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Chapter 2 Motion in One Dimension Distance, Speed The average speed 𝑣= 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑒𝑑 𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑒𝑙𝑎𝑝𝑠𝑒𝑑 = Δ𝑑 Δ𝑡 Why average? Speed up, slow down, stop… (board) Units are … CT2-1 Quantities in Motion • Position • Velocity • Acceleration x v a Need reference point for these e.g. position of car from where? Position – x of xva Defined in terms of a frame of reference A choice of coordinate axes 1D: x-axis - ,Origin, + (board) Motion diagrams & x vs t (all do this) 1. Car at rest at origin 2. Car moves to the right at 10 m/s 3. Car at rest speeds up to 10 m/s to the right 4. Car moving to the right at 10 m/s slows down to a stop 5. Car speeds up to the right, cruises, slows to a stop Motion diagrams & x vs t 1. Car moves to the right at 10 m/s 2. Car moves to the left at 10 m/s 3. Car moves to the left at 20 m/s Displacement Defined as the change in position Dx x x f i f stands for final and i stands for initial Units are meters (m) in SI D = ‘final’ – ‘initial’ Average Velocity: v of xva The average velocity is rate at which the displacement occurs vaverage Dx xf xi Dt tf ti Acceleration – a of xva Changing position velocity Changing velocity acceleration Dv vf vi a Dt tf ti Units are… Avg v vs inst v (board) avg () = slope of line between A&B, etc instantaneous () = slope of tangent at A Instantaneous v, a (board) time interval approaches zero Dt 0 v lim Dt 0 Dx Dt 𝑑𝑥 𝑣= 𝑑𝑡 Similarly, instantaneous acceleration 𝑑𝑣 𝑎= 𝑑𝑡 Derivative Slope v = Slope of x-t graph a = Slope of v-t graph Average () between 2 t’s: t1 & t2 Instantaneous () at a given t Review of slope (+,-,0 Slope = Dy/Dx v = Dx/Dt board ) Let’s practice some…CQs CT 2-2,2-3,2-4,2-5 CT 2-6, 2-7 This slide intentionally blank Kinematic Equations Used in situations with uniform acceleration v vo at 1 Dx vt vo v t 2 1 2 Dx vot at 2 v 2 vo2 2aDx Derivation (board) Start with definition of ‘a’, calc… v vo at Start with definition of ‘v’, calc… 1 2 Dx v o t at 2 Graphical Interpretation Interpret Dx=… (board) Free fall – only gravity acting g = 9.80 m/s² g is always directed downward When estimating, use g 10 m/s2 Toward the center of the earth Close to earth’s surface, g constant Air resistance: we’ll neglect this, why? Skydiving, baseball, sheet of paper… Constant accel (contd) ANOTHER CASE: SKI SLOPE CT 2-8,2-9,2-10 (motion diagrams, {x,v,a} vs t graphs) CT 2-11,2-12,2-13, 2-14 Chapter 2, Problem 25 An electric vehicle starts from rest and accelerates at a rate of 2.4 m/s2 in a straight line until it reaches a speed of 23 m/s. The vehicle then slows at a constant rate of 1.2 m/s2 until it stops. (a) How much time elapses from start to stop? (b) How far does the vehicle move from start to stop? Draw {a,v,x} vs t Chapter 2, Problem 45 (a) With what speed must a ball be thrown vertically from ground level to rise to a maximum height of 27 m? (b) How long will it be in the air?