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”??ConclusionVelocity is not the same at all times.It is changingA changing velocity is called an acceleration.dtdvatvtvvainitialfinalaverageAcceleration in PicturesConstant VelocityConstant AccelerationAverage AccelerationAcceleration is the rate of change of the velocity.Dimensions are L/T2SI 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 20atvvvvatvtvafinitialf0Instantaneous AccelerationThe instantaneous acceleration is the limit of the average acceleration as t approaches 0220limx xxtv dv d xat dt dtD �D= = =DCalculus22dtxddtdvadtdxvInstantaneous Acceleration -- graphThe slope of the velocity vs. time graph is the accelerationThe green line represents the instantaneous accelerationThe blue line is the average accelerationAcceleration and Velocity, 1When an object’s velocity and acceleration are in the same direction, the object is speeding upWhen an object’s velocity and acceleration are in the opposite direction, the object is slowing downExampleAcceleration and velocity are in opposite directionsAcceleration 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 …20020000000021210attvxxattvxxtdtadtvdxatdtdtvdxatvdtdxatvvttxxOne more calculus trick: the the chain rulechain ruleaxvvvvxxavdvdxavdvadxdxdvvdtdxdxdvdtdvaffvvxxf20for x)(21)(2020202000Kinematic Equations -- summary from the book.Kinematic EquationsThe kinematic equations may be used to solve any problem involving one-dimensional motion with a constant accelerationYou may need to use two of the equations to solve one problemMany times there is more than one way to solve a problemKinematic Equations, specificFor constant a,Can determine an object’s velocity at any time t when we know its initial velocity and its accelerationDoes not give any information about displacementKinematic Equations, specificFor constant acceleration,The average velocity can be expressed as the arithmetic mean of the initial and final velocities2xi xfxv vv+=Kinematic Equations, specificFor constant acceleration,Gives final position in terms of velocity and accelerationDoesn’t tell you about final velocity212f i xi xx x v t a t= + +Graphical Look at Motion – displacement – time curveThe slope of the curve is the velocityThe curved line indicates the velocity is changingTherefore, there is an accelerationGraphical Look at Motion – velocity – time curveThe slope gives the accelerationThe straight line indicates a constant accelerationThe 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 ObjectsA freely falling object is any object moving freely under the influence of gravity alone.It does not depend upon the initial motion of the objectDropped – released from restThrown downwardThrown upwardAcceleration of Freely Falling ObjectThe acceleration of an object in free fall is directed downward, regardless of the initial motionThe magnitude of free fall acceleration is g = 9.80 m/s2g decreases with increasing altitudeg varies with latitude9.80 m/s2 is the average at the Earth’s surfaceIn the English/American system, g=32 ft/sec2.Acceleration of Free Fall/Rise, cont.We will neglect air resistanceFree fall motion is constantly accelerated motion in one dimensionUse 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 CalculusDisplacement equals the area under the velocity – time curveThe limit of the sum is a definite integral0lim ( )fintxn n xttnv t v t dtD �D
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