chapter-2-motion-along-a-straight-linePPT优秀课件

Copyright 2008 Pearson Education Inc.,publishing as Pearson Addison-WesleyPowerPoint Lectures forUniversity Physics,Twelfth Edition Edited by Richard Hugh D.Young and Roger A.FreedmanLectures by James PazunChapter 2Motion Along a Straight LineGoals for Chapter 2To study motion along a straight lineTo define and differentiate average and instantaneous linear velocityTo define and differentiate average and instantaneous linear accelerationTo explore applications of straight-line motion with constant accelerationTo examine freely falling bodiesTo consider straight-line motion with varying acceleration2.1 Displacement,time,and the average velocity Displacement:change in position,it is a vector quantity.Its direction is from start to end.x=x2 x1 Average x-velocity:the displacement,x,divided by the time interval t.Time:change in timet=t2 t11212ttxxtxvxavDistance and Average Speed Distance:length of the path,it depends on the path.It is a scalar quantity.It has no direction.Average x-speed:the distance traveled s divided by the time interval t.It is a scalar.Average speed vs.average velocityWhen Alexander Popov set a world record in 1994 by swimming 100.0 m in 46.74 sec,his average speed was(100.0 m)/(46.74 s)=2.139 m/s.but because he swam four lengths in a 25 meter pool,he started and ended at the same point and he had zero total displacement and zero average velocity!example What is the average velocity of the car?Position at t1=1.0 sPosition at t2=4.0 sx1=19 mx2=277 m vav-x=(277 m 19 m)/(4.0 s 1.0 s)=86 m/s The average velocity is positive because it is moving in the positive direction.Note:you can choose any way as+.+displacementP-T graph of the cart(s)X(m)x1=19mx2=277mSlope=vav-xxtt=1 st=4 sCheck your understanding 2.1Each of the following automobile trips takes one hour.The positive x-direction is to the east.A travels 50 km due east.B travels 50 km due westC travels 60 km due east,then turns around and travels 10 km due westD travels 70 km due east.E travels 20 km due west,then turns around and travels 20 km due east.Rank the five trips in order of average x-velocity from most positive to most negative.Which trips,if any,have the same average x-velocity?For which trip,if any,is the average x-velocity equal to zero?4,1,3,5,21,35Practice 2.2In an experiment,a shearwater(a seabird)was taken from its nest,flown 5150 km away,and released.The bird found its way back to its nest 13.5 days after release.If we place the origin in the nest and extend the+x-axis to the release point,what was the birds average velocity in m/sFor the return flight?1.For the whole episode,from leaving the nest to returning?-4.42 m/s0 m/sPractice 2.4Starting from a pillar,you run 200 m east(the+x-axis)at an average speed of 5.0 m/s,and then run 280 m west at an average speed of 4.0 m/s to a post.Calculate Your average speed from pillar to post,1.You average velocity from pillar to post.4.4 m/s-0.72 m/sPractice 2.6Two runners start simultaneously from the same point on a circular 200 m track and run in the same direction.One runs at a constant speed of 6.20 m/s,and the other runs at a constant speed of 5.50 m/s.When will the fast one first“lap”the slower one and how far from the starting point will each have run?1.When will the fast one overtake the slower one for the second time,and how far from the starting point will they be at that instant?286 s,1770 m,1570 m572 s,3540 m,3140 mPractice 2.8A Honda Civic travels in a straight line along a road.Its distance x from a stop sign is given as a function of time t by the equation x(t)=t2 t3,where =1.50 m/s2 and =0.0500 m/s3.Calculate the average velocity of the car for each time interval:t=0 to t=2.00 s;t=0 to t=4.00 sa.t=2.00 s to t=4.00 s.example A cat runs along a straight line(the x-axis)from point A to point B to point C,as shown.The distance between points A and C is 5.00 m,the distance between points B and C is 10.0 m,and the positive direction of the x-axis points to