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# University Physics I PHYS 2050

WMU

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This 164 page Class Notes was uploaded by Zula Tromp on Wednesday September 30, 2015. The Class Notes belongs to PHYS 2050 at Western Michigan University taught by Lisa Paulius in Fall. Since its upload, it has received 38 views. For similar materials see /class/216930/phys-2050-western-michigan-university in Physics 2 at Western Michigan University.

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Date Created: 09/30/15

The Laws of Motion Chapter 5 Sir Isaac Newton 1642 1727 Formulated basic laws of mechanics Discovered Law of Universal Gravitation Invented form of calculus Many observations dealing with light and op cs Newton s First Law The Law of Inertia An object in motion will stay in motion with constant speed and direction an object at rest will stay at rest unless acted upon by an outside force Newton s Second Law the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass in The net force is the cause of change in motion as measured by the acceleration c Mathematically 2Fm speeds much lower than the speed of light Newton s Third Law If two objects interact the force IEABexerted by object A on object B is equal in magnitude and opposite in direction to the force EA exerted by object B on object A AB gt Note on notation FAB is the force exerted byA on B Classes of Forces Ulllzu39l tm39t vx 7va if 6 7 7 V i ii mm quot1 X39 39L s t l 7 quotii iu39i Q xv 7 2 Ir Contact forces involve physical contact between two objects a bc Field forces act through empty space No physical contact is required def r il i 2007 Thomson Higher Education Units of Force a The SI unit of force is the newton N 1N1kgmSZ o The US Customary unit of force is a pound lb 31 lb 1 slugfts2 01 N 11 lb Inertial Frames c Any reference frame that moves with constant velocity relative to an inertial frame is itself an inertial frame o A reference frame that moves with constant velocity relative to the distant stars is the best approximation of an inertial frame c We can consider the Earth to be such an inertial frame although it has a small centripetal acceleration associated with its motion Inertia and Mass 0 The tendency of an object to resist any attempt to change its velocity is called inertia 0 Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity More About Mass 0 Mass is an inherent property of an object 0 Mass is independent of the object s surroundings 0 Mass is independent of the method used to measure it 0 Mass is a scalar quantity 0 The SI unit of mass is kg Weight Weight is the gravitational force exerted on the object o Weight will vary with location 0 Example gt gt Wearth 180 lb Wmoon 30 lb Q mearth 2 kg mmoon 2 kg More About Forces a b C o A spring can be used to calibrate the magnitude of a force o Doubling the force causes double the reading on the spnng c When both forces are applied the reading is three times the initial reading More About Newton s 2quot Law 2E is the net force a This is the vector sum of all the forces acting on the one mass of interest Be careful to add only forces acting on that object o Newton s Second Law can be expressed in terms of components EFX m ax c EFy m ay c EFZ m az A baseball is struck by a bat While the ball is in the air what objects exert forces on the ball 9 QFFP P PP N Eanh Bat Air Bat Air Earth Bat Earth Air Earth Bat Air There are no forces on the ball The answer depends on whether the ball is going up going down or at its highest point Vector Nature of Forces Forces are vectors so you must use the rules for vector addition to find the net force acting on an object d nnnnnnnnnnnnnnnnnnnnn on Vector Nature of Forces Two forces 450 N at 10 and 400 N at 30 are applied to a car in an effort to accelerate it What is the magnitude of the resultant of these two forces CL k A 79 m Find the direction of the resultant force If the car has a mass of 3000 kg what acceleration does it have Ignore friction 20N 20N Consider the 3 forces shown in the diagram 40N Will the box experiancc acceleration 1 Yes downward 2 Yes upward 3 No it is balanced 4 UnablEV to determine without the angle A ball initially moves horizontally with ve locity 17 as shown t is than sn uck by a stick After leaving the stick the ball moves vertically with a eiu ity E which has the same magnitude as 5 vf 71 Which of the fullmving vectors best repre sents the direction 0f the average force that the stick exerts an the all i tquot 5 None ofchese graphs is correct Gravitational Force a The gravitational force IE9 is the