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Chapter 3 Kinematics in Two Dimensions We will now apply the tools for dealing with motion in one dimension to motion occurring in multiple dimensions Actually this sounds a lot worse than it really is We ll start with recalling the kinematics equations written in vector form 17 13 17 Remember 5 0 ms2 for this equation i r l l 1i2 Arvit at Chapter 3 Recall the arrows are simply there to remind you these equations can be written for all dimensions You may hear these referred to as component equations ffi r VVi r xxi vxt yyivyt vx vm axt vy vyyiayt AFVirl t2 2 Axv tla 2 A v tla 2 xi 2 x 3quot yo 2 y Chapter 3 The only new concept in this chapter is the idea that the standard components in this class are independent That means that motion in the Xdimension does not affect the motion in the ydimension and Viceversa FIGURE 410 One ball is released from rest at the same instant that another ball is shot horizontally to the right Their vertical motions are identical Chapter 3 3 Now the only real trick is to be able to apply these equations correctly in the context of a particular problem The best thing to do is continue to organize your data but do so keeping the component information separate from each other You may want to create a table like the one in the book X Direction Data YDirection Data xilelvilvflaxlt yilyflvilvflavlt Just keep in mind that some kinematics information may be the same for each component Time is the variable that is most often shared by each component Chapter 3 4 Example 1 Chapter 3 10 A bird watcher meanders through the woods walking 050 km due east 075 km due south and 215 km in a direction 350 north of west The time required for this trip is 250 h Determine the magnitudes of the bird watcher39s a displacement and b average velocity Chapter 3 Example 2 Chapter 3 64 On a spacecraft two engines fire for a time of 565 s One gives the craft an acceleration in the x direction of ax 510 ms2 while the other produces an acceleration in the y direction of ay 730 ms2 At the end of the firing period the craft has velocity components of vx 3775 ms and vy 4816 ms Find a the magnitude and b the direction of the initial velocity EXpress the direction as an angle with respect to the x aXis Chapter 3 6 Projectile Motion One of the most popular examples of motion in two dimensions is projectile motion An object is launched with an initial velocity at some angle with respect to the horizontal Kinematics allows us to predict how high and how far the object will travel Just remember the assumptions and restrictions we are using namely we are ignoring air resistance Maxvmum haghl R Range Chapter 3 Once the object is free of Whatever launched it the only force acting on the object is the force of gravity We will talk about forces in Chapter 4 however this means the acceleration of gravity is present ONLYin the vertical direction However with the absence of air resistance there is no acceleration in the horizontal direction Translation the horizontal component of the object s velocity will NOT change after it is launched In other words vix and vfx are the SAME and as a result you can only use de nition of constant velocity for the xdirection Putting this information into the tables X Direction Data YDirection Data xilelvilvflaxlt yilyflvilvflajlt 0 g Chapter 3 Visualizing the Velocity Vector A H um 115 ms I 2 y v 39v 17 V I vv v i x 339 VI v0 I vo I y 9 v 0 I 39 x 39 0 Wk vyl 9 R i i v Chapter 3 Projectile Motion further analysis Let s now apply these ideas to projectile motion problems Chapter 3 Example 3 Chapter 3 21 The drawing shows two planes each dropping an empty fuel tank At the moment of release each plane has the same speed of 135 ms and each tank is at the same height of 200 km above the ground Although the speeds are the same the velocities are different at the instant of release because one plane is ying at an angle of 150 above the horizontal and the other is ying at an angle of 150 below the horizontal Find the a magnitude and b direction of the velocity with which the fuel tank hits the ground if it is from plane A Find the c magnitude and 1 direction of the velocity with which the fuel tank hits the ground if it is from plane B In each part give the direction as a positive angle with respect to the horizontal 7 Fueltank 15 h Plane A plane B Chapter 3 1 1 Example 4 Chapter 3 17 A hotair balloon is rising straight up with a speed of 30 ms A ballast bag is released from rest relative to the balloon when it is 95 m above the ground How much time elapses before the ballast bag hits the ground Chapter 3 12 Example 5 Chapter 3 29 A diver runs horizontally with a speed of 120 ms off a platform that is 100 111 above the water What is his speed just before striking the water Chapter 3 Example 6 Chapter 3 text example 3 The drawing shows an airplane moving horizontally with a constant velocity of 115 ms at an altitude of 1050 m The direction to the right has been