the right.The time to run from A to B is 20.0 s,and the time from B to C is 8.00 s.BCAWhat is the average speed of the cat between points A and C?What is the average velocity of the cat between points A and C?Example-Walking 1/2 the time vs.Walking 1/2 the distance Tim and Rick both can run at speed vr and walk at speed vw,with vw vr.They set off together on a journey of distance D.Rick walks half of the distance and runs the second half.Tim walks half of the time and runs the other half.a)Draw a graph showing the positions of both Tim and Rick versus time.b)Write two sentences explaining who wins and why.c)How long does it take Rick to cover the distance D?d)Find Ricks average speed for covering the distance D.e)How long does it take Tim to cover the distance?txDD/2tTim tTimtRick Tim wins because he takes short time to cover the same distance as Rick.a.solution)11(222wrwrRickvvDvDvDtd.wrwrRickRickvvvvtDv)(2c.wrTimTimwTimrvvDttvtvD2)2()2(e.16Vectors V1 and V2 shown above have equal magnitudes.The vectors represent the velocities of an object at times t1 and t2,respectively.The average acceleration of the object between time t1 and t2 wasZeroDirected northDirected westDirected north of eastDirected north of west2.2 Instantaneous velocity Instantaneous velocity is defined as the velocity at any specific instant of time or specific point along the path.Instantaneous velocity is a vector quantity,its magnitude is the speed,its direction is the same as its motions direction.How long is an instant?In physics,an instant refers to a single value of time.To find the instantaneous velocity at point P1,we move the second point P2 closer and closer to the first point P1 and compute the average velocity vav-x=x/t over the ever shorter displacement and time interval.Both x and t become very small,but their ratio does not necessarily become small.In the language of calculus,the limit of x/t as t approaches zero is called the derivative of x with the respect to t and is written dx/dt.P1P2 The instantaneous velocity is the limit of the average velocity as the time interval approaches zero;it equals the instantaneous rate of change of position with time.dtdxtxvtx0limA cheetah is crouched 20 m to the east of an observers vehicle.At time t=0 the cheetah charges an antelope and begins to run along a straight line.During the first 2.0 s of the attack,the cheetahs coordinate x varies with time according to the equation x=20 m+(5.0 m/s2)t2.Find the displacement of the cheetah between t1=1.0 s and t2=2.0 sFind the average velocity during the same time interval.Find the instantaneous velocity at time t1=1.0 s by taking t=0.1 s,then t=0.01 s,then t=0.001 s.a.Derived a general expression for the instantaneous velocity as a function of time,and from it find vx at t=1.0 s and t=2.0 s Example 2.1 Example 2.1 Average and instantaneous velocityAverage and instantaneous velocities in x-t graph Secant line average velocitytangent line instantaneous velocityexample Which car starts later?When does A&B pass each other?Which car reaches 200 km first?Calculate average speed of A and B.The automobiles make a 5 hour trip over a total distance of 200 km.The Derivative akaThe SLOPE!Suppose an eccentric pet ant is constrained to move in one dimension.The graph of his displacement as a function of time is shown below.tx(t)t+tx(t+t)ABAt time t,the ant is located at Point A.While there,its position coordinate is x(t).At time(t+t),the ant is located at Point B.While there,its position coordinate isx(t+t)The secant line and the slopeSuppose a secant line is drawn between points A and B.Note:The slope of the secant line is equal to the rise over the run.tx(t)t+tx(t+t)ABThe slope of the secant line is average velocityThe“Tangent”lineREAD THIS CAREFULLY!If we hold POINT A fixed while allowing t to become very small.Point B approaches Point A and the secant approaches the TANGENT to the curve at POINT A.tx(t)t+tx(t+t)ABtx(t)t+tx(t+t)ABWe are basically ZOOMING in at point A where upon