force that the earth exerts on an object o This force is directed toward the center of the ea h 0 From Newton s Second Law Q gt Fg m we will refer to it as the Weight Fg mg Normal Force l o The normal force is always perpendicular to the surface hence the name o It resists an object pushing into the surface c We will be assuming our objects are completely a nnnnnnnnnnnnnnnnnnnnnn a n rigid Normal Force 0 Carefulll The normal force is not always equal to the gravitational force on the object o For example in this case 2Fyn Fg FO andnFgF uuuuuuuuuuuuuuuuuuuuuuuuu o n may also be less than Fg Free Body Diagrams 0 Draw the object as a point mass 0 Draw vectors for only the forces acting on that mass do not include forces it exerts on other objects do not include velocities accelerations etc 0 Try to indicate relative magnitudes by the length of the vectors when possible a label angles Draw a free body diagram for the man when the elevator is at rest What is the reading on the scale 1nsgtmg 2nsm9 3hsltmg Draw a free body diagram for the man when the elevator is accelerating upward at a05g up What is the reading on the scale 1nsgtmg 2nsmg 3lnsltm9 Draw a free body diagram for the man when the elevator is accelerating downward at a05g down What is the reading on the scale 1nsgtmg 2nsmg 3lnsltm9 Draw a free body diagram for the man when the elevator is moving up with v 5ms What is the reading on the scale 1nsgtmg 2nsm9 3nsltm9 Tension forces from ropes Unless otherwise stated we will assume the ropes have no mass 0 Direction is H to the rope o Direction is away from object You can NOT PUSH with a rope o For any uninterrupted length of massless rope the tension will be equal and opposite at the two ends or the rope a By uninterrupted we mean no knots or not rubbing over something with friction Example of tension V th interrupted ropes the knot is an interruption so the tension in the ropes can be different on the different sides a eznm Thomson ngner Education A person is pulling a 100 kg box across a frictionless floor with a force of 3OON as shown a Determine the magnitude and direction normal force of the floor on the box bDetermine the acceleration of the box 40 c How would these values change if she were pushing down the 100 kg box across a frictionless floor the same force of 3OON as shown Three blocks are stacked as shown below Assume the blocks are in a vacuum How many forces are acting on the bottom block m3 1 One force Two forces Three forces Four forces Five forces Six forces More than six forces No forces act on the block 53 9 Cannotbedetermined NOU gtWN Newton s Third Law 0 If two objects interact the force EZ exerted by object 1 on object 2 is equal in magnityde and opposite in direction to the force F21 exerted by object 2 on object 1 F12 39F21 a 3 Note on notation FAB is the force exerted byA on B Newton s Third Law Alternative Statements 0 Forces always occur in pairs A single isolated force cannot exist o The action force is equal in magnitude to the reaction force and opposite in direction 0 The action and reaction forces act on different objects and must be of the same type ActionReaction Examples 1 o The force IE12 exerted by object 1 on object 2 is equal in magnitude and opposite in direction to E1 exerted by object 2 on object 1 39 F12 39F21 ActionReaction Examples 2 o The normal force table on monitor is the reaction of the force the monitor exerts on the table a Normal means perpendicular in this case 0 The action Earth on monitor force is equal in magnitude and opposite in direction to the reaction force the force the monitor exerts on the Earth ltagt Particles in Equilibrium o If the acceleration of an object that can be modeled as a particle is zero the object is said to be in equilibrium 0 Mathematically the net force acting on the object is zero Equilibrium Example 1a 0 A lamp is suspended from a chain of negligible mass 0 The forces acting on the lamp are a the downward force of gravity 4 the upward tension in the chain 0 Applying equilibrium gives V 2Fy09TFgOeTFg 1 uuuuuuuuuuuuuuuuuuuuuuu an Equilibrium Example 1b T and F9 9 Not an actionreaction pair 3 Bothqact on the lamp T and T39 o Actionreaction forces a Lamp on chain and chain on lamp T39 and Tquot o Actionreaction forces a Chain on ceiling and ceiling on chain 0 Only the forces acting on the lamp are included in the free body gt diagram Fg uuuuuuuuuuuuuuuuuuuuuuu an Equilibrium Example 2a 0 Example 54 o Conceptualize the traffic light a Assume cables don t break 6 Nothing is moving 0 Categorize as an equilibrium problem a No movement so acceleration is zero e Model as