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## Linear Motion

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**q**q Linear Motion Angular Motion TRANSLATIONONLY! ROTATIONONLY! Object rotates about a fixed point (axis) Object maintains angular orientation (q) but this point does NOT have to lie within the object measured in meters - SI unit other units - inches, feet, miles, centimeters, millimeters measured in radians – SI Unit other units – degrees, revolutions Rectilinear motion - if path of one point on object is a straight line Curvilinear motion – if path of one point on object is curved**Types of Motion**• General • combination of linear and angular motion • translation and rotation B B A A**Kinematics is the study of motion without regard for the**forces causing the motion or …the description of motion • there are three basic kinematic variables • position, velocity and acceleration • the position of an object is simply its location in space • changes in position can be described by distance or displacement • the velocity of an object is how fast it is changing its position • the acceleration of an object is how fast the velocity is changing**Frame 1**(x1,y1) (x2,y2) Y (x4,y4) (x5,y5) (x3,y3) (0,0) X Acquisition of Position Data Position data is often acquired by digitizing the x and y coordinates from film or video. Velocities and accelerations are calculated from the position data.**Position and Displacement (d)**• Position (s) is the location of an object in space • units: m, cm, km, in, ft, mi • Displacement (Ds= sf - si) is the change in position of an object s2 displacement = d d d = s1 – s2 s1**Y**s2 = (x2,y2) d d = s1 – s2 s1 = (x1,y1) X • Problem: • how do you describe s1 and s2? • If you put the arrow on graph paper you describe position with x- & y-coordinates**Y**s2 = (x2,y2) d d = s1 – s2 s1 = (x1,y1) X • d = s1 – s2 • d = (x1,y2) – (x2,y2) How do you do this? • Realize that displacement is a vector so you must determine either the Cartesian or polar coordinates**Two choices to describe vector**• CartesianCoordinates (dx,dy) • dx = x2 – x2 = distance in the x-direction • dy = y2 – y1 = distance in the y-direction • PolarCoordinates (d,q) • “How far and in which direction” q = measured directly from graph**Y**s2 d d = s1 – s2 s1 X Second problem: Since this movement occurs over time, displacement (as a vector) does NOT represent changes in the direction of movement well. For example – what if s1 represents you at Building A and s2 represents you at Building B 10 minutes later.**Assuming this city is like most cities you have to walk up**and down city blocks and not through buildings. Y s2 d s1 X**Y**s2 d s1 X So your actual route is around the buildings, traveling up and down city blocks. dy dx**Thus the actual distance you covered is more than**displacement represents distance = the length of your travel in the x-direction (dx) plus the length of your travel in the y-direction (dy) BUT since we are only concerned with the length of travel we don’t distinguish between directions Y s2 distance = dx + dy d dy s1 dx X**dx + dy = distance =**Note: use “ ” for length d dy dx Distance ( ) • distance is the length of the path traveled • it is a scalar - “How far” • units: same as displacement**Example - Distance vs. Displacement**N leg 3 = 2 miles leg 2 = 3 miles Total DISTANCE Traveled = 2 miles + 3 miles + 2 miles = 7 miles leg 1 = 2 miles**Describing Displacement**Describing Displacement First Method (Cartesian) 3 miles East 4 miles North (3, 4) miles put ‘horizontal’ coordinate 1st put ‘vertical’ coordinate 2nd N displacement vector**Displacement Magnitude**Second Method (Polar) 1st - calculate length of displacement vector N displacement vector 4 miles q 3 miles**Displacement Direction**2nd – Calculate the angle using trigonometric relationships N displacement vector 4 miles q 3 miles**Displacement Vector(Polar Notation)**Describe the displacement vector by its length and direction N displacement vector 4 miles q 3 miles**distance**Speed = time Average Speed • speed is a scalar quantity • it is the rate of change of distance wrt time • units: same as velocity**(80,40)**0.5 s (60,10) (0,0) What is the average speed of the basketball?**Average Velocity (v)**• rate of change of displacement wrt time • velocity is a vector quantity • “How fast and in which direction” • units: m/s, km/hr, mi/hr, ft/s NOTE: displacement (d) is a vector so must obey rules of vector algebra when computing velocity.**When two velocities act on an object you find the net or**resultant effect by adding the velocities. • Because velocity is a vector you can’t simply add the numbers. • Instead – you must use vector algebra to add the velocities. In this example the boat is propelled to the right by its motor while the river’s current carries it towards the top of the picture. This describes 2 velocities**Other examples of velocities that can be added together**include the wind direction when flying.**Adding Velocities**Use the laws of vector algebra. Example - the path of the swimmer is determined by the vector sum of the swimmer’s velocity and the river current’s velocity.**Example:**vswimmer = 2 m/s vriver = 0.5 m/s What is the swimmer’s resultant velocity?