Chapter 2 Motion Along 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)

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

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

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

Velocity and speed

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

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

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

Chapter 2 Problem 15 An object moves along the x axis according to the equation x(t) = (3.00 t t ) 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.

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

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

Case of constant acceleration

To help you solve problems EquationsMissing variables

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.

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

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?

Alternative derivation Using definitions and initial conditions we obtain

Graphical representation

Graphical integration

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

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

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

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

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

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