chosen as the x direction and upward is the y direction The plane releases a quotcare package that falls to the ground along a curved trajectory Ignoring air resistance determine the time required for the package to hit the ground Chapter 3 14 Example 7 Chapter 3 13 Suppose now that this plane is traveling with a horizontal velocity of 230 ms If all other factors remain the same determine the time required for the package to hit the ground Chapter 3 15 Example 8 Chapter 3 16 A golfer imparts a speed of 303 ms to a ball and it travels the maximum possible distance before landing on the green The tee and the green are at the same elevation a How much time does the ball spend in the air b What is the longest quothole in onequot that the golfer can make if the ball does not roll when it hits the green Chapter 3 l6 Example 9 Chapter 3 33 An Olympic long jumper leaves the ground at an angle of 23 and travels through the air for a horizontal distance of 87 m before landing What is the takeoff speed of the jumper Chapter 3 Example 10 Chapter 3 63 A golf ball rolls off a horizontal cliff with an initial speed of 114 ms The ball falls a vertical distance of 155 m into a lake below a How much time does the ball spend in the air b What is the speed v of the ball just before it strikes the water Chapter 3 18 Example 11 Chapter 3 24 A majorleague pitcher can throw a ball in excess of 410 ms If a ball is thrown horizontally at this speed how much will it drop by the time it reaches a catcher who is 170 111 away from the point of release Chapter 3 l9 Example 12 Chapter 3 61 A dolphin leaps out of the water at an angle of 35 above the horizontal The horizontal component of the dolphin39s velocity is 77 ms Find the magnitude of the vertical component of the velocity Chapter 3 20 Example 13 Chapter 3 62 A bullet is fired from a ri e that is held 16 m above the ground in a horizontal position The initial speed of the bullet is 1100 ms Find a the time it takes for the bullet to strike the ground and b the horizontal distance traveled by the bullet Chapter 3 21 Example 14 Chapter 3 66 The highest barrier that a projectile can clear is 135 m when the projectile is launched at an angle of 150 above the horizontal What is the projectile39s launch speed Chapter 3 22 Example 15 Chapter 3 20 A car drives horizontally off the edge of a cliff that is 54 m high The police at the scene of the accident note that the point of impact is 135 m from the base of the cliff How fast was the car traveling when it drove off the cliff Chapter 3 23 Example 11 Four charges labeled q 1 q q 3 and q4 are placed on the corners of a rectangle of length 100 meter and width of 0500 meter as shown in the picture What is the force on charge q located inthe middle ofthe rectangle ifq 300 pC q 150 14C q 120 pC q3 1200 pC and q4 1500 C CD Eekmm 5 L Ir eilfaxg 0 quot FLt era5 7 4 1 Q L Lm it a 393 9M Vy f39l 1 it 39 i u 2 7 N01 1 5706 3 6LSc I i r i 6 6 V F K 1114714 PH 1 F1 39 x un r Aquot 522 Ykl 1 rquot P f I K 39 r L i F L d 193 an In w we may w m3 121 9 he 1 J 143 pew ch Xawpmm39l s mad WI 11 S fgaw M Cakrlowiwls So F 739 5 726 F 169621 N 9 w 2lt PM 39rz39 aw SL39AG 7 ste 19y unjan TLtf afw 7 i V f 702939 r7 Or Show1 4 a Show 0312 ML 36 7 E1 N L O gau r S O B 5wz Z 5M9 1 USc 13914 a eld Kw EH 5 6 an agedm 6 171350 9 t O NYK Example 3 A uniform magnetic eld points out of the page A new charged particle is moving upwards in this magnetic eld The force the charged particle feels is 1 to the rioht 2 to the left 3 upwards 04 4 downwards 5 5 into the page 4 6 outofthepage P 7VXK 7 the partlcle feels no force from the magnetlc eld nth MwJ rule LN 1 r 39viquotcwj Q 6 La W st H in 4M 61 in J chaf c Example 4 A uniform magnetic eld points to the left A positively charged particle is moving to the right in this magnetic eld The force the charged particle feels is 1 to the right a 2 totheleft 6quot 3 upwards lt 5 4 downwards 7 5 into the page 61 out of the page d A 5 the particle feels no force from the magneftgelg T7 y x Z Whg t A gQ 920 160 REA539 CHAPTER 18 Chapter 181 The Origin of Electricity Electric charge is an intrinsic property of some atomic and subatomic particles The most common atomic particles associated with charge are the proton and electron Electrons have been associated with negative charge and protons with positive charge Choosing which object was positive or negative was an arbitrary choice The pain of that initial choice still haunts general physics today What is important is the understanding that two different charges eXist and that the electron and the proton possess different kinds of charge However both the proton and electron possess the same magnitude of electric charge e This is the smallest amount of charge to be discovered so far and as a result charge is only found in multiples of e Electric charge has the SI unit of the coulomb C e 160gtlt10 19 C Like charges reQel unlike charges attract jelectrically neutral an object has the same amount of positive