a particle in equilibrium Equilibrium Example 2b 0 Analyze 0 Need two freebody diagrams 9 Apply equilibrium equation to the light 0 Apply equilibrium equations to the knot Equilibrium Example 2 c o Analyze cont o Find T3 from applying equilibrium in the ydirection to the light 0 Find T1 and T2 from applying equilibrium in the x and ydirections to the knot 0 Finalize 0 Think about different situations and see if the results are reasonable Inclined Planes o Forces acting on the object o The normal force acts perpendicular to the plane 0 The gravitational force acts straight down 0 Choose the coordinate system with X along the incline and y perpendicular to the incline 0 Replace the force of gravity with its components Forces of Friction 0 When an object is in motion on a surface or through a viscous medium there will be a resistance to the motion 0 This resistance is called the force of friction Explore Forces of Friction o Vary the applied force a Note the value of the frictional force Compare the values a Note what happens when the can starts to move gum nnnnnnnnnnnnnnnnnnnnnn u n Friction interactive AF 0516htm Forces of Friction o p depends on the surfaces in contact 0 fstatic is generally greater than fkinetic a direction is opposite the direction of slipping motion and H to the surface 0 p z independent of the area of contact Static Friction 0 Static friction acts to keep the object from movmg o If IE increases so does is o IfIE decreases so doesfs 0 f8 5 18 n o the equal sign is only valid when the surfaces are on the verge of slipping Kinetic Friction o The force of kinetic friction acts when the object is in motion 0 Although uk can vary with speed we shall neglect any such vana ons 39 fkHkn Friction Example o A 10kg block sits on an plane inclined at 25 The coefficient of kinetic friction between the block and the plane is p006 Ifthe block starts from rest how far does it move in 3s A block sits motionless on an inclined plane held in place by friction The plane could be tilted even more upwards and the block would remain motionless 9 Consider the following statements The magnitude of the frictional force equals AI 9 sin 9 II The magnitude of the frictional force is VI less than AI 9 sin 6 The magnitude of the frictional force is greater than 5 AI 9 sin 9 Static friction must vanish since the block is not moving The magnitude of the frictional force equals its AI 9 cos 0 The magnitude of the frictional force is less than its I 9 cos 9 Multiple Objects a When multiple objects are connected or in contact apply Newton s laws to 1 the system as a whole andor 2 to each individual object o Whichever you use to solve the problem the other approach can be used as a check Multiple Objects Conceptualize Two objects in contact With outside forces acting Determine a o the acceleration 31 a n2 0 Magnitudes of P12 P21 L F 21 32 x 39quotJ L mfg b Reverse the direction of the applied force F what changes Multiple Objects Example 1 o First treat the system as a whole EFX msystemax o Apply Newton s Laws to the individual blocks Solve for unknowns Example The mass of the heavier block is 20 kg and the mass of the lighter blocks is 10 kg and the magnitude of the force of the connecting string on the smaller block is 22 N Assume g 98 Lns2 and the horizontal surface on which the obj eets slide is frictionv 22N F Determine the force F What if now we have friction 02 N The mass of the heavier block is 20 kg and the mass of the lighter blocks is 10 kg and the magnitude of the force of the connecting string on the smaller block is 22 N Assume g 98 11152 and the horizontal surface on which the obj eets slide is frictionv Determine the force F Fuzzy dice I use the fuzzy dice hanging my car as an accelerometer All I look at is the angle they make with the vertical If the of the dice with the vertical is 30 what is the acceleration of my car Does it depend on the mass of the dice 30 GQOOOO OOQGOOO OOQQOO 0863 O ProblemSolving Hints Newton s Laws 0 Conceptualize 0 Draw a diagram 0 Choose a convenient coordinate system for each object o Categorize o Is the model a particle in equilibrium 0 If so 2F O 0 Is the model a particle under a net force 0 If so 2F m a ProblemSolving Hints Newton s Laws cont 0 Analyze O D D G 390 390 Draw freebody diagrams for each object Include only forces acting on the object Find components along the coordinate axes Be sure units are consistent Apply the appropriate equations in component form Solve for the unknowns 0 Finalize 390 0 Check your results for consistency with your freebody diagram Check extreme values PuHeys o Pulleys act to link