Motion inOne Dimension

50403020100t (s)

-50-400405030x (m)

FEDCBA

BC

D

EF

10 20 30 40 500

20

40

60

0

-20

-40

-60t (s)

x (m)

A

50403020100t (s)

-50-400405030x (m)

FEDCBA

BC

D

F

10 20 30 40 500

20

40

60

0

-20

-40

-60t (s)

x (m)

A

E

Δx = xfinal-xinitial

= -40m - 0m = -40 m

Δx/Δt =-40 m/10 s

= -4 m/s

Instantaneous velocity

�

v(t) =dx

dt

10 20 30 40 500

2

4

0

-2

-4

-6

v (m/s)

Forward motion

v > 0

10 20 30 40 500

2

4

0

-2

-4

-6

v (m/s)

Extreme position

v = 0

10 20 30 40 500

2

4

0

-2

-4

-6

v (m/s)

backward motion

v < 0

10 20 30 40 500

2

4

0

-2

-4

-6

v (m/s)

Extreme position

v = 0

x

t

v

t t t

v v

t t t

v vv

A B

D E

C

F

x

t

v

t t t

v v

t t t

v vv

A B

D E

C

F

x

t

vt

t t

v v

t

t t

v vv

A B

D E

C

F

v

t

x

t t t

x x

t t t

x xx

A B

D E

C

F

Analytic positions

�

x(t) = 2t2! 4t !1

Q: what is v(t)?

�

v(t)=dx

dt= 4t ! 4

Analytic positions

�

x(t) = 2t2! 4t !1

Q: what is avg. vel. between t=-1 & t=1?

Δx = x(1)-x(-1) = -3 - 5 = -8

�

v =!x

!t="8

2= "4 m/s

Analytic positions

�

x(t) = 2t2! 4t !1

�

v =!x

!t="8

2= "4 m/s

AccelerationAverage acceleration

Instantaneous acceleration

�

a =!v

!t

�

a =dv

dt

�

v(t) = 2t2! 4t !1

Q: what is a(t)?

�

a(t)=dv

dt= 4t ! 4

x

t

t

v

t

a

• When is the particle speeding up?• When is the particle slowing down?• In which direction is it accelerating at its extreme position?

v

t

a

t t t

a a

t t t

a aa

A B

D E

C

F

v

t

a

t t t

a a

t t t

a aa

A B

D E

C

F

a

t

v

t t t

v v

t t t

v vv

A B

D E

C

F

Physics Activity #2: 1d kinematics

Physics Activity #2: 1d kinematics

3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50

0

1

2

3

4

5

6

7

8

just bik

ing

just w

alking

5

Positio

n

Time

0 10 20 30 40 50 60

0

1

2

3

4

5

6

4:45 h3:45 h

Positio

n

Time (min)

Norm Cliff

0 10 20 30 40 50 60

0

2

4

6

8

10

12V

elo

city

Time

Normvel Cliffvel