and negative charge This is the same as saying the object has a net charge of zero This is n0t the same thing as having no charge Chapter 182 Charged Objects and the Electric Force Law of Conservation of Electric Charge During any process the net electric charge of an isolated system remains constant is conserved Chapter 183 Conductors and Insulators Electrical conductors a material where electrons are free to move from atom to atom Most metals are usually very good conductors of electric charge Electrical insulators a material where electrons are tightly bound to their nucleus and are very unlikely to move to another atom It s not that electrical insulators don t conduct charge they just conduct charge very poorly Chapter 184 Charging by Contact and by Induction Charging by contact Ebonite rod L Metal sphere f Insulated stand of 5 i L J Charging by induction i 7 Ebonite rod gt3 y Metal sphere Grounding Wire Insulated stand h a if 7 Connection 1 quot i 39 to ground Example 1 Chapter 18 5 Consider three identical metal spheres A B and C Sphere A carries a charge of 5 q Sphere B carries a charge of q Sphere C carries no net charge Spheres A and B are touched together and then separated Sphere C is then touched to sphere A and separated from it Last sphere C is touched to sphere B and separated from it a How much charge ends up on sphere C b What is the total charge on the three spheres b before they are allowed to touch each other and c after they have touched Example 2 Chapter 18 4 Four identical metallic objects carry the following charges 108 697 407 and 974 C The objects are brought simultaneously into contact so that each touches the others Then they are separated a What is the final charge on each object b How many electrons or protons make up the final charge on each object Chapter 185 Coulomb s Law Charges whether they have the same sign or different sign experience forces due to the presence of other charges Coulomb s Law describes this force Coulomb s Law the magnitude of the electrostatic force 15E exerted by one point charge q 1 on another point charge 2 is directly proportional to the magnitudes of the charges and inversely proportional to the square of the distance r between them 13 quilqzl E r Here k is a proportionality constant whose value in SI units is I 2 899xlO9NC T The direction of the electrostatic force is directed along the line joining the charges and is attractive if the charges have opposite signs and repulsive if the charges have the same sign Mathematically the electrostatic force is very similar to the gravitational force from Chapter 4 The only mathematical difference is that the gravitational force is always attractive in this class Whereas the electrostatic force can be attractive or repulsive Problem solving hintsclari cations When using Coulomb s Law make sure to only use the magnitudes of the charges The concept of like charges repel and unlike charges attract will tell you the direction to use for 15E in things like freebody diagrams Example 3 Chapter 18 60 In a vacuum two particles have charges of q and 612 Where q1 35uC They are separated by a distance of 026 m and particle 1 experiences an attractive force of 34 N What is 61 magnitude and sign In the case where there are more than two charges I d strongly suggest the use of subscripts to keep things straight Here s the convention I ll be using for problems with more than two charges Starting with Coulomb s Law from the text I ll start by dropping the absolute value signs because when using coulomb s law the magnitude of the force only depends on the magnitude of the charge It s the relationship of the two charges that determines the direction of the coulombicforce FE leq2 r2 Then I ll label all the forces and distances to avoid confusing which forces and distances I m talking about To illustrate this consider the arrangement of the three charges shown below If I want to know the net force on charge 61 1 I have to worry about the force on charge q 1 from charge q and 3 612 O 611 13 F 16611612 F 16611613 1 2 r2 13 r2 12 13 Here the subscripts on F112 are to be read as the electrostatic force on charge 611 due to charge 61 and the subscripts on F1 3 are to be read as the electrostatic force on charge 611 due to charge Q3 Similarly the subscripts on r112 denote the distance between charge 611 and charge 61 and the subscripts on r113 denote the distance between charge 611 and 3 Example 4 Chapter 18 18 The drawing shows an equilateral triangle each side of which has a length of 200 cm Point charges are fixed to each corner as shown The 400 uC charge experiences a net force due to the charges qA and q3 This net force points vertically downward in the drawing and has a magnitude of 405 N Determine the magnitudes and algebraic signs of the charges qA and q3 400 HC 11 Example 5 Chapter 18 17 The drawing shows three point charges xed in place The charge at the origin has a value of q1 800MC the other two have identical magnitudes but opposite