directions that were independent before 0 massless and frictionless pulleys do not change the tension in a rope all they do is change the direction of the forces Later when we do pulleys with mass we will see the tension in the rope changes on either side of the pulley PuHeys When solving problems with pulleys 1 Always pick a direction 1St and clearly label it 2 Draw free body diagrams for each object 3 Do 2FXmaX ZFymay for each object 4 Pay attention to which directions are linked by the pulley MA10 kg MB 30 kg Assume the pulley is massless and frictionless and that there is no friction between mA and the table Find a and T in the rope on both sides of the pulley MA10 kg MB 30 kg Which of the following statements is true TATB Tor F i MA10 kg MB 30 kg Which of the following statements is true TMAg Tor F i MA10 kg MB 30 kg Which of the following statements is true TMBg Tor F i MA10 kg MB 30 kg Which of the following statements is true nA MAG Tor F i MA10 kg MB 30 kg What is the acceleration a MA10 kg MB 30 kg What is the tension T MA10 kg MB 30 kg What could be changed in this problem Atwood s Machine m2 8kg M15kg Find a While driving your pickup down the street your neighbor s yappy little dog leaps in front of your windshield Not wanting to hurt the poor little darling you decide to accelerate your car so it remains stuck to your windshield How fast must you accelerate so the dog does not slide down what will it depend on Assume your windshield is vertical and that p03 1 A it 1 If m 5kg find the reading on the scale for each case In 0 let the angle be 30 Enter your answer for case a Next we will start Circular Motion Ch 44 amp 45 Ch 61amp 62 Chapter 2 Motion in One Dimension Goals for learning 0 Understand how we describe position and motion in 1D 0 Definitions of and difference between o Displacement amp distance a Speed amp velocity 0 acceleration 0 Differences between average and instantaneous values 0 Difference between vectors and scalars in 1D Skills to develop 0 Translate between word descriptions and graphical representations of motion 0 Develop alternate methods for representing motion Kinematics the description of motion Does not address the causes of the motion 0 3 basic types of motion 0 Translational o Rotational o Vibrational Particle Model assume you can treat a complicated object as if it can be shrunk down to a point has mass but infinitesimal size For what types of problems would this not work Describing Position Choose a coordinate system c In 1D this is a straight line typically the x or the y axis o The choice of the origin is arbitrary pick it to make your problem as simple as possible o Choose which direction is positive 1 Dimensional Motion Movie 0 AF 0201htm Pictorial Representation Graph x m 60 I s 10 20 30 40 50 a nnnnnnnnnnnnnnnnnnnnnn cm o The positiontime graph shows the motion of the particle car Data Table TABLE 21 Position of the Car at Various Times Position t s x m 0 30 10 52 20 38 30 0 40 37 50 53 g Using Alternative Representations an excellent strategy for understanding a problem 0 Picture 0 Graph 0 Data Table w TABLE 21 Position of the Car at Various Times Position t s x m 0 30 10 52 20 38 30 0 8 40 37 0 10 20 30 40 50 50 53 b Our goal is often a mathematical representation for example x 30m50mst3m32t2 Displacement AX the change in position AX E Xfina 39 Xinitia SI units are meters m AX can be positive or negative Note not the same as the distance Distance d The total distance traveled is the sum of the absolute values of all the displacements for each stage of motion d2lei SI units are meters m distance is always positive x m Examples Find d and AX for the car moving between Points i From A to B ii From B to C o 10 20 30 40 50 I L iiiFrom A to D a gggggggggggggggggggg m b What types of differences can you have between d and Ax What causes the differences xi Two children start at one end of a street the origin run to 9 the other end then head back On the way back Joan is ahead of Mike Which statement is correct about the distances run and the displacements from the origin a Joan has run a greater distance and her displacement is greaterthan Mike39s b Mike has run a greater distance and his displacement is greaterthan Joan39s c Mike has run a greater distance but his displacement is less than Joan39s d Joan has run a greater distance but her displacement is less than Mike39s e Mike has run a shorter distance and his displacement is less than Joan39s Vectors and Scalars Scalar magnitude only Vectors magnitude and direction In 1 D we use and signs to indicate directions What type of quantity is the distance What type