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Chapter 2 Motion in a Straight Line Linear motion In this chapter we will consider moving objects: • Along a straight line • With every portion of an object moving in the same direction and at the same rate (particle-like motion) Distance, position, and displacement • Distance a total length of the path traveled regardless of direction (SI unit: m) • In each instance we choose an origin – a reference point, convenient for further calculations • Position of an object is described by the shortest distance from the origin and direction relative to the origin • Displacement – a change from position x1 to position x2 x x2 x1 Velocity and speed • Average speed - a ratio of distance traveled (over a time interval) to that time interval (SI unit: m/s) • Average velocity - a ratio of displacement (over a time interval) to that time interval x x2 x1 v t t 2 t1 • Instantaneous velocity – velocity at a given instant x dx v lim t 0 t dt • Instantaneous speed – a magnitude of an instantaneous velocity Velocity and speed Velocity and speed Instantaneous velocity x v lim t 0 t • The instantaneous velocity is the slope of the line tangent to the x vs. t curve • This would be the green line • The light blue lines show that as Δt gets smaller, they approach the green line Acceleration • Average acceleration - a ratio of change of velocity (over a time interval) to that time interval (SI unit = (m/s)/s = m/s2) v v2 v1 a t t 2 t1 • Instantaneous acceleration – a rate of change of velocity at a given instant 2 d dx d x v dv a lim 2 t 0 t dt dt dt dt Acceleration • The blue line is the average acceleration • The slope (green line) of the velocity-time graph is the acceleration v a lim t 0 t Chapter 2 Problem 43 You allow 40 min to drive 25 mi to the airport, but you’re caught in heavy traffic and average only 20 mi/h for the first 15 min. What must your average speed be on the rest of the trip if you’re to make your flight? Case of constant acceleration • Average and instantaneous accelerations are the same • Conventionally ti 0 t f t • Then v v f vi v f vi aa t t 0 t f ti v f vi at Case of constant acceleration • Average and instantaneous accelerations are the same • Conventionally • Then ti 0 t f t x x f xi x f xi x f xi v t v t t f ti t 0 vi (vi at ) v1 v2 vi v f at vi v 2 2 2 2 2 at x f xi vi t 2 Case of constant acceleration 2 at x f xi vi t 2 dx v dt v f vi at dv a dt Case of constant acceleration Case of constant acceleration To help you solve problems Equations Missing variables v f vi at x f xi at 2 x f xi vi t 2 vf v f vi 2a( x f xi ) 2 2 x f xi (vi v f )t 2 at 2 x f xi v f t 2 t a vi Chapter 2 Problem 36 You’re driving at speed v0 when you spot a stationary moose on the road, a distance d ahead. Find an expression for the magnitude of the acceleration you need if you’re to stop before hitting the moose. Case of free-fall acceleration • At sea level of Earth’s mid-latitudes all objects fall (in vacuum) with constant (downward) acceleration of a = - g ≈ - 9.8 m/s2 ≈ - 32 ft/s2 • Conventionally, free fall is along a vertical (upward) y-axis v f vi gt gt y f yi vi t 2 2 Chapter 2 Problem 37 You drop a rock into a deep well and 4.4 s later hear a splash. How far down is the water? Neglect the travel time of sound. Graphical representation Graphical representation Graphical representation Graphical representation Graphical representation Graphical representation Graphical representation Graphical representation Graphical representation Questions? Answers to the even-numbered problems Chapter 2 Problem 14: (a) 24 km (b) 9.6 km/h (c) - 16 km/h (d) 0 km (e) 0 km/h Answers to the even-numbered problems Chapter 2 Problem 24: 15 m/s2 Answers to the even-numbered problems Chapter 2 Problem 26: 31 s Answers to the even-numbered problems Chapter 2 Problem 28: 100 m Answers to the even-numbered problems Chapter 2 Problem 38: 11 m/s Answers to the even-numbered problems Chapter 2 Problem 42: 3.8 m/s2