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Kinematics IIRemember this graph?ConclusionAcceleration in PicturesAverage AccelerationExample – Constant acceleration a=1 m/s2Instantaneous AccelerationCalculusInstantaneous Acceleration -- graphAcceleration and Velocity, 1ExampleSome math. Assume constant accelerationMore …One more calculus trick: the chain ruleKinematic Equations -- summary from the book.Kinematic EquationsKinematic Equations, specificSlide 18Slide 19Graphical Look at Motion – displacement – time curveGraphical Look at Motion – velocity – time curveGraphical Look at Motion – acceleration – time curveLet’s get real and throw something out of a building and watch it fall.Freely Falling ObjectsAcceleration of Freely Falling ObjectAcceleration of Free Fall/Rise, cont.Free Fall ExampleSlide 28Motion Equations from CalculusKinematics IIJanuary 14, 2019Remember this graph?smsmmmvA/75.3830sec83060:smsmmvD.5.410450:WHAT ABOUT “B”??ConclusionVelocity is not the same at all times.It is changingA changing velocity is called an acceleration.dtdvatvtvvainitialfinalaverageAcceleration in PicturesConstant VelocityConstant AccelerationAverage AccelerationAcceleration is the rate of change of the velocity.Dimensions are L/T2SI units are m/s²Example – Constant acceleration a=1 m/s2Time (s) Velocity (m/s)0 01 12 23 34 45 56 67 78 89 910 1011 1112 1213 1314 1415 1516 1617 1718 1819 1920 20atvvvvatvtvafinitialf0Instantaneous AccelerationThe instantaneous acceleration is the limit of the average acceleration as t approaches 0220limx xxtv dv d xat dt dtD �D= = =DCalculus22dtxddtdvadtdxvInstantaneous Acceleration -- graphThe slope of the velocity vs. time graph is the accelerationThe green line represents the instantaneous accelerationThe blue line is the average accelerationAcceleration and Velocity, 1When an object’s velocity and acceleration are in the same direction, the object is speeding upWhen an object’s velocity and acceleration are in the opposite direction, the object is slowing downExampleAcceleration and velocity are in opposite directionsAcceleration is uniform (blue arrows maintain the same length)Velocity is decreasing (red arrows are getting shorter)Positive velocity and negative accelerationSome math. Assume constant accelerationatvva tvvattad taa dtvvdvadtd vadtdvftvvtff 00000)0(constant0More …20020000000021210attvxxattvxxtdtadtvdxatdtdtvdxatvdtdxatvvttxxOne more calculus trick: the the chain rulechain ruleaxvvvvxxavdvdxavdvadxdxdvvdtdxdxdvdtdvaffvvxxf20for x)(21)(2020202000Kinematic Equations -- summary from the book.Kinematic EquationsThe kinematic equations may be used to solve any problem involving one-dimensional motion with a constant accelerationYou may need to use two of the equations to solve one problemMany times there is more than one way to solve a problemKinematic Equations, specificFor constant a,Can determine an object’s velocity at any time t when we know its initial velocity and its accelerationDoes not give any information about displacementKinematic Equations, specificFor constant acceleration,The average velocity can be expressed as the arithmetic mean of the initial and final velocities2xi xfxv vv+=Kinematic Equations, specificFor constant acceleration,Gives final position in terms of velocity and accelerationDoesn’t tell you about final velocity212f i xi xx x v t a t= + +Graphical Look at Motion – displacement – time curveThe slope of the curve is the velocityThe curved line indicates the velocity is changingTherefore, there is an accelerationGraphical Look at Motion – velocity – time curveThe slope gives the accelerationThe straight line indicates a constant accelerationThe zero slope indicates a constant accelerationGraphical Look at Motion – acceleration – time curveLet’s get real and throw something out of a building and watch it fall.This is called free fall.Freely Falling ObjectsA freely falling object is any object moving freely under the influence of gravity alone.It does not depend upon the initial motion of the objectDropped – released from restThrown downwardThrown upwardAcceleration of Freely Falling ObjectThe acceleration of an object in free fall is directed downward, regardless of the initial motionThe magnitude of free fall acceleration is g = 9.80 m/s2g decreases with increasing altitudeg varies with latitude9.80 m/s2 is the average at the Earth’s surfaceIn the English/American system, g=32 ft/sec2.Acceleration of Free Fall/Rise, cont.We will neglect air resistanceFree fall motion is constantly accelerated motion in one dimensionUse the kinematic equations to solve problems.Free Fall ExampleLet’s drop an object from the topof this building by simply releasing it.How long will it take to get to the ground?How fast will it be going when it gets to theground?What color is the object?Issues –Where is the origin?IS y the same thing as x?? How can that be???Use Transparencies …..Motion Equations from CalculusDisplacement equals the area under the velocity – time curveThe limit of the sum is a definite integral0lim ( )fintxn n xttnv t v t dtD �D


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UCF PHY 2048 - Set 2 - Kinematics II

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