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

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 x1 to position x2

12 xxx

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

t

xvavg

t

xv

t

0

lim

12

12

tt

xx

dt

dx

Acceleration

• Average acceleration (vector) - a ratio of change of velocity (over a time interval) to that time interval (SI unit = (m/s)/s = m/s2)

• Instantaneous acceleration (vector) – a rate of change of velocity at a given instant

2

2

dt

xd

t

vaavg

12

12

tt

vv

t

va

t

0

limdt

dv

dt

dx

dt

d

Case of constant acceleration

• Average and instantaneous accelerations are the same

• Conventional notation

• Then tt 2

01 xx 01 txx 2

01 vv vv 2

t

vaa avg

12

12

tt

vv

0

0

t

vv

atvv 0


• Average and instantaneous accelerations are the same

• Conventional notation

• Then

t

xvavg

00

t

xx tvxx avg 0

22021 vvvv

vavg

20

atv

12

12

tt

xx

2

)( 00 atvv

2

2

00

attvxx

tt 2

01 xx 01 txx 2

01 vv vv 2


dt

dxv

dt

dva

atvv 0

2

2

00

attvxx



To help you solve problems

Equations Missing variables

2

2

00

attvxx

atvv 0

)(2 02

02 xxavv

2

)( 00

tvvxx

2

2

0

atvtxx

v

0xx

t

a

0v

Alternative derivation

Using definitions and initial conditions

we obtain

dt

dva

2

2

00

attvxx

dtadv dt

dxv vdtdx

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

gtvv 0

2

2

00

gttvyy

Graphical representation







Graphical integration

1

0

)(01

t

t

dttavv

1

0

)(01

t

t

dttvxx

dt

dxv

dt

dva

Answers to the even-numbered problems

Chapter 2:

Problem 12: (a) – 6 m/s.(b) negative x direction (c) 6 m/s(e) Yes(f) No


Chapter 2:

Problem 16: 5.9 m


Chapter 2:

Problem 28: (a) 2.5 s.


Chapter 2:

Problem 42: (a) 3.70 m/s.(b) 1.74 m/s (c) 0.154 m


Chapter 2:

Problem 68: (a) 5.00 m/s(b) 1.67 m/s2

(c) - 7.5 m(d) - 3 s

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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