signs q2 500yC and q3 500MC a Determine the net force exerted on g by the other two charges b If g had a mass of 150 g and it were free to move what would be its acceleration y 42 1 I 130 r9 39 230 39 x 117 gsm I l 130m l 3 Example 6 Chapter 18 24 There are four charges each with a magnitude of 165 C Two are positive and two are negative The charges are fixed to the corners of a 0496m square one to a corner in such a way that the net force on any charge is directed toward the center of the square Find the magnitude of the net electrostatic force experienced by any charge Example 7 Four charges labeled q 1 Q2 q3 and q4 are placed on the corners of a square of length 200 meters as shown in the picture What is the force on charge q located in the middle of the square if q 300 uC q 120 uC q2 120 uC q3 500 uC and q4 500 uC 2quot Example 8 Four charges labeled q 1 Q2 Q3 and q4 are placed on the corners of a square of length 150 meters as shown in the picture What is the force on charge q located in the middle of the square if q 300 uC q 100 uC q2 100 uC q3 500 uC and q4 500 uC Example 9 Four charges labeled q 1 Q2 Q3 and q4 are placed on the corners of a square of length 250 meters as shown in the picture What is the force on charge q located in the middle of the square if q 300 uC q 120 uC q2 700 uC q3 700 uC and q4 700 uC Example 10 Four charges labeled q1 q2 q3 and q4 are placed on the comers of a rectangle oflength 200 meters and width of 100 meter as shown in the picture What is the force on charge q located in the middle ofthe rectangle ifq 300 C q 120 pC q2 120 C Q3 500 LC and q4 500 C Example 11 Four charges labeled q1 q2 q3 and q4 are placed on the comers of a rectangle oflength 100 meter and width of 0500 meter as shown in the picture What is the force on charge q located in the middle ofthe rectangle ifq 300 pC q 150 C q2 120 C q3 1200 LC and q4 1500 uC Example 12 Four charges labeled q1 q2 q3 and q4 are placed on the comers of a rectangle of length 300 meters and width of 125 meters as shown in the picture What is the force on charge q located in the middle ofthe rectangle ifq 500 C q 120 pC q2 500 C Q3 120 LC and q4 500 C l I r l Ballistic Pendulum a Completely Inelastic Collision A ballistic pendulum is used to determine speeds of projectiles and consists of a penetrable material usually wood or Styrofoam connected by a string as shown below If a projectile of mass 500 g undergoes an inelastic collision with the block of wood mass of 275 g and the combination of the bullet and block undergo a change of height of 150 cm what was the initial speed of the projectile rm vlrl 045 lifmm l A l in no mlmzr r 4 C 2L v01 V 3 I n 39 quot1 X39o f iLj V L I f r v 9 26 a i D MCULLET mum L WELv kV uxk 1391112 CLJC c 1 a 7 LMV Dryquot rwc39wl at V L kl JIML CM 56 w lt39 lt lt a 3 HPY anjp Rquot jS Aiaf r 1 quoti VJ 9 Arr 3f a w L F P 7 c 9 39H 39 7 6 4 ctl Pit 51 5 zquot 14 l rv of L an J a 1 14 393 quotIV i x y 391 2 wrr39 iv lbw l 9quot Iquot but u 4 l f F I 2 J i A h i 7 V r X 5 f L 39 9 91 0 6 v I I7l c Oi k 39 Fr 4 H 939 r 3 1 Ra v 3 Shir 3f39 11 b2 abuta MatHF Mani J A i im rm 1 v n v39 v quot VGULK Bum quotAeLuk 03 ng 39Oo0 lcy 41757 MMv1 vquot 39 Iii ylgLJLLET 6 eye k3 quot iEULi ml l QMf Example 3 Chapter 6 1 The brakes of a truck cause it to slow down by applying a retarding force of 30 gtlt103 N to the truck over a distance of 850 In What is the work done by this force on the truck Is the work positive or negative Why 7 I 5 c GWE quotIquot J 5V r i r 0 50 dawn fagxiii 4 05 LJC 39 5 f 7 i v J V 39 DU NU kmcv v LotM7 1 wb HLES 7 EUR 05 56 o o M 05 jgd V Jim ampLq 1 2 L Mo 3 TLC LJ IL I Mrjo7 1gt1 Lt nasg if a 85EYLS dquotquotfe2 a Pomt A tie aft705 V hitLy 7 Example 7 Chapter 7 25 Kevin has a mass of 87 kg and is skating with in Iine skates He sees his 22 kg yodngeri 2 7 brother up ahead standing on the sidewalk with his back turned Coming up from behind he grabs his brother and rolls off at a speed of 24 ms Ignoring iction nd Kevin s speed just before he grabbed his brother SENT L E but i 7 v 39 gt V 393 quot A F a U C1 PL I No HZHL M 3 V1 Md iw i g 6quot6 g i r MIN W t 33 a i P Pr mm 9 m i 7 f g 39 MK Vt n wimvi Mt 7 If h 3 d a M 13A rquot I 39 539 r ng g Sivifji 35 Mg i 1 L g um V L 7 w mfg Example 15 Work Done by the Gravitational Force amp a Closed Path Suppose a 500 kg mass is moved from A along the red path to B and then along the green path back to A Find the work done on the mass by the gravitational force Let IE mom3239 k 500m B Hag X39JJKU 9 Vin Willa Wu hGRJ 500In a A O u w Asgt5vs r552 976 F as on MW 2 rJ ECOSBh 1 6 41 6430 mg 431256 makwcmswwa mw 3934 23039 3 quot b 3 P m r v y l O U Vb 06 g 5 U a 1 in 2 g 39 H UZEMs m 45 1 60 7 raoksl lel jt szwwW50quot 2 Umm 7 we 1 r uky 4 Chapter 3 Kinematics in Two Dimensions We will now apply the tools for dealing with motion in one dimension to motion occurring in multiple dimensions Actually this sounds a lot worse than it really is We ll start with recalling the