inspection the line“APPEARS”straight.Thus the secant line becomes a TANGENT LINE.The slope of the tangent line is _ velocity.The derivativeMathematically,we just found the slope!line tangent of slope)()(limlinesecant of slope)()(01212ttxttxttxttxxxyyslopetLim stand for“_ and it shows the t approaches zero.As this happens the top numerator approaches a finite#.This is what a derivative is.A derivative yields a NEW function that defines the rate of change of the original function with respect to one of its variables.The above example shows the rate of change of x with respect to time.In most Physics books,the derivative is written like this:Mathematicians treat as a SINGLE SYMBOL which means find the derivative.It is simply a mathematical operation.The derivative is the slope of the line tangent to a point on a curve.dtdxexample Consider the function x(t)=3t+2;What is the time rate of change of the function(velocity)?This is actually very easy!The entire equation is linear and looks like y=mx+b.Thus we know from the beginning that the slope(the derivative)of this is equal to 3.We didnt even need to INVOKE the limit because the t is cancel out.Regardless,we see that we get a constant.Example Consider the function x(t)=kt3,where k=proportionality constant.What happened to all the ts?They went to ZERO when we invoked the limit!What does this all mean?22322033223033003)3()()(33 lim)()(3)(3lim)()(lim)()(lim)(kttkttttkttttttttkttkttkttxttxdttdxttttThe MEANING?For example,if t=2 seconds,using x(t)=kt3=(1)(2)3=8 meters.The derivative,however,tell us how our DISPLACEMENT(x)changes as a function of TIME(t).The rate at which Displacement changes is also called VELOCITY.Thus if we use our derivative we can find out how fast the object is traveling at t=2 second.Since dx/dt=3kt2=3(1)(2)2=12 m/s233)(ktdtktdTHERE IS A PATTERN HERE!4534125)(4)(2)(ktdtktdktdtktdktdtktdExample x=5,Derivative of a constant0)(CdxdWhy?dtdx33Power Rule1)(nnxnxdxdx=t5Example x=t-5x=t?dtdx?dtdx?dtdx34Constant Multiplier)()(xfdxdcxfcdxdExample?dtdxx=4t535Addition and Subtraction Rule)()()()(xgdxdxfdxdxgxfdxdThe derivative of the sum(or difference)of two or more functions is the sum(or difference)of the derivatives of the functions.x=2t5+3t-1Example?dtdxChain ruleIf x is a function of f,and f is a function of t,so indirectly,x is a function of t:x(f(t)Example dtdfdfdxdtdx?dtdx215)32(ttx21532fxttfClass workFind the derivatives(dx/dt)of the following functionx=t3x=1/t=t-1x=(6t3+2/t)-21.x=16t2 16t+4Average velocity vs.instantaneous velocity ExampleA Honda Civic travels in a straight line along a road.Its distance x from a stop sign is given as a function of time t by the equation x(t)=t2 t3,where =1.50 m/s2 and =0.0500 m/s3.Calculate the average velocity of the car for the time interval:t=0 to t=4.00 s;Determine the instantaneous velocity of the car at t=2.00 s and t=4.00 s.example An object is moving in one dimension according to the formula x(t)=2t3 t2 4.find its velocity at t=2 s.example The position of an object moving in a straight line is given by x=(7+10t 6t2)m,where t is in seconds.What is the objects velocity at 4 seconds?Example An object moves vertically according to y(t)=12 4t+2t3.what is its velocity at t=3 s?example An objects motion is given by the equation x(t)=2+4t3.what is the equation for the objects velocity?v(t)=12t2Follow the motion of a particle The motion of the particle may be described from x-t graph.Questions The graph above shows velocity v versus time t for an object in linear motion.Which of the following is a possible graph of position x versus time t for this object?Test your understanding 2.2According to the graphRank the values of the particles x-velocity vx at the points P,Q,R,and S from most positive to most negative.At which points is vx positive?At which points is vx negative?At which points is vx zero?a.Rank the values of the particles speed at the points P,Q,R,and S from fastest to slowest.PRQ,SR,P,Q=SExample 2.10A physics professor leaves her house