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Kinematics Equations Kinematics Kinematic Equations Lana Sheridan De Anza College Jan 9, 2020 Last time • kinematic quantities • graphs of kinematic quantities Overview • relating graphs • derivation of kinematics equations 2.4 Acceleration 33 down! It is very useful to equate the direction of the acceleration to the direction of a force because it is easier from our everyday experience to think about what Pitfall Prevention 2.4 effect a force will have on an object than to think only in terms of the direction of Negative Acceleration Keep in the acceleration. mind that negative acceleration does not necessarily mean that an object is slowing down. If the acceleration is Q uick Quiz 2.4 If a car is traveling eastward and slowing down, what is the direc- negative and the velocity is nega- tion of the force on the car that causes it to slow down? (a) eastward (b) west- tive, the object is speeding up! ward (c) neither eastward nor westward Pitfall Prevention 2.5 Deceleration The word deceleration has the common popular connota- From now on, we shall use the term acceleration to mean instantaneous accelera- tion of slowing down. We will not tion. When we mean average acceleration, we shall always use the adjective average. use this word in this book because Because v 5 dx/dt, the acceleration can also be written as it confuses the definition we have x given for negative acceleration. dv d dx d 2x a 5 x 5 5 (2.12) x dt dt dt dt 2 That is, in one-dimensional motion, the accelerationa b equals the second derivative of x with respect to time. Conceptual Example 2.5 Graphical Relationships Between x, vx, and ax The position of an object moving along the x axis varies with time as in Figure 2.8a. Graph the velocity versus time and the acceleration versus time for the object. Acceleration vs. Time Graphs SOLUTION x The velocity at any instant is the slope of the tangent to the x–t graph at that instant. Between t 5 0 and t 5 tᎭ, the slope of the x–t graph increases uniformly, so the velocity increases linearly as shown in Figure 2.8b. Between tᎭ and tᎮ, the slope of the x–t graph is con- stant, so the velocity remains constant. Between tᎮ and t tᎭ tᎮ t Ꭿ t൳ t൴ t൵ t൳, the slope of the x–t graph decreases, so the value of a the velocity in the v –t graph decreases. At t , the slope x ൳ vx of the x–t graph is zero, so the velocity is zero at that instant. Between t൳ and t൴, the slope of the x–t graph and therefore the velocity are negative and decrease uni- t tᎭ tᎮ tᎯ t൳ t൴ t൵ formly in this interval. In the interval t൴ to t൵, the slope of the x–t graph is still negative, and at t൵ it goes to zero. b Finally, after t൵, the slope of the x–t graph is zero, mean- ax ing that the object is at rest for t . t൵. The acceleration at any instant is the slope of the tan- gent to the v –t graph at that instant. The graph of accel- t x t t t eration versus time for this object is shown in Figure 2.8c. tᎭ Ꭾ ൴ ൵ The acceleration is constant and positive between 0 and c tᎭ, where the slope of the vx–t graph is positive. It is zero Figure 2.8 (Conceptual Example 2.5) (a) Position–time graph between tᎭ and tᎮ and for t . t൵ because the slope of the v –t graph is zero at these times. It is negative between for an object moving along the x axis. (b) The velocity–time graph x for the object is obtained by measuring the slope of the position– tᎮ and t൴ because the slope of the vx–t graph is negative time graph at each instant. (c) The acceleration–time graph for during this interval. Between t൴ and t൵, the acceleration the object is obtained by measuring the slope of the velocity–time is positive like it is between 0 and tᎭ, but higher in value graph at each instant. because the slope of the vx–t graph is steeper. Notice that the sudden changes in acceleration shown in Figure 2.8c are unphysical. Such instantaneous changes cannot occur in reality. Question What does the area under an acceleration-time graph represent? 36 Chapter 2 Motion in One Dimension In Figure 2.10c, we can tell that the car slows as it moves to the right because its 36 Chapter 2 Motiondisplacement in One Dimension between adjacent images decreases with time. This case suggests the car moves to the right with a negative acceleration. The length of the velocity arrow Chapter 2 Motion in One Dimension 36 decreasesIn Figure in 2.10c, time weand can eventually tell that thereaches car slows zero. as From it moves this todiagram, the right we because see that its the displacementacceleration betweenand velocity adjacent arrows images are notdecreases in the withsame time. direction. This case The suggests car is moving the car withIn moves Figurea positive to the2.10c, velocity, right