kinematics equations written in vector form Remember 0 ms2 for this equation i2 A 1 r vit at Chapter 3 Recall the arrows are simply there to remind you these equations can be written for all dimensions You may hear these referred to as component equations yyivr Chapter 3 The only new concept in this chapter is the idea that the standard components in this class are independent That means that motion in the Xdimension does not affect the motion in the ydimension and Viceversa FIGURE 410 One ball is released from rest at the same instant that another ball is shot horizontally to the right Their vertical motions are identical Chapter 3 3 Now the only real trick is to be able to apply these equations correctly in the context of a particular problem The best thing to do is continue to organize your data but do so keeping the component information separate from each other You may want to create a table like the one in the book X Direction Data YDirection Data xilelvilvflaxlt yilyflvilvflavlt Just keep in mind that some kinematics information may be the same for each component Time is the variable that is most often shared by each component Chapter 3 4 Example 1 Chapter 3 10 A bird watcher meanders through the woods walking 050 km due east 075 km due south and 215 km in a direction 350 north of west The time required for this trip is 250 h Determine the magnitudes of the bird watcher39s a displacement and b average velocity Chapter 3 Example 2 Chapter 3 64 On a spacecraft two engines fire for a time of 565 s One gives the craft an acceleration in the x direction of ax 510 ms2 while the other produces an acceleration in the y direction of ay 730 ms2 At the end of the firing period the craft has velocity components of vx 3775 ms and vy 4816 ms Find a the magnitude and b the direction of the initial velocity EXpress the direction as an angle with respect to the x aXis Chapter 3 6 Projectile Motion One of the most popular examples of motion in two dimensions is projectile motion An object is launched with an initial velocity at some angle with respect to the horizontal Kinematics allows us to predict how high and how far the object will travel Just remember the assumptions and restrictions we are using namely we are ignoring air resistance Maxvmum hagm RRan e Chapter 3 Once the object is free of Whatever launched it the only force acting on the object is the force of gravity We will talk about forces in Chapter 4 however this means the acceleration of gravity is present ONLYin the vertical direction However with the absence of air resistance there is no acceleration in the horizontal direction Translation the horizontal component of the object s velocity will NOT change after it is launched In other words vix and vfx are the SAME and as a result you can only use de nition of constant velocity for the xdirection Putting this information into the tables X Direction Data YDirection Data xilelvilvflaxlt yilyflvilvflavlt 0 Chapter 3 Visualizing the Velocity Vector A E um 115m5 lE gt r 115 M le gh ms n v Chapter 3 CHAPTER 2 Motion in ONE Dimension Definitions variable meaning units symbol SI X yor r distance m 2 y or Z displacement m V speed ms 7 velocity ms Z acceleration ms2 t time s magnitude A change in nalinitial subscript meaning 0 i initial f final Chapter 2 Chapter 2 Displacement vs Distance DDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD DDDDDDDDDDDDDDDDDDDD Z Example 1 Chapter 2 1 A Whale swims due east for a distance of 69 km turns around and goes due west for 18 km and finally turns around again and heads 37 km due east a What is the total distance traveled by the Whale b What is the magnitude of the displacement of the Whale Chapter 2 An object goes from one point in space to another After it arrives at its destination its displacement is than its distance traveled 1 either greater than or equal to 2 always greater than D always equal to 4 either smaller than or equal to LII always smaller than C either smaller or larger Chapter 2 Velocity vs Speed Both quantities are changes in an object s position over a certain period of time However one is a vector and one is a scalar You must be able to remember which is which because this small difference has enormous consequences Which is which Axxfxi A f l V i V T At tf tl t tf zl Acceleration Acceleration is simply a change in an object s velocity over a period of time However realize the importance of this simple definition A17 17f i 5 r f z i a Chapter 2 Chapter 2 Average Velocity and Acceleration It s normal to let I be 0 the stopwatch will always start at zero and Ito be understood as Ifthe time when the stopwatch stops so don t be surprised to see these equations written as Instantaneous Velocity and Acceleration As the interval A gets smaller and smaller 7gt a A d a A17 N V E gt E a r E Since this is an algebrabased course we won t be doing any calculus What this means for you is that we can clean up some of the notation We will be dealing only with average speeds and