is the displacement Note although the distance is always positive in general scalars can be positive or negative For example temperature is a scalar no direction but you can have T 12 F We also want to describe how J an object s position changes in time Velocity displacement per unit time Speed distance per unit time 0 Both have dimensions are length time UT 0 The SI units are ms British system fts average velocity displacementtime over a large time interval Ax xf xi Vx avg At if ti Graphically this is the slope of the line in the position time graph Graph 60 1s 0 10 20 30 4O 50 a 2 D D 7 m H E d t From tA to tB vav average slope of xt line from A to B Average Speed distancetime over a finite time interval 1 t S avg o Speed is a scalar quantity 0 Since both d and t are a positive numbers s is always positive as well Find vav and sav for i t O to 23 9 ii t 23 to 53 lt iiitOs to 53 ivt5s to 553 v tOs to 833 ONPCDODO Think about it Can the mangitude of vav gt sav a A Yes b B No Explain and if yes under what circumstances Think about it Can the magnitude of vav lt sav Explain and if yes under what circumstances A Yes B No Instantaneous Velocity velocity at a given point in time o The limit of the average velocity as the time interval approaches zero Ax dx v 39 x lAltEQAt dt 0 The instantaneous velocity indicates what is happening at every point of time Instantaneous Velocity graph 60 Interactive Figure b nnnnnnnnnnnnnnnnnnnnnnn an o vtA slope X vs tcurve at time tA green line 0 as MAB gets smallervav between A and B blue lines approaches the instantaneous v at point A Instantaneous Speed 0 The instantaneous speed is the magnitude of the instantaneous velocity Sinst lvinstl o Scalar quantity Vocabulary Note 0 Velocity v and speed s will indicate instantaneous values 0 I will explicity mention average when the average velocity vav or average speed sav is intended Average Acceleration a the rate of change of the velocity gt Avx anf Viaf axavg At tf tl o Dimensions are LT2 a Si units are ms2 0 In 1D can be used to indicate direction Instantaneous Acceleration Instantaneous Acceleration graph 0 a slope of vt curve ij39 o The green line represents the vm instantaneous acceleration at time tf o The blue line is the average acceleration aaaaaaaaaaaaaaaaaaaaaaaaa an Graphical Comparison 0 Given the displacementtime graph a o vX slope of xt o ax slope of vxt PosMon x The velocity is defined as v dv dzx The acceleration is defined as ax E dtz lfl know a as a function of time how would I find v How would I find x Practice various forms A particle moves along the Laxis so that its velocity at any time t 2 0 is given by 1325 3t2 7 2t 7 1 The position t is 5 for t 2 a Find a polynomial expression for the position of the particle at any time t 2 0 L3 72 it3 z m3 3J3797t3 42 4 27t73 a ia3 o If we are given a plot of vt how do we determine the displacement o How would we find the acceleration Area 11x anglquot UXIL avg 1 lim vazn ffvxtdt ti Atn gt0 n o Displacement equals the area under the velocity time curve a definite integral Test Graphical Interpretations Ux U94 0 Mt V t t a b C ax ax l d 6 f 0 Which plot shows at for the vt curve shown in plot a d Test Graphical Interpretations Ux U94 0 Mt V t t a b C ax ax l d 6 f 0 Which plot shows vt for the at curve shown in plot d d Deriving the Kinematic Equations in 1D PosMon x dx The veIOCIty IS defined as vx dt dv dzx The accelerat on 8 def ned as x 2 39 39 39 dt dt Kinematic Equations Calculus Form with Constant Acceleration dv a x 3 o If a constant dt vx f axdt gt vx axt C vx axt v0 We will derive the kinematic equations for the special case of constant acceleration aconstant DO NOT USE THEM IF a constant Kinematic Equations xt for aconst dx v 2 x dr x fvxdt gt x fv0 axtdt xv0taxt2 C plug in t0xC so x x0 v0taxt2 Kinematic Equations xt for aconst G xx0 v0taxt2 C2 vx v0axt We can derive a third equation by solving Eq 2 fort and plugging into Eq 1 a v v 2axx x0 Let s do some examples of problems with constant a A car traveling at 7390 111 ELC C CIE39I ELtC S 25 111quot82 t o reach a speed of 12 0 111 5 HOW long does it take for this acceleratioll to C 13111quot Two ants race across a table 50 cm long One travels at 4 cm s and the other at 2 01115 When the rst one crosses the nish line how far behind is the second one A truck covers 350 m in 850 5 while smoome slowing down to nal speed 300 ms a nd its on39ginal speed b nd its acceleration A Elf travelian at a mnslam Ei BEtl 13f 405 urn395 35ng trrmgpar 39Elli mutmw le hi dm blamier El hullhm rm EE39i Emlfl altar the spewing car DEELE E39E rm EerIIEJErEIr l E EELS mt rm l 