Lecture 5Kinematics in 2 DSpecial Case 1: FreefallTrajectory with constant acceleration along the verticalAnother trajectorySlide 6Exercise 1 & 2 Trajectories with accelerationExercise 1 Trajectories with accelerationExercise 2 Trajectories with accelerationExercise 3 Relative Trajectories: Monkey and HunterSchematic of the problemCase 2: Uniform Circular Motion Circular motion has been around a long timeGeneralized motion with only radial acceleration Uniform Circular MotionUniform Circular Motion (UCM) is common so we have specialized termsAngular displacement and velocityCircular motion also has a radial (perpendicular) componentWhat if w is linearly increasing …Non-uniform Circular MotionAngular motion, signsCircular MotionMass-based separation with a centrifugeRelative motion and frames of referenceRelative VelocityRecapPhysics 207: Lecture 5, Pg 1Lecture 5Lecture 5Goals:Goals: Address systems with multiple accelerations in 2-Address systems with multiple accelerations in 2-dimensions (including linear, projectile and circular motion)dimensions (including linear, projectile and circular motion) Discern different reference frames and understand how Discern different reference frames and understand how they relate to particle motion in stationary and moving framesthey relate to particle motion in stationary and moving frames Recognize different types of forces and know how they act Recognize different types of forces and know how they act on an object in a particle representationon an object in a particle representation Identify forces and draw a Identify forces and draw a Free Body DiagramFree Body DiagramAssignment: HW2, (Chapters 2 & 3, due Wednesday)Read through Chapter 6, Sections 1-4Physics 207: Lecture 5, Pg 2Kinematics in 2 DKinematics in 2 DThe position, velocity, and acceleration of a particle moving in 2-dimensions can be expressed as: rr = x ii + y jj vv = vx ii + vy jj aa = ax ii + ay jj22dtxdax22dtydaydtdxvxdtdyvy)( txx )( tyy Special Cases: 1. ax=0 ay= -g2. Uniform Circular MotionPhysics 207: Lecture 5, Pg 3Special Case 1: FreefallSpecial Case 1: Freefallx and y motion are separate and t is common to bothNow: Let g act in the –y direction, v0x= v0 and v0y= 0 yt04xt = 04ytx04x vs ty vs tx vs yconst. )(0xxvtvxtxtgvtvtgtvytyyyy022100)()(Physics 207: Lecture 5, Pg 4Trajectory 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 curve!Acceleration may or may not be!Example ProblemGiven How far does the knife travel (if no air resistance)?yt = 04xx vs y00v & rPhysics 207: Lecture 5, Pg 5Another trajectoryAnother trajectoryx vs yt = 0t =10Can you identify the dynamics in this picture?How many distinct regimes are there?Are vx or vy = 0 ? Is vx >,< or = vy ?yxPhysics 207: Lecture 5, Pg 6Another trajectoryAnother 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?Physics 207: Lecture 5, Pg 7Exercise 1 & 2Exercise 1 & 2Trajectories with accelerationTrajectories with accelerationA rocket is drifting sideways (from left to right) 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 (i.e., acceleration is constant) is fired at point B and left on for 2 seconds in which time the rocket travels from point B to some point C Sketch the shape of the path from B to C. At point C the engine is turned off. Sketch the shape of the path after point CPhysics 207: Lecture 5, Pg 8Exercise 1Exercise 1Trajectories with accelerationTrajectories with accelerationA. AB. BC. CD. DE. None of theseBCBCBCBCACBDFrom B to C ?Physics 207: Lecture 5, Pg 9Exercise 2Exercise 2Trajectories with accelerationTrajectories with accelerationA. AB. BC. CD. DE. None of theseCCCCACBDAfter C ?Physics 207: Lecture 5, Pg 10Exercise 3Exercise 3Relative Trajectories: Monkey and HunterRelative Trajectories: Monkey and HunterA. go over the monkey.B. hit the monkey.C. go under the monkey.All free objects, if acted on by gravity, accelerate similarly.A 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 …Physics 207: Lecture 5, Pg 11Schematic of the problemxB(t) = d = v0 cos tyB(t) = hf = v0 sin t – ½ g t2xM(t) = d yM(t) = h – ½ g t2Does yM(t) = yB(t) = hf? Does anyone want to change their answer ?(x0,y0) = (0 ,0)(vx,vy) = (v0 cos , v0 sin Bullet v0g(x,y) = (d,h)hfWhat happens if g=0 ?How does introducing g change things? MonkeyPhysics 207: Lecture 5, Pg 12Case 2: Uniform Circular MotionCase 2: Uniform Circular MotionCircular motion has been around a long timeCircular motion has been around a long timePhysics 207: Lecture 5, Pg 13Generalized motion with only Generalized motion with only radialradial acceleration acceleration Uniform Circular MotionUniform Circular MotionChanges only in the direction of vaa = 0aaaaaa=+A particle doesn’t speed up or slow down! aar||aatavvPhysics 207: Lecture 5, Pg 14Uniform Circular Motion (UCM) is common so we have specialized termsArc traversed s = rTangential velocity vtPeriod, T, and frequency, fAngular position, Angular velocity, Period (T): The time required to do one full revolution, 360° or 2 radiansFrequency (f): 1/T, number of cycles per unit timeAngular velocity or speed = 2f = 2/T, number of radians traced out per unit time (in UCM average and instantaneous will be the same)rvtsPhysics 207: Lecture 5, Pg 15Angular displacement and velocity Arc traversed s = rin time t then s = rso s / t = ( / t) rin the limit t 0one gets ds / dt = d/ dt r vt = r ≡ d/ dt if is constant, integrating = d/ dt, we obtain: = + tCounter-clockwise is positive, clockwise is negativervtsPhysics 207: Lecture 5, Pg 16Circular motion also has a
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