accelerations Thus the formulas you will see on the equation sheet will look like Chapter 2 Acceleration Deceleration and Negative Acceleration What s the difference Well Acceleration is a change in velocity Usually people think it means an object is speeding up This is not the case because acceleration is a very general term Negative acceleration is more specific it is something that depends on your choice of a coordinate system In this case the object is experiencing an acceleration in either the 2 or y directions or maybe a bit of both Deceleration means an object is slowing down Conceptually this means the velocity and acceleration vectors point in opposite directions This is the only way an object will slow down Chapter 2 8 Example 2 A vehicle moves to the right at 15 ms After a short time the vehicle s velocity decreased This means that the vehicle s velocity and acceleration vectors point 1 in the same direction 2 in opposite directions 3 not enough information Chapter 2 9 Example 3 Chapter 2 7 A tourist being chased by an angry bear is running in a straight line toward hisher car at a speed of 435 ms The car is a distance d away The bear is 305 m behind the tourist running at 505 ms The tourist reaches the car safely What is the maximum possible value for 0 Answer Chapter 2 10 Example 4 A bicyclist makes a trip that consists of two parts each in the same direction due east along a straight road During the first part she rides for 22 minutes at an average speed of 72 ms During the second part she rides for 36 minutes at an average speed of 51 ms a How far has the bicyclist traveled during the entire trip b What is the average speed of the bicyclist for the trip Chapter 2 l l Example 5 A runner accelerates to a velocity of 536 ms due east in 28 seconds HisHer average acceleration is 0550 msz also directed due east What is was hisher velocity when heshe began accelerating Answer Chapter 2 Example 6 A dog is running in a park and travels 500 m due east before turning and traveling 100 m at 300 south of east What is the dog s displacement Answer What distance does the dog travel Answer Chapter 2 l3 Example 7 A car is traveling with a velocity of 100 ms due east 400 seconds later the car s velocity is 500 ms due east What is the average acceleration of the car Answer Chapter 2 The assumption used to derive the kinematics equations in section 24 and 25 is 1 constant acceleration 2 constant velocity 3 constant speed 4 none of the above Chapter 2 15 Chapter 2 Kinematics Equations the real ones X f Xcomponent 2 Axv tla t2 me 2 x x v tia t2 x me me x ycomponent 2 V tiat y y yw Ly 2 y 1 2 Ay vigyt j ayt 17i7Zz t Xcomponent v x vi x axt f ycomponent V vat fay Ly Y From the MeriamWebster Dictionary Kinematics a science that deals with motion apart from considerations of mass and force Equation Number Equation 24 Vf viat 1 27 x vivft 1 2 28 xf xivil al 29 v vi22an Chapter 2 Using the Kinematics Equations How to Choose Equation Variables Number Equation X a vf t vi 24 vfwat v v v v 27 xvivft I I I l 28 xfxiviz az 2 I I I I 29 v vi2Zan I I I I Other useful stuff Quadratic equations AX2BXC0 B IBZ 4AC X 2A Enforce that X is the variable that is quadratic in the next overhead that variable is t Chapter 2 18 Apply the solution for quadratic equations to x xi vilalz AXZ BX C 0 math Chapter 2 xf f 1 2 xivil al phy51cs Example 8 Chapter 2 13 A motorcycle has a constant acceleration of 25 ms2 Both the velocity and acceleration of the motorcycle point in the same direction How much time is required for the motorcycle to change its speed from a 21 to 31 ms and b 51 to 61 ms Chapter 2 20 Example 9 Chapter 2 22 a What is the magnitude of the average acceleration of a skier who starting from rest reaches a speed of 80 ms when going down a slope for 50 s b How far does the skier travel in this time Answer Chapter 2 21 Example 10 Chapter 2 67 A jetliner traveling northward is landing with a speed of 690 ms Once the jet touches down it has 750 m of runway in which to reduce its speed to 610 ms Compute the average acceleration magnitude and direction of the plane during landing Answer Chapter 2 22 Example 11 Chapter 2 53 A cement block accidentally falls from rest from the ledge of a 530mhigh building When the block is 140 m above the ground a man 20 m tall looks up and notices that the block is directly above him How much time at most does the man have to get out of the way Answer Chapter 2 23 Example 12 Chapter 2 29 Suppose a car is traveling at 200 ms and the driver sees a traffic light turn red After 0530 s has elapsed the reaction time the driver applies the brakes and the car decelerates at 700 msz What is the stopping distance of the car as measured from the point Where the driver first notices the red light Chapter 2 24 Example 13 Chapter 2 20 A cart is driven by a large propeller or fan which can accelerate or decelerate the cart The cart starts out at the position x 0 m with an initial velocity of 50 ms and a constant acceleration due