we with canbut atellwith negative that a negative the acceleration. car acceleration. slows as The it moves (This length totype of the the of right velocitymotion because arrowis exhib its - decreasesdisplacementited by a carin time thatbetween andskids eventually adjacent to a stop images afterreaches its decreases zero.brakes From are with applied.)this time. diagram, This The case wevelocity see suggests that and the accelthe - accelerationcareration moves are to inandthe opposite rightvelocity with directions.arrows a negative are Innot acceleration. terms in the of same our The earlier direction. length force ofThe discussion,the car velocity is moving imaginearrow withdecreasesa force a positive pulling in timevelocity, on and the but eventuallycar with opposite a negative reaches to the acceleration. zero. direction From (This itthis is moving:typediagram, of motionit weslows see isdown. thatexhib the- itedacceleration Each by a car purple that and skids accelerationvelocity to a arrowsstop arrow after are its innot brakesparts in the (b)are same andapplied.) direction.(c) of The Figure velocityThe 2.10car and is moving theaccel same- erationwithlength. a positive are Therefore, in oppositevelocity, these but directions. diagramswith a negative In represent terms acceleration. of our motion earlier (This of forcea particletype discussion, of undermotion constant imagine is exhib accel - - aitederation. force by pullinga This car that important on skids the car to analysis aopposite stop after model to itsthe brakeswill direction be arediscussed itapplied.) is moving: in theThe it next velocityslows section. down. and accel- erationEach arepurple in opposite acceleration directions. arrow Inin termsparts (b)of our and earlier (c) of forceFigure discussion, 2.10 is the imagine same length.aQ force uick Therefore,pulling Quiz 2.5 on Whichthesethe car diagrams one opposite of the represent to following the direction motion statements itof is a moving:particle is true? under it (a)slows constantIf adown. car accelis trav-- eration. Eacheling This purpleeastward, important acceleration its acceleration analysis arrow model mustin willparts be be eastward.(b) discussed and (c) (b) in of Ifthe Figurea nextcar is section.2.10 slowing is the down, same length.its acceleration Therefore, thesemust bediagrams negative. represent (c) A particle motion with of a constantparticle under acceleration constant accelcan - Qeration. uicknever QuizThis stop 2.5important and Which stay stopped. analysisone of the model following will be statements discussed isin true? the next (a) If section. a car is trav- eling eastward, its acceleration must be eastward. (b) If a car is slowing down, Q itsuick acceleration Quiz 2.5 Whichmust be one negative. of the following(c) A particle statements with constant is true? acceleration (a) If a car iscan trav- nevereling eastward,stop and stay its accelerationstopped. must be eastward. (b) If a car is slowing down, x 2.6its acceleration Analysis must Model: be negative. Particle (c) A particle with constant acceleration can never stop and stay stopped. Slope ϭ vxf Under Constant Acceleration x 2.6If the accelerationAnalysis of Model:a particle varies Particle in time, its motion can be complex and difficult Slope ϭ v Constant Accelerationx Graphs xf Underto analyze. Constant A very common Acceleration and simple type of one-dimensional motion, however, is i x that2.6 in Analysiswhich the acceleration Model: is Particle constant. In such a case, the average acceleration Slope ϭ vxi If the acceleration of a particle varies in time, its motion can be complex and difficult Slope ϭ vxf t a over any time interval is numerically equal to the instantaneous acceleration a t toUnder xanalyze.,avg AConstant very common Acceleration and simple type of one-dimensional motion, however, is x xi at any instant within the interval, and the velocity changes at the same rate through- thatIf the in acceleration which the acceleration of a particle isvaries constant. in time, In itssuch motion a case, can the be averagecomplex acceleration and difficult aSlope ϭ vxi out the motion. This situation occurs often enough that we identify it as an analysis t a over any time interval is numerically equal to the instantaneous acceleration a x t tox,avg analyze. A very common and simple type of one-dimensional motion, however, xis i atmodel: any instant the particle within the under interval, constant and theacceleration. velocity changes In the at discussion the same rate that through follows,- we vSlopex ϭ v that in which the acceleration is constant. In such a case, the average acceleration a xi outgenerate the motion.
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