and walks along the side walk toward campus.After 5 min it starts to rain and she returns home.According to the graph,at which of the labeled points is her velocityZero?Constant and positive?Constant and negative?Increasing in magnitude?a.Decreasing in magnitude?IVIVIIIIIexample Which pair of graphs represents the same 1-dimensional motion?A.B.C.D.exampleThe graph represents the relationship between distance and time for an object.What is the instantaneous speed of the object at t=5.0 seconds?a.t=2.0 seconds?01.5 m/sexampleAccording to the graph,the acceleration of the object must beZeroConstant and positiveConstant and negativeIncreasinga.decreasingtdo2.3 average and instantaneous acceleration The average acceleration of the particle as it moves from P1 to P2 is a vector quantity,whose magnitude equals to the change in velocity divided by the time interval.Velocity describes how fast a bodys position change with time.Acceleration describes how fast a bodys velocity change,it tells how speed and direction of motion are changing.12vvtvttvvaavg1212Instantaneous accelerationdtdvtvat0lim22)(dtxddtdxdtdadtdxvThe instantaneous acceleration is the limit of average acceleration as the time interval approaches zero.Average and instantaneous acceleration Example 2.3Suppose the x-velocity vx of a car at any time t is given by the equation:vx=60 m/s+(.50 m/s2)t2Find the change in x-velocity of the car in the time interval between t1=1.0 s and t2=3.0 s.Find the average x-acceleration between t1=1.0 s and t2=3.0 s.Derive an expression for the instantaneous x-acceleration at any time,and use it to find the x-acceleration at t=1.0 s and t=3.0 s.4.0 m/s 2.0 m/s2 a=(1.0 m/s3)t;1.0 m/s2;3.0 m/s2example The position of a vehicle moving on a straight track along the x-axis is given by the equation x(t)=t2+3t+5 where x is in meters and t is in seconds.What is its acceleration at time t=5 s?(2 m/s2)Finding acceleration on a vx-t graph and ax-t graphAverage acceleration can be determined by v-t graphFinding the acceleration on v-t graphA graph of and t may be used to find the acceleration.Average acceleration:the slope of secant line.Instantaneous acceleration:the slope of a tangent line at point.Caution:The sign of acceleration and velocitya is in the same direction as vv:posa:pos.v:neg.a:neg.a is in the opposite direction as vv:posa:neg.v:neg.a:pos.We can obtain an objects position,velocity and acceleration from it v-t graphFinding acceleration on a x-t graphOn a x-t graph,the acceleration is given by the curvature of the graph.Curves up from the point:acceleration is positivestraight or not curves up or down:acceleration is zeroCurves down:acceleration is negativeExampleVtta The figure is graph of the coordinate of a spider crawling along the x-axis.Graph its velocity and acceleration as function of time.60Check your understanding 2.3Refer to the graph,At which of the points P,Q,R,and S is the x-acceleration ax positive?At which points is the x-acceleration ax negative?At which points does the x-acceleration appear to be zero?1.At each point state whether the speed is increasing,decreasing,or not changing.P:v is not change;Q:v is zero,changing from pos.to neg.,first decrease in pos.then increase in neg.,R:v is neg.,constant;S:v is zero,changing from neg.to pos.,first decrease in neg.then increase in pos.,SQP,R2.4 motion with constant accelerationGiven:derive:vx=vx0+axt(assume t0=0)x=x0+vx0+axt2vx2 vx02=2ax(x x0)txxvxav020 xxxavvvvtvvaxxxav0Motion with constant acceleration vx-t graphA horizontal line indicate the slope=0,a=0Since ax=v/t;v=axt which is represented by the area.a-t graphThe area indicate the change in velocity during tKinematics equations for constant accelerationExample 2.4A motorcyclist heading east through a small Iowa city accelerates after he passes the signpost marking the city limits.His acceleration is a constant 4.0 m/s2.At time t=0 he is 5.0 m east of the signpost,moving east at 15 m/s.Find his position and velocity at time t=2.0 