**vR = (2 m/s)2 + (0.5 m/s)2**vR = 2.06 m/s q = 14 Example - Solution 50 m 0.5 m/s 2 m/s vR**Average Speed and Velocity**• average speed has a greater magnitude than average velocity unless there are no direction changes associated with travel • in sports • average speed is often more important than average velocity**1996 Olympic Marathon**Men 2:12:36 Josia Thugwane - RSA Women 2:26:05 Fatuma Roba - ETH Distance 26 miles + 385 yards 26 miles * 1.61 km/mile = 41.86 km 385 yards * 0.915 m/yd = 352 m Total = 41.86 km + .35 km = 42.21 km**Average Speed & the Marathon**• marathon example (cont.) t = 2:12:36 t=2 hrs (3600s/1 hr) + 12 min (60 s/ 1min) + 36 s = 7,956 s t = 2:26:05 = 8,765 s**Average Speed and the Marathon**• average speed = distance/time speed = 42,210m/7956 s = 5.3 m/s speed = 42,210/8765 s = 4.8 m/s average velocity???**Average vs. Instantaneous**• average velocity is not very meaningful in athletic events where many changes in direction occur • e.g. marathon • start and end in same place so**Instantaneous Values**• instantaneous velocity (v) is very important • specifies how fast and in what direction one is moving at one particular point in time • magnitude of instantaneous velocity is exactly the same as instantaneous speed**Average vs. Instantaneous Speed**1991 World Championships - Tokyo**Average Acceleration (a)**• rate of change of velocity with respect to time • “How fast the velocity is changing” • acceleration is a vector quantity • units: m/s/s or m/s2 , m·s-2, ft/s/s**Average Acceleration**v0.0 = 0 m/s v2.5 = 5 m/s v5.0 = 0 m/s**1st interval**Note: velocity is positive and acceleration is positive.**2nd interval**Note: velocity is positive but acceleration is negative.**t = 3 seconds**+ direction final initial a vi = 5 m/s vf = 8 m/s Six Cases of Acceleration 1 - speed up in positive direction = positive accel. Calculate average acceleration!**t = 3 seconds**+ direction final initial a vf = 5 m/s vi = 8 m/s Six Cases of Acceleration 1 - speed up in positive direction = positive accel. 2 - slow down in positive direction = negative accel. Calculate average acceleration!**Six Cases of Acceleration**1 - speed up in positive direction = positive accel. 2 - slow down in positive direction = negative accel. 3 - speed up in negative direction = negative accel. t = 3 seconds + direction initial final a vi = -5 m/s vf = -8 m/s Calculate average acceleration! What is happening to speed?, velocity?**Six Cases of Acceleration**1 - speed up in positive direction = positive accel. 2 - slow down in positive direction = negative accel. 3 - speed up in negative direction = negative accel. 4 - slow down in negative direction = positive accel. t = 3 seconds + direction final initial vi = -8 m/s vf = -5 m/s Calculate average acceleration! What is happening to speed?, velocity?**initial**final Six Cases of Acceleration 1 - speed up in positive direction = positive accel. 2 - slow down in positive direction = negative accel. 3 - speed up in negative direction = negative accel. 4 - slow down in negative direction = positive accel. 5 - reverse directions from pos to neg = negative accel. t = 3 seconds + direction a vi = +1 m/s vf = -1 m/s Calculate average acceleration!**+ direction**t = 3 seconds initial final a vf = -1 m/s vi = +1 m/s Six Cases of Acceleration 1 - speed up in positive direction = positive accel. 2 - slow down in positive direction = negative accel. 3 - speed up in negative direction = negative accel. 4 - slow down in negative direction = positive accel. 5 - reverse directions from pos to neg = negative accel. 6 - reverse directions from neg to pos = positive accel. Calculate average acceleration!**Human Response to Sustained g’s**In certain activities people experience + & - accelerations. By standardizing these accelerations to the normal acceleration on earth (-9.8 m/s/s) you get an idea of how much force they are experiencing • 6-9 Gs: "Increased chest pain and pressure; breathing difficult, with shallow respiration from position of nearly full inspiration; further reduction in peripheral vision, increased blurring, occasional tunneling, great concentration to maintain focus; occasional lacrimation; body, legs, and arms cannot be lifted at 8 G; head cannot be lifted at 9 G." • 9-12 Gs: "Breathing difficulty severe; increased chest pain; marked fatigue; loss of peripheral vision, diminution of central acuity, lacrimation." • 15 Gs:"Extreme difficulty in breathing and speaking; severe vise-like chest pain; loss of tactile sensation; recurrent complete loss of vision. Data primarily from: Bioastronautics Data Book, second edition, 1973, NASA)**Relationships Betweens, v,& a**• v is the rate of change of s wrt time • a is the rate of change of v wrt time • consider a graph of s vs. time • s on vertical axis • time on horizontal axis • rate of change is interpreted as the slope**q**Slope of a Curve • “Slope” = number which describes the steepness of a line • rise/run • Note: this is the definition for the tangent of q, opposite / adjacent**Changes in the slope**• positive slope • up and to the right • negative slope • down and to the right • quick change • very steep slope • slow change • very flat slope**Position**Time (s) The slope of the position by time curve is the velocity.**Position**Time (s) Note that this is the average velocity during the period from 5 seconds to 10 seconds.