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1

Chapter 2

Motion along a straight line

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Kinematics & Dynamics

Kinematics:

Description of Motion without regard to its cause.

Dynamics:

Study of principles that relate motion to its cause.

Basic physical variables in kinematics and dynamics:

Time

Position

Displacement

Velocity

Acceleration

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3

Part 1

Time, Position, Velocity, Acceleration

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Time

Time: that part of existence which is measured in seconds, minutes, hours, days, weeks, months, years, etc., or this process considered as a whole

Cambridge Dictionary

Time: something that is measured in minutes, hours, years etc using clocks

Clocks: an instrument in a room or in a public building that shows what time it is

Longman Dictionary of Contemporary English

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5

Time

”Time is like a river flowing.” – Newton

”Zeit ist wass enie Uhr mass.” – Einstein

(”Time is what a clock measures.”)

“Time is space between events” – Feynman

"Time is that quality of nature which keeps events from happening all at once” – Anonymous

Time is neither young nor old

a beginning or an end,

now or forever;

time is the empty distance in between.

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Position and Displacement

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7

Trajectory

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Velocity

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9

Velocity and Speed

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Instantaneous velocity

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11

Acceleration

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Analytic representation

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13

Velocity and acceleration

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One-Dimensional (1-D) Motion

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15

Part 2

Motion with constant Acceleration

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Constant Acceleration

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17

Calculating position

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Summary table of results

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19

1D motion with constant acceleration a

⎪⎩

⎪⎨⎧

++=

+=2

00

0

21 attvxx

atvv the system of equations with six variables

max: two unknowns

case 1: we may eliminate time from the system

)(2 020

2 xxavv −+=

case 2: we may eliminate acceleration from the system

tvvxx )(21

00 ++=

20

example ⎪⎩

⎪⎨⎧

++=

+=2

00

0

21 attvxx

atvv

11

21

example ⎪⎩

⎪⎨⎧

++=

+=2

00

0

21 attvxx

atvv

22

example ⎪⎩

⎪⎨⎧

++=

+=2

00

0

21 attvxx

atvv

)(2 020

2 xxavv −+=

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23

problem 2.22 ⎪⎩

⎪⎨⎧

++=

+=2

00

0

21 attvxx

atvv

The catapult of the aircraft carrier USS Abraham Lincoln accelerates an F/A-18 Hornet jet fighter from rest to a takeoff speed of 173 mph in a distance of 307 ft. Assume constant acceleration.

1. Calculate the acceleration of the fighter in m/s.

2. Calculate the time required for the fighter to accelerate to takeoff speed.

Given: calculations

)(2 020

2 xxavv −+=

smmphvmftx

mxsmv

/33.770.17357.93307

0.0/0.0

0

0

====

== sm

xxvva /0.32

)(2 0

20

2

=−−

=

24

problem 2.36 ⎪⎩

⎪⎨⎧

++=

+=2

00

0

21 attvxx

atvv

At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of 3.2 m/s2 . At the same instant a truck, traveling with a constant speed of 20.0 m/s , overtakes and passes the car.How far beyond its starting point does the car overtake the truck?

Given:

x0 = 0.0 mv0t= 20.0 m/sv0c= 0.0 m/sxc = xtat = 0.0 m/s2

ac= 3.2 m/s2

)(2 020

2 xxavv −+=

c

tt

c

t

cctt

avx

avt

taxtvx

20

0

20

2

25.0

=

=

===

2 objects = 4 equationstime is the same

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25

Part 3

Free Fall

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Free fall ⎪⎩

⎪⎨⎧

−+=

−=2

00

0

21 gttvyy

gtvv

14

27

example

my 0.300 =

smv /0.100 =

⎪⎩

⎪⎨⎧

−+=

−=2

00

0

21 gttvyy

gtvv

28

Solution (cont)

my 0.300 =

smv /0.100 =

⎪⎩

⎪⎨⎧

−+=

−=2

00

0

21 gttvyy

gtvv

)(2 020

2 xxavv −+=

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29

The second question

my 0.300 =

smv /0.100 =

⎪⎩

⎪⎨⎧

−+=

−=2

00

0

21 gttvyy

gtvv

30

Solution for the second question

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31

Discussion of result for question 2

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The third question

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33

Being practical: 1D motion with const. a

⎪⎩

⎪⎨⎧

++=

+=2

00

0

21 attvxx

atvv1. initial value problem:knowing initial conditions (x0,v0) and acceleration a, one may find (x,v) at any moment in time.

2. “who cares about time”then the system of equation can be reduced to a simple equation

3. two object problemin this case we need co consider a system of 4 equation with the same time variables. In most problems we need only a couple equations from the system

)(2 020

2 xxavv −+=

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

++=

+=

++=

+=

2202022

2022

2101011

1011

21

21

tatvxx

tavv

tatvxx

tavv

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example

The engineer of a passenger train traveling at 25.0 m/s sights a freight train whose caboose is 200 m ahead on the same track. The freight train is traveling at 15.0 m/s in the same direction as the passenger train.

The engineer of the passenger train immediately applies the brakes, causing a constant acceleration of -0.10 m/s2, while the freight train continues with constant speed.

1 .Will the cows nearby witness a collision?

2. If so, where will it take place?

21

01

01

/1.0

/0.250.0

sma

smvmx

−=

==

22

02

02

/0.0

/0.150.200

sma

smvmx

=

==

18

35

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

++=

+=

++=

+=

2202022

2022

2101011

1011

21

21

tatvxx

tavv

tatvxx

tavv

21

01

01

/1.0

/0.250.0

sma

smvmx

−=

==

22

02

02

/0.0

/0.150.200

sma

smvmx

=

==

⎪⎪

⎩

⎪⎪

⎨

⎧

+==

++=

+=

tvxxvv

tatvx

tavv

02022

022

21011

1011

210.0

tvxtatv

xxmeanscollision

02022

101

21

21

+=+

=

solutions:

t1 = 22.5 s

t2 = 177.0 s

negative solutions?