to the fan The direction to the right is positive The cart reaches a maximum position of x 125 m where it begins to travel in the negative direction Find the acceleration of the cart Chapter 2 25 Example 14 Chapter 2 23 The left ventricle of the heart accelerates blood from rest to a velocity of 26 cms a If the displacement of the blood during the acceleration is 20 cm determine its acceleration in cms2 b How much time does blood take to reach its nal velocity Chapter 2 26 Now for the good stuff Example 15 Chapter 2 24 A cheetah is hunting Its prey runs for 30 s at a constant velocity of 90 ms Starting from rest What constant acceleration must the cheetah maintain in order to run the same distance as its prey runs in the same time Chapter 2 27 Example 16 Chapter 2 17 A car is traveling along a straight road at a velocity of 360 ms when its engine cuts out For the next twelve seconds the car slows down and its average acceleration is 51 For the next siX seconds the car slows down further and its average acceleration is 52 The velocity of the car at the end of the eighteensecond period is 280 ms The ratio of the average acceleration values is 5152 150 Find the velocity of the car at the end of the initial twelvesecond interval Chapter 2 28 Example 17 Chapter 2 28 A race driver has made a pit stop to refuel After refueling he leaves the pit area with an acceleration Whose magnitude is 60 ms2 and after 40 s he enters the main speedway At the same instant another race car that is on the speedway and traveling at a constant speed of 700 ms overtakes and passes the entering car If the entering car maintains its acceleration how much time is required for it to catch the other car Chapter 2 29 Example 18 Chapter 2 31 A car is traveling at a constant speed of 33 ms on a highway At the instant this car passes an entrance ramp a second car enters the highway from the ramp The second car starts from rest and has a constant acceleration What acceleration must it maintain so that the two cars meet for the first time at the next eXit which is 25 km away Chapter 2 30 Example 19 Chapter 2 72 A drag racer starting from rest speeds up for 402 m with an acceleration of 170 msz A parachute then opens slowing the car down with an acceleration of 610 msz How fast is the racer moving 350 m after the parachute opens Chapter 2 31 Equation Variables Number Equation X a vf t Vi 24 Vf Vi 5 1 27 x vivft 1 2 28 xfxivit at 29 v vi220Ax Equation Variables Number Equation X a vf t Vi 24 Vf Vi 5 1 27 x vivft 1 2 28 xfxivit at 29 v vi220Ax Equation Variables Number Equation X a vf t Vi 24 Vf Vi 5 1 27 x vivft 1 2 28 xfxivit at 29 v vi22an Chapter 2 32 Section 26 Free Fall In section 26 two of the main assumptions are I ignore air resistance J velocity is constant U small drop distance 4 1amp2 52amp3 C 1amp3 712amp3 00 there are none Chapter 2 33 Just after an object is dropped the magnitude of the acceleration it experiences is l 98 ms2 2 0 ms2 3 always increasing 4 always decreasing Chapter 2 34 If you drop an object in the absence of air resistance it accelerates downward at 98 msz If instead you throw it downward its downward acceleration after release is 1 less than 98 ms2 2 98 ms2 3 more than 98 ms2 Chapter 2 3 5 Example 20 Chapter 2 38 From the top of a cliff a person uses a slingshot to re a pebble straight downward which is in the negative direction The initial speed of the pebble is 90 ms a What is the acceleration magnitude and direction of the pebble during the downward motion b After 050 s how far beneath the cliff top is the pebble Chapter 2 36 Example 21 Chapter 2 45 From her bedroom Window a girl drops a waterfilled balloon to the ground 60 m below If the balloon is released from rest how long is it in the air Chapter 2 37 Objects being thrown up but not in the I m sick sense Problem solving help This goes for all problems but may be very helpful for objects being acted on by gravity only Give a rough sketch with a labeled axis This will help you assign the correct directions and signs to variables Include given information Put these on the object s trajectory in your sketch Label anything else that will be assumed given from rest special points those sorts of things Chapter 2 3 8 AssumptionsRestrictions thus far 0 Acceleration is constant This must be true to use any kinematics equations 0 For an object in free fall ignore air resistance 0 For an object in free fall the distance of the fall will be small compared to the earth s radius Chapter 2 39 You are throwing a ball straight up in the air At the highest point the ball s l velocity and acceleration are zero 2 velocity is nonzero but its acceleration is zero 3 acceleration is nonzero but its velocity is zero 4 velocity and acceleration are both nonzero Chapter 2 3 40 A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed Neglecting air resistance the ball to hit the ground below the cliff with the greater speed is the one initially thrown l upward 2 downward 3 neither the both hit at the