Page 1Physics 207 – Lecture 3Physics 207: Lecture 3, Pg 1Physics 207, Lecture 3Assignment: 1. For Thursday: Read Chapter 3 (carefully) through 4.42. Homework Set 2 available at noon Goals (finish Chap. 2 & 3) Understand the relationships between position, velocity & acceleration in systems with 1-dimensional motion and non-zero acceleration (usually constant) Solve problems with zero and constant acceleration (including free-fall and motion on an incline) Use Cartesian and polar coordinate systems Perform vector algebraPhysics 207: Lecture 3, Pg 2Position, velocity & acceleration for motion along a line If the position x is known as a function of time, then we can find both the instantaneous velocity vxand instantaneous acceleration axas a function of time!tvxtxaxt22vadtxddtdxx==dtdxx=v] offunction a is [ )(txtxx∆∆=Page 2Physics 207 – Lecture 3Physics 207: Lecture 3, Pg 3Going the other way…. Particle motion with constant acceleration The magnitude of the velocity vector changes A particle with smoothly decreasing speed:vv11vv00vv33vv55vv22vv44aaaaaaaaaaaavt0vf= vi+ a ∆t= vi+ a (tf- ti)a ∆t = area under curve = ∆v (an integral)tva∆∆≡rravgtvat∆∆≡→∆rr0instlimati∆t0ttfPhysics 207: Lecture 3, Pg 4And given a constant acceleration we can integrate to get explicit v and axaxvxttttxxx∆+= avv02210 a v0ttxxxx∆+∆+=consta=x22vadtxddtdxx==dtdxx=v] offunction a is [ )(txtxx∆∆=Page 3Physics 207 – Lecture 3Physics 207: Lecture 3, Pg 5Key point: If the position x is known as a function of time, then we can find both velocity vvxtxt)( 10tvdtxtt∫=dtdxvx=)(txx= “Area” under the v(t) curve yields the change in position Algebraically, a special case, if the velocity is a constantthen x(t)=v t + x0Physics 207: Lecture 3, Pg 6Position, displacement, velocity & acceleration All are vectors and so vector algebra is a must ! These cannot be used interchangeably (different units!)(e.g., position vectors cannot be added directly to velocity vectors) But we can determined directions “Change in the position” vector gives the direction of the velocity vector “Change in the velocity” vector gives the direction of the acceleration vector Given x(t) vx(t) ax(t) Given ax(t) vx(t) x(t)vrarPage 4Physics 207 – Lecture 3Physics 207: Lecture 3, Pg 7Rearranging terms gives two other relationships For constant acceleration: From which we can show (caveat: a constant acceleration))v(v21v)x(x2av vxx(avg)x 0x2x2x00+=−=−txxx∆+= avv02210 a v0ttxxxx∆+∆+=consta=xaxtPhysics 207: Lecture 3, Pg 8Free Fall When any object is let go it falls toward the ground !! The force that causes the objects to fall is called gravity. This acceleration on the Earth’s surface, caused by gravity, is typically written as “little” g Any object, be it a baseball or an elephant, experiences the same acceleration (g) when it is dropped, thrown, spit, or hurled, i.e. g is a constant. 2210 g v)(0ttytyy∆−∆+=∆ga-y=Page 5Physics 207 – Lecture 3Physics 207: Lecture 3, Pg 9Gravity facts: g does not depend on the nature of the material ! Galileo (1564-1642) figured this out without fancy clocks & rulers! Feather & penny behave just the same in vacuum Nominally, g = 9.81 m/s2At the equator g = 9.78 m/s2At the North pole g = 9.83 m/s2Physics 207: Lecture 3, Pg 10When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path?A. Both v = 0 and a = 0B. v ≠ 0, but a = 0C. v = 0, but a ≠ 0D. None of the aboveyExercise 1Motion in One DimensionPage 6Physics 207 – Lecture 3Physics 207: Lecture 3, Pg 11In driving from Madison to Chicago, initially my speed is at a constant 65 mph. After some time, I see an accident ahead of me on I-90 and must stop quickly so I decelerate increasingly “fast” until I stop. The magnitude of my acceleration vs time is given by,A. B. C. atExercise 2 More complex Position vs. Time Graphs• Question: My velocity vs time graph looks like which of the following ?vtvvPhysics 207: Lecture 3, Pg 12Exercise 3 1D FreefallA. vA< vBB. vA= vBC. vA> vB Alice and Bill are standing at the top of a cliff of height H. Both throw a ball with initial speed v0, Alice straight down and Bill straight up. The speed of the balls when they hit the ground arevAand vBrespectively..((Neglect air resistance.)vv00vv00BillAliceAliceHHvvAAvvBBPage 7Physics 207 – Lecture 3Physics 207: Lecture 3, Pg 13Exercise 3 1D Freefall : Graphical solutionAlice and Bill are standing at the top of a cliff of Alice and Bill are standing at the top of a cliff of heightheightHH. Both throw a ball with initial speed. Both throw a ball with initial speedvv00, , Alice straightAlice straightdowndownand Bill straightand Bill straightupup. . vxtcliffback atcliffturnaroundpointgroundgroundv0-v0vground∆v= -g ∆tidentical displacements(one + and one -)Physics 207: Lecture 3, Pg 14The graph at right shows the The graph at right shows the yyvelocity versus velocity versus timetimegraph for a graph for a ball. Gravity is acting downward ball. Gravity is acting downward in the in the --yydirection and the direction and the xx--axis axis is along the horizontal. is along the horizontal. Which explanation Which explanation best fitsbest fitsthe the motion of the ball as shown by motion of the ball as shown by the velocitythe velocity--time graph below?time graph below?A. The ball is falling straight down, is caught, and is then thrown straight down with greater velocity. B. The ball is rolling horizontally, stops, and then continues rolling. C. The ball is rising straight up, hits the ceiling, bounces, and then falls straight down. D. The ball is falling straight down, hits the floor, and then bounces straight up. E. The ball is rising straight up, is caught and held for awhile, and then is thrown straight down. Home Exercise,1D FreefallPage 8Physics 207 – Lecture 3Physics 207: Lecture 3, Pg 15Problem Solution Method:Five Steps:1) Focus the Problem- draw a picture – what are we asking for?2) Describe the physics- what physics ideas are applicable- what are the relevant variables known and unknown3) Plan the solution- what are the relevant physics equations4) Execute the plan- solve in terms of variables- solve in terms of numbers5) Evaluate the answer- are the
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