m /s( iˆ ˆj) 1 km 3600 s The path lenth d of this part of the rae is 1/4 the irumferene of a irle: km d π 1.96 km. 4 d 1.96 km The time of trael is t hr 78 s 90 km / hr The aerae aeleration is: 5 m /s( iˆ ˆj) a ˆ ˆ a 0.31 m/s ( i j) t 78 s EFLECT The aerae aeleration is direted toward the south-west diretion whih jibes with the piture and the trajetor of the ar around the trak. The manitude of the aeleration is 0.45 ms or about 5% the aeleration due to rait on the Earth.

5 63. OGANIZE AND PLAN Gien the initial eloit we an use the relationships deried in question 11 to find the rane and the time of fliht. The rane is ien b 0 T. The time of fliht was deried to be 0 The maimum heiht an be determined b onsiderin that half the time time of fliht is spent oin up and half is spent oin down when the launh eleation is the same as the landin eleation. When the projetile is at the ape of the path the omponent of the eloit in the ertial diretion is zero. Usin equation 3.19a with zero initial eloit in 1 the diretion of aeleration ( t ) and takin the oriin of the oordinate sstem as the ape. The distane of the fall in Known: iˆ ˆj i OLVE The rane is: The time of fliht is: 7 m/s(os45 sin45 ) The maimum heiht is: 0. T t will ie us the maimum heiht. 00 (7) m /s 74 m 9.8 m/s 0 (7) m /s T 3.9 s 9.8 m/s s 9.8 m /s ( ) 19 m t EFLECT The distane between home-plate and the outfield wall is approimatel 400 ft. The rane found aboe is onsistent with a fl-ball hit into the outfield. The time of fliht is also onsistent with the eperiene of wathin a ball ame and timin the time from bat to loe of a fl-ball.

6 66. OGANIZE AND PLAN Gien the initial eloit we an use the relationships deried in question 11 to find the rane and the time of fliht. The rane is ien b 00 sinos where is the launh anle from the horizontal and is the speed. We are asked to determine if an initial eloit and launh anle will result in a rane that is less than 5 m. We shall etrat 0 and 0 from the ien information: i (0 m/s,15 ) to obtain whether or not the ball will be in or out of the ourt. Gien a rane and a launh anle, determination of the speed is just alebrai manipulation of the relationship aboe: sinos and 0 0 m /s os m/s Preliminar alulations: 0 0 m/s sin m/s OLVE Proof that the ball will land in pla: The rane of the struk tennis ball is 5.18 m/s19.3 m/s 0 m 5 m 9.8 m/s The maimum eloit that will result in a ball in pla is: ma 9.8 m /s 5 m m/s sin15os15 EFLECT Another appliation of the rane formula. It is interestin to note that m/s differene in the eloit of the ball omin off the raket makes the different between in and out of the ourt. Great tennis plaers must hae done er well in their phsis lasses.

7 We hae deeloped the rane equation ien the horizontal and ertial omponents of the eloit: 70.OGANIZE AND PLAN 0 0 This problem is stated in a wa that allows a straihtforward appliation of this equations. OLVE The rane of the lon jumper is: 3.85 m/s7.5 m/s 5.89 m 9.8 m/s EFLECT The worlds reord for women s lon jump in 198 was 5.98 meters. Toda is 7.5 meters. A distane of 5.89 m is a respetable and reasonable distane. 8.OGANIZE AND PLAN The equation for the entripetal aeleration is a limit on the allowed alue for a 1.0 ms leadin to the inequalit: olin for : r a ma a min ma a ma ar. We are ien Notes on units: We will standardize the units in the problem to meters and seonds. OLVE m 1 hr 100 km hr m s km 3600 s (7.8 ms) min 773 m min a 1 ms Minimum radius of urature is EFLECT A larer radius of urature an aommodate faster speeds or lower entripetal aeleration.