1MB hil ll rril flu atlm the tar Hmlgrahmg at a Exam3mm ratE 1390 mm Him lung lms it 13m quotJuly It awr mm mg mr39 0 Go over Example 28 from text i E 39 quot39 u l I V 5 1 A magcling car 335595 a hidden hammer Susan and Jerome are playing soccer Suzie takes control of the ball while standing directly on the goal line She immediately starts running fonNard with an acceleration of 6 fts2 At the moment the catch is made Jerome is 20 yards away and heading directly toward Suzie with a steady speed of 15 fts If neither deviates from a straightahead pathwhere will the two comde Hint Start by drawing a picture Judy and Martha mortal enemies are driving their cars on the same stretch of road one night Judy is traveling 12 ms due East and Martha is traveling 5 ms due West They spot each other and enter in a deadly game of chicken Determine the distance each of them travels before they collide 839 A train travels between stations 1 and 2 as shown in the gure The engineer of the train is instructed to start from rest at station 1 and accelerate uniformly between points A and B then coast with a uniform velocity be tween points B and C and nally accelerate uniformly between points C and D until the train stops at station 2 The distances AB BC and CD are all equal and it takes 500 min to travel between the two stations As sume that the uniform accelerations have the same magnitude even when they are opposite in direction Station A Station B I39 l 1 if 1 A B C D a How much of this 500 min period does the train spend between points A and B 39 r The graph shows the velocity versus time graph for a ball Which explanation best fits the motion of the ball as shown by the graph 1 The ball is falling is caught and is thrown down with greater velocity 2 The ball is rolling stops and then continues rolling 3 The ball is rising hits the ceiling and falls down 4 The ball is falling hits the floor and bounces up 5 The ball is rising is caught and then is thrown down o 3 I I an i I I o o l III ii In 1quot III III 0 l in II t a it 1 an 4 39 l I I I l i I I I i I I a t l I l l i Q 1 Five motion diagrams in which points represent the positions of an object at equal time intervals are shown above Which statement is correct a A has the greatest speed and the greatest acceleration b C has decreasing speed c D slows down and then speeds up d D speeds up and then slows down e E has a greater speed than A Chapter 4 Motion in Two Dimensions Note we only cover secions 41 to 43 now We save sections 44amp45 for Ch 6 when we discuss circular motion Motion in Two Dimensions 0 Using or signs is not sufficient to fully describe motion in more than one dimension 0 Vectors can be used to more fully describe motion 0 Will look at vector nature of quantities in more detail 0 Need to define displacement velocity and acceleration in multiple Dimensions Position and Displacement 0 position vector in rect coordinates y 17xiyjzk A I tl tf o displacement I change in its position 1 Path of gt 39 16 rf partlc Arn n arr an aaaaaaaaaaaaa an Average Velocity o average velocity 4 A x x0A 0 A Vavrf 1W YJJ At tf to tf to 0 direction of vav is direction A 0 vav is independent of the path taken because it is dependent on the displacement also independent of the path Instantaneous Velocity Direction of at 7 AFd7 JEAN dt dxX dy z dt dt a 2mm Thamm nnnnnnnnnnnnn an Instantaneous Velocity cont 0 direction of at any point is tangent to the path at that point and in the direction of motion 0 magnitude ofVis the speed c The speed is a scalar quantity Average Acceleration Average Acceleration cont 0 As particle moves y direction of Av in i velocity is found by 1 vector subtraction 6 m7 vf v 39 a V AV 5i 0 Direction of aav is along AV 0 x a my mamaquot man rrrrrrr an Instantaneous Acceleration a AV av aEIImA Ifa AteO dv dVy quotl dt dt Producing An Acceleration 0 Various changes in 17 may produce an acceleration o magnitude can change 0 direction can change 0 Both may change simultaneously Finding Kin Eq for general cases at constant 2 Given that a particle starts at a 2 3 And has a velocity vector given by 17 5 2t2i 7t 2t3j a Determine the units on the constants so the equations are dimensionally correct b Determine the acceleration as a function of time c The position of the particle at t2s Kinematic Equations for 2D Motion 0 When 2D motion has a constant a we can derive Kin Eq similar to 1D 0 Motion in 2D can be modeled as two independent motions in two perpendicular directions usually labeled xampy c Any influence in the y direction does not affect the motion in the x direction 0 Time is the 1 thing that links both