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# UW-Madison PHYSICS 207 - Physics 207 – Lecture 7 Notes

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Page 1Physics 207 – Lecture 7Physics 207: Lecture 7, Pg 1Physics 207, Physics 207, Lecture 7, Sept. 26Lecture 7, Sept. 26Agenda:Agenda:Assignment: For Wednesday read Chapter 7 Assignment: For Wednesday read Chapter 7 MP Problem Set 3 due tonightMP Problem Set 3 due tonightMP Problem Set 4 available nowMP Problem Set 4 available nowMidTermMidTermThursday, Oct. 4, Chapters 1Thursday, Oct. 4, Chapters 1--7, 90 minutes, 7:157, 90 minutes, 7:15--8:45 PM8:45 PMRooms: Rooms: B102 & B130 in B102 & B130 in Van Van VleckVleck..• Chapter 6 (Dynamics II) Motion in two (or three dimensions) Frames of reference• Start Chapter 7 (Dynamics III) Circular MotionPhysics 207: Lecture 7, Pg 2Chapter 6: Chapter 6: Motion in 2 (and 3) dimensionsMotion in 2 (and 3) dimensions, , Dynamics IIDynamics II Recall instantaneous velocity and acceleration These are vector expressions reflecting x, y and z motionrr = rr(t)vv = drr / dtaa = d2rr / dt2Physics 207: Lecture 7, Pg 3KinematicsKinematics The position, velocity, and acceleration of a particle in 3-dimensions can be expressed as:rr = x ii + y jj + z kkvv = vx ii + vyjj + vz kk (ii ,jj ,kk unit vectors )aa = ax ii + ay jj + az kk All this complexity is hidden away in this compact notationrr = rr(t)vv = drr / dtaa = d2rr / dt2ad xdtx=22ad ydty=22ad zdtz=22vdxdtx=vdydty= vdzdtz=)(txx=yyt=()zzt=()Physics 207: Lecture 7, Pg 4Special CaseSpecial CaseThrowing an object with x along the horizontal and y along the vertical. x and y motion both coexist and t is common to bothLet g act in the –y direction, v0x= v0 and v0y= 0yt04xt = 04ytx04x vs ty vs tx vs yPhysics 207: Lecture 7, Pg 5A different trajectoryA different trajectoryx vs yt = 0t =10Can you identify the dynamics in this picture?How many distinct regimes are there?yxPhysics 207: Lecture 7, Pg 6A different trajectoryA different trajectoryx vs yt = 0t =10Can you identify the dynamics in this picture?How many distinct regimes are there?0 < t < 3 3 < t < 7 7 < t < 10 I. vx= constant = v0; vy= 0 II. vx= vy= v0 III. vx= 0 ; vy= constant < v0yxWhat can you say about the acceleration?Page 2Physics 207 – Lecture 7Physics 207: Lecture 7, Pg 7Lecture 7, Lecture 7, Exercises 1 & 2Exercises 1 & 2Trajectories with accelerationTrajectories with acceleration A rocket is moving sideways in deep space, with its engine off, from A to B. It is not near any stars or planets or other outside forces.  Its constant thrust engine (force is constant) is fired at pointB and left on for 2 seconds in which time the rocket travels from point B to some point C  Sketch the shape of the pathfrom B to C.  At point C the engine is turned off. Sketch the shape of the path after point CPhysics 207: Lecture 7, Pg 8Lecture 7, Lecture 7, Exercise 1Exercise 1Trajectories with accelerationTrajectories with accelerationA. AB. BC. CD. DE. None of theseBCBCBCBCACBDFrom B to C ?Physics 207: Lecture 7, Pg 9Lecture 7, Lecture 7, Exercise 2Exercise 2Trajectories with accelerationTrajectories with accelerationA. AB. BC. CD. DE. None of theseCCCCACBDAfter C ?Physics 207: Lecture 7, Pg 10Trajectory with constant Trajectory with constant acceleration along the vertical acceleration along the vertical How does the trajectory appear to different observers ? What if the observer is moving with the same x velocity?xt = 04yx vs ytx04x vs tPhysics 207: Lecture 7, Pg 11Trajectory with constant Trajectory with constant acceleration along the vertical acceleration along the vertical This observer will only see the y motionxt = 04yx vs yxt = 04yy vs tIn all inertial reference frames everyone sees the same accelerationWhat if the observer is moving with the same x velocity?Physics 207: Lecture 7, Pg 12Lecture 7, Lecture 7, Exercise 3Exercise 3Relative Trajectories: Monkey and HunterRelative Trajectories: Monkey and HunterA. Go over the monkeyB. Hit the monkeyC. Go under the monkeyA hunter sees a monkey in a tree, aims his gun at the monkey and fires. At the same instant the monkey lets go. Does the bullet?Page 3Physics 207 – Lecture 7Physics 207: Lecture 7, Pg 13Trajectory with constant Trajectory with constant acceleration along the vertical acceleration along the vertical What do the velocity and accelerationvectors look like? Velocity vector is always tangent to the trajectory curve!Acceleration may or may not be!yt = 04xx vs yPhysics 207: Lecture 7, Pg 14Instantaneous VelocityInstantaneous Velocity But how we think about requires knowledge of the path. The direction of the instantaneous velocity is along a line that is tangent to the path of the particle’s direction of motion.v The magnitude of the instantaneous velocity vector is the speed, s. (Knight uses v)s = (vx2+ vy2+ vz2)1/2Physics 207: Lecture 7, Pg 15Average Acceleration: ReviewAverage Acceleration: Review The average acceleration of particle motion reflects changes in the instantaneous velocity vector (divided by the time interval during which that change occurs). a The average acceleration is a vector quantity directed along ∆vPhysics 207: Lecture 7, Pg 16Instantaneous AccelerationInstantaneous Acceleration The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero The instantaneous acceleration is a vector with components parallel (tangential) and/or perpendicular (radial) to the tangent of the path Changes in a particle’s path may produce an acceleration The magnitude of the velocity vector may change The direction of the velocity vector may change(Even if the magnitude remains constant) Both may change simultaneously (depends: path vs time)Physics 207: Lecture 7, Pg 17Motion along a path Motion along a path ( displacement, velocity, acceleration )( displacement, velocity, acceleration ) 3-D Kinematics : vector equations:rr = rr(t)vv = drr / dtaa = d2rr / dt2Velocity :v = drr / dtvav= ∆rr / ∆tAcceleration :a = dvv / dtaav= ∆vv / ∆tvv2∆vv--vv1vv2yxpathvv1Physics 207: Lecture 7, Pg 18General 3General 3--D motion with nonD motion with non--zero acceleration:zero acceleration: Uniform Circular Motion (Ch. 7) is one specific case:aavpathand timetaaaaaa = 0Two possible options:Change in the magnitude of vChange in the direction of vaa= 0aa= 0aaaaaa=+Animationhttp://romano.physics.wisc.edu/winokur/fall2006/AF_0418.htmlPage 4Physics 207 – Lecture 7Physics 207: Lecture 7, Pg 19Lecture 7,Lecture 7,Exercise

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