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Chapter 2 Motion Along a Straight Line

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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)

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Types of physical quantities In physics, quantities can be divided into such general categories as scalars, vectors, matrices, etc. Scalars – physical quantities that can be described by their value (magnitude) only Vectors – physical quantities that can be described by their value and direction

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Distance, position, and displacement Distance (scalar) 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 (vector) is described by the shortest distance from the origin and direction relative to the origin Displacement (vector) – a change from position x i to position x f

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Velocity and speed Average speed (scalar) - a ratio of distance traveled (over a time interval) to that time interval (SI unit: m/s) Average velocity (vector) - a ratio of displacement (over a time interval) to that time interval Instantaneous velocity (vector) – velocity at a given instant Instantaneous speed (scalar) – a magnitude of an instantaneous velocity

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Velocity and speed

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Instantaneous velocity 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

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Acceleration Average acceleration (vector) - a ratio of change of velocity (over a time interval) to that time interval (SI unit = (m/s)/s = m/s 2 ) Instantaneous acceleration (vector) – a rate of change of velocity at a given instant

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Acceleration The slope (green line) of the velocity-time graph is the acceleration The blue line is the average acceleration

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Chapter 2 Problem 15 An object moves along the x axis according to the equation x(t) = (3.00 t 2 - 2.00 t + 3.00) m, where t is in seconds. Determine (a) the average speed between t = 2.00 s and t = 3.00 s, (b) the instantaneous speed at t = 2.00 s and at t = 3.00 s, (c) the average acceleration between t = 2.00 s and t = 3.00 s, and (d) the instantaneous acceleration at t = 2.00 s and t = 3.00 s.

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Case of constant acceleration Average and instantaneous accelerations are the same Conventionally Then

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Case of constant acceleration Average and instantaneous accelerations are the same Conventionally Then

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Case of constant acceleration

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To help you solve problems EquationsMissing variables

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Chapter 2 Problem 28 A particle moves along the x axis. Its position is given by the equation x = 2 + 3t - 4t 2, with x in meters and t in seconds. Determine (a) its position when it changes direction and (b) its velocity when it returns to the position it had at t = 0.

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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/s 2 ≈ - 32 ft/s 2 Conventionally, free fall is along a vertical (upward) y-axis

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Chapter 2 Problem 38 A ball is thrown directly downward, with an initial speed of 8.00 m/s, from a height of 30.0 m. After what time interval does the ball strike the ground?

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Alternative derivation Using definitions and initial conditions we obtain

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Graphical representation

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Graphical integration

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Answers to the even-numbered problems Chapter 2 Problem 4: (a) 50.0 m/s (b) 41.0 m/s

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Answers to the even-numbered problems Chapter 2 Problem 6: (a) 27.0 m (b) 27.0 m + (18.0 m/s)∆t + (3.00 m/s 2 )(∆t) 2 (c) 18.0 m/s

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Answers to the even-numbered problems Chapter 2 Problem 12: (b) 1.60 m/s 2 ; 0.800 m/s 2

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Answers to the even-numbered problems Chapter 2 Problem 20: (a) 6.61 m/s (b) −0.448 m/s 2

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Answers to the even-numbered problems Chapter 2 Problem 38: 1.79 s

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Answers to the even-numbered problems Chapter 2 Problem 48: (b) 3.00 × 10 −3 s (c) 450 m/s (d) 0.900 m

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