directions Projectile Motion Diagram Projectile Motion 0 An object may move in both the X and y directions simultaneously 0 The form of twodimensional motion we will deal with is called projectile motion Assumptions of Projectile Motion 0 freefall acceleration is constant over the range of motion gt ay 98 ms2 0 The effect of air friction is negligible X an object in projectile motion will follow a parabolic path Projectile Motion Diagram A 065 kg rock is projected from the edge of the top of a building with an initial Velocity of 1013 111 s at an angle 54 above the horizontal The building is 905 in in height Assume The ground is level and that the side of the building is Vertical The accelorw tion of gravity is 98 insZ t what horizontal distance x from the base ofthe building will the rock strike the ground 39 39uvovvo oooooo was o X direction 003 54 0587 0 on 0587 105ms 617ms 0 Y dir sin54 0809 0 Vyo 0809 105 849 ms 0 O hvyOt12gt2 a 4399 Xf39 I 617247 ma 2 849 152399 905 Solutions 0745 t 247 o In this same problem 0 What is the ball s velocity at the maximum height 9 What is the magnitude of the ball s velocity just before it hits the ground 9 What is the direction of the ball s velocity just before it hits the ground 9 You have gotten a job at the circus over spring break Unfortunately for you your get the job of being shot out of the canon You read the manual and notice that you will be shot out of the canon at a speed of 33ms The canon is xed so that it Will shoot you out at an angle of 50 with respect to the ground The muzzle of the canon is 15 m above the ground The tent is 60 m Wide Where should you tell them to position the net on the other side of the tent to catch you To start an avalanche on a mountain slope an artillery shell is fired with an initial velocity of 350 ms at 530 above the horizontal It explodes on the mountainside 390 s after firing What are the x and y coordinates of the shell where it explodes relative to its firing point On the back of a moving truck what happens Z39Ell39l l l ll It Bf Vii 3i fram s par smn Juan EDIIEQE Ert HE BMW v I thE had fth mink q i53quotlnng A re ghter 500 m away from a burning building directs a stream of water from a rehose at an anglr of 300quot above the horizohml as in Figure P418 If Lhc initial speed of the stream i5 400 ms at whm height does the water strike the building Santa is on the roof of a 15 m tall building The roof is icy so he starts sliding down the roof with a constant acceleration of 5 ms2 along the roo ine If the distance he slides is 12 m determine how far from the base from the 3 building he will land 12m 5 15m A grasshopper jumps a horizontal distance of 130m from rest with an initial velocity at an angle of 370 with respect to the horizontal a Find the initial speed of the grasshopper b Find the maximum height reached by the grasshopper A ball is thrown and follows the parabolic path shown Air friction is negligible oiut Q is the highest point on the path Points P and R are the same height above the ground How do the speeds of the ball at the three llts compare 5b lt IWPH HFRH M lt IWQH lt HFRH nli lt H170 lt HEPH Bali lt IWRH lt HEPH pli HFRH lt M 395 Pictured below are eight arrows that have been shot horizontally ie straight out by archers on platforms All of the arrows are identical but they have been shot at different speeds from platforms of varying height Specific values for the speeds and of varying heights are given in the figures All of the arrows miss the targets and hit the ground Rank these arrows from longest to shortest on the basis of how long it takes the arrows to hit the ground That is put first the arrow that will take the longest time from being shot to hitting the ground and put last the arrow that will take the shortest time 5 ms 8 ms 8 ms 5 ms A lt 0 A B C D 10 ms 8 We 10 ms 5 ms 2 m 23 m 3 m 30 m E F G H Longest 1 2 3 4 5 6 7 8 Shortest 0r all of the arrows hit the ground at the same time A home run in a baseball game is hit in such a way that the ball just clears the wall which surrounds the out eld challenging t fz l k zinm H a What is the initi u sywu m ulv w b How long does it take the ball to reach the wall Shoot the Monkey Demonstration 9 Given The battleship and enemy ships A and B lie along a straight line Neglect air friction A battleship si111ulta1reously res two shells with the same muzzle velocity at these two enemy ships battleship A B If the shells follow the parabolic trajectories shown in the gure which ship gets hit rst 1 l3 2 A 3 both at the same time 4 need more i11for111atio11

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