s.a.Where is the motorcyclist when his velocity is 25 m/s?Example 2.5A motorist traveling with a constant speed of 15 m/s passes a school crossing corner,where the speed limit is 10 m/s.Just at the motorist passes,a police officer on a motorcycle stopped at the corner starts off in pursuit with constant acceleration of 3.0 m/s2.How much time elapses before the officer catches up with the motorist?What is the officers speed at that point?a.What is the total distance each vehicle has traveled at that point?Test your understanding 2.4Four possible vx-t graphs are shown for the two vehicles in example 2.5.which graph is correct?If we ignore air friction and the effects due to the earths rotation,all objects fall at the constant acceleration.The constant acceleration of a freely falling body is called the acceleration due to gravity,and we use letter g to represent its magnitude.Near the earths surface g=9.81 m/s/s=32 ft/s/sOn the surface of the moon,g=1.6 m/s/sOn the surface of the sun,g=270 m/s/sa=-g2.5 Free Falling BodiesExample 2.6A one-euro coin is dropped from the Leaning Tower of Pisa.It starts from rest and falls freely.Compute its position and velocity after.1.0 s.2.0 s,and 3.0 s.Example 2.7You throw a ball vertically upward from the roof of a tall building.The ball leaves your hand at a point even with the roof railing with an upward speed of 15.0 m/s;the ball is then in free fall.On its way back down,it just misses the railing.At the location of the building,g=9.80 m/s2.findThe position and velocity of the ball 1.00 s and 4.00 s after leaving your handThe velocity when the ball is 5.00 m above the railingThe maximum height reached and the time at which it is reacheda.The acceleration of the ball when it is at its maximum height.Velocity and acceleration at the highest pointExample 2.8Find the time when the ball in Example 2.7 is 5.00 m below the roof railing.Check your understanding 2.5If you toss a ball upward with a certain initial speed,it falls freely and reaches a maximum height h at time t after it leaves your hand.If you throw the ball upward with double the initial speed what new maximum height does the ball reach?1.If you throw the ball upward with double the initial speed,how long does it take to reach its maximum height?4h2t In the case of straight-line motion,if the position x is a known function of time,we can find vx=dx/dt to find x-velocity.And we can use ax=dvx/dt to find the x-acceleration as a function of time In many situations,we can also find the position and velocity as function of time if we are given function ax(t).2.6 velocity and position by integrationFinding v(t)and x(t)when given a(t)The“AREA”In v-t graph,the area under the line represent displacement.How ever,if acceleration is not constant,how can we determine x(t)?t1t2v(m/s)t(s)x=areat1t2v(m/s)t(s)x=areat1v(m/s)t(s)t1v(m/s)t1v(m/s)v(t)t2v(m/s)Zoom indtdxvtxvt0limWe have learned that the rate of change of displacement is defined as the VELOCITY of an object.Consider the graph belowdttvdx)(t1t2v(m/s)t(s)tt1t2v(m/s)t(s)tt1t2v(m/s)t(s)tt1v(t)t2v(m/s)t(s)tdtArea=v(t)dtTOTAL DISPLACEMENT=Area=v(t)dtThe“Integral”the area The temptation is to use the conventional summation sign“S.The problem is that you can only use the summation sign to denote the summing of DISCRETE QUANTITIES and NOT for something that is continuously varying.Thus,we cannot use it.When a continuous function is summed,a different sign is used.It is called an Integral,and the symbol looks like this:When you are dealing with a situation where you have to integrate,realize:WE ARE GIVEN:the derivative already WE WANT:The original function x(t)So what are we basically doing?WE ARE WORKING BACKWARDS!OR FINDING THE ANTI-DERIVATIVEExampleAn object is moving at velocity with respect to time according to the equation v(t)=2t.a)What is the displacement function?Hint:What was the ORIGINAL FUCNTION BEFORE the“derivative?was taken?b)How FAR did it travel from t=2s to t=7s?dttdtvtx)2()(2)(ttx44927)2()(2