Chapter 2 Motion in a Straight Line

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

12 xxx

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

• Instantaneous velocity – velocity at a given instant

• Instantaneous speed – a magnitude of an instantaneous velocity

t

xv

t

xv

t

0

lim

12

12

tt

xx

dt

dx

Velocity and speed

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

t

xv

t

0

lim

Acceleration

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

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

2

2

dt

xd

t

va

12

12

tt

vv

t

va

t

0

limdt

dv

dt

dx

dt

d

Acceleration

• The blue line is the average acceleration

• The slope (green line) of the velocity-time graph is the acceleration

t

va

t

0

lim

Chapter 2Problem 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

• Then

0it

t

vaa

if

if

tt

vv

0

t

vv if

atvv if

tt f


• Average and instantaneous accelerations are the same

• Conventionally

• Then

0it tt f

t

xv

0

t

xx if tvxx if

2221 fi vvvv

v

2

atvi

if

if

tt

xx

2

)( atvv ii

2

2attvxx iif


dt

dxv

dt

dva

atvv if

2

2attvxx iif



To help you solve problems

Equations Missing variables

2

2attvxx iif

atvv if

)(222ifif xxavv

2

)( tvvxx fiif

2

2attvxx fif

fv

if xx

t

a

iv

Chapter 2Problem 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

gtvv if

2

2gttvyy iif

Chapter 2Problem 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









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


Chapter 2

Problem 24:

15 m/s2


Chapter 2

Problem 26:

31 s


Chapter 2

Problem 28:

100 m


Chapter 2

Problem 38:

11 m/s


Chapter 2

Problem 42:

3.8 m/s2

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Chapter 2 Motion in a Straight Line