same speed Chapter 2 41 Example 22 Chapter 2 68 A hotair balloon is rising upward with a constant speed of 250 ms When the balloon is 300 m above the ground the balloonist accidentally drops a compass over the side of the balloon How much time elapses before the compass hits the ground Chapter 2 42 Example 23 Chapter 2 39 Two identical pellet guns are fired simultaneously from the edge of a cliff These guns impart an initial speed of 300 ms to each pellet Gun A is fired straight upward with the pellet going up and then falling back down eventually hitting the ground beneath the cliff Gun B is fired straight downward In the absence of air resistance how long after pellet B hits the ground does pellet A hit the ground Chapter 2 43 Example 24 Chapter 2 46 A pellet gun is fired straight downward from the edge of a cliff that is 15 m above the ground The pellet strikes the ground with a speed of 27 ms How far above the cliff edge would the pellet have gone had the gun been fired straight upward Chapter 2 44 Example 25 Chapter 2 43 An astronaut on a distant planet wants to determine its acceleration due to gravity The astronaut throws a rock straight up with a velocity of 15 ms and measures a time of 200 s before the rock returns to his hand What is the acceleration magnitude and direction due to gravity on this planet positive up negative down Chapter 2 45 Graphs of Position vs Time Acceleration is constant but 0 At tf l l t xfxivt 16 12 E l 3 IAX 8 I8m 9 I 5 I Le 4 At28 O O 1 2 3 4 Time ts ymxb gt ybmx xfxivt Moral of the story if you are looking at a graph of position versus time and you see a line the slope of that line is the velocity of the object Chapter 2 46 Graphs of Position vs Time What else does the slope mean 1200 800 Position x m 400 Positive gt velocity 7 1 Zero veIOCIty 1 NA 5 it lt Negative velocity J At 200 s 400 200 600 800 1000 At400 s Time s 1200 1400 1600 0 Moral of the story the slope of the line i will tell you the object s relative velocity This is because the i signs give you a relative direction either toward or away from some spot Chapter 2 1800 47 Graphs of Position vs Time Acceleration is constant but 75 O 1 2 xf xivit at 1 2 1 2 0 x x vt at 0 at vt x x l f l 2 2 l l f 0 At2 BtC Object Experincing Constant Acceleration 45 7 a 40 35 7 1 g 30 7 1 c 25 7 2 a t 20 i r m a E 15 7 Q t 10 7 4 t quot 5 i r v t v Q t quot 0 39I A 1111 w w w w 0 05 1 15 2 25 3 Time s Chapter 2 48 Graphs of Position vs Time Acceleration is constant but 75 O 0 lat2v1 x x 2 l l f 0At2BtC Position m HNNwwbb U39IOU39IOU39IOU39I 10 Object Experincing Constant Acceleration y 4905x2 6E14x 3E14 05 1 15 2 Time s Chapter 2 49 Graphs 0f Velocity vs Time Velocity 1 ms Time 1 s ymxb ybmx V ViLlI f Moral of the story if you are looking at a graph of velocity or speed versus time and you see a line the slope of that line is the acceleration of the object Chapter 2 50 Graphs of Velocity vs Time Is there anything else to these types of graphs Oh yeah there is dudes amp dudettes 500 400 300 200 Velocity ms l00 15 20 25 30 Time h 0 Moral of the story the area under the curve of a velocity versus time graph is the displacement of the object Chapter 2 51 Example 26 During which segment is the magnitude of the acceleration the greatest Least Zero 100 80 60 40 Velocity 1 ms 20 O 10 2O 3O 4O 5O 60 Time 1 5 What is the magnitude of the object s displacement during this 60 second period Keep two significant figures Chapter 2 52 Example 27 During which segment is the speed the greatest least zero 500 400 300 200 Position x km gt l00 0 05 10 15 20 25 30 Time I h Chapter 2 53 Example 28 Chapter 2 60 A bus makes a trip according to the position versus time graph shown below What is the average velocity of the bus during each segment Express your answer in kmhr 500 400 300 200 Position x km gt l00 0 05 10 15 20 25 30 Time h Chapter 2 5 4 Example 29 Chapter 2 57 A snowmobile moves according to the velocity versus time graph shown below What is the average acceleration during each segment Roughly How far did the snow mobile travel 100 Jgt 03 00 O O 0 Velocity 1 ms O O 10 2O 30 4O 5O 60 Time 1 5 Chapter 2 55 Example 30 Chapter 2 61 A bus makes a trip according to the position versus time graph shown below What is the average acceleration in kmhrz of the bus for the entire 35 hour period shown in the graph 400 300 B C E 5 R 200 9 g A 1 100 0 0 05 10 15 20 25 30 35 Time h Chapter 2 Example 25 A basketball player makes a jump shot The 0600 kg ball is released at a height of 200 m above the oor with a speed of 720 ms The ball goes through the net 310 m above the oor at a speed of 420 ms What is the work done on the ball by air resistance I ML Y y K 755 k g f jo l 9591 1 WA L T 1 Fricl am 9 Non cussmime Eva v Jinnp5 lmL a will quot471 W hmg n4ij LE3 i 2 OIGOO L10g2 739de 2 0L59XAlj TYgl wpzadm 1 l r 7x78 395 r I M M Wyeth Mons Me3 7 1393 Lee 02 WW f we ion 1 Lta pe HA wit 5mm Lewda grb39ah w my day you JunkI
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