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ACCELLERATION -Acceleration shows how fast velocity changes -Acceleration is the “velocity of velocity” dt dx t x v t x v t 0 lim dt dv t v a t v a t 0 lim

ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

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Page 1: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

ACCELLERATION-Acceleration shows how fast velocity changes

-Acceleration is the “velocity of velocity”

dt

dx

t

xv

t

xv

t

0lim

dt

dv

t

va

t

va

t

0lim

Page 2: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

Uniform motion

t

xv

0

0

ttt

xxx

00 ttvxx

Uniform acceleration

t

va

0

0

ttt

vvv

00 ttavv

t

x(t)

t

v(t)

v(t)

t0 t

vv0 Δx

2 0 if

2

2

2

2

2

000

2

00

2

0

000

attvxxttt

tatvxx

tatvx

ttavv

tvv

x

Δx

Δt

tΔt

Page 3: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

For any motion:

For uniform acceleration:

tvx

2200 vv

v Δt vv

Δx

Velocity Acceleration

dttvxx

dt

dxtv

t

t

0

0 dttavtv

dt

dva

t

t

0

0

Uniform accelerationUniform motion

00

const

ttvxx

v

2

20

000

0000

00

00

ttattvxx

dtttavxdttvxx

ttav vconst at

t

t

t

Page 4: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

Base equations for 1D uniform acceleration

ttt

tavv

tatvxx

0 if

2

0

0

2

002 equations

2 quantities can be found

-What to remember?

-How to use?

Two useful equations that can be derived from the base equations

tvv

x

2

.1 0 See how it was derived on previous slide

20

202 vvxxa

20

20000000

2

00

0

222

2

.2

vvvvvvvvvvvvatvatxxa

attvxx

atvv

Page 5: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

1. speeds up all the time.

2. slows down all the time.

3. speeds up part of the time and slows down part of the time.

4. moves at a constant velocity.

time

Example: A train car moves along a long straight track. The graph shows the position as a function of time for this train. The graph shows that the train:

Steepness of slope is decreasing

time

posi

tion

Positive Acceleration = a smile

time

posi

tion

Negative Acceleration = a frown

Page 6: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

Example: The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true?

1. At time t0, both trains have the same velocity.

2. Both trains speed up all the time.

3. Both trains have the same velocity at some time before t0.

4. Somewhere on the graph, both trains have the same acceleration.

t0t1

Same slope at t = t1

Page 7: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

Posi

tion

Velo

city

Acc

ele

rati

on

v = slope of x(t)dtdx

v

2

2

dtxd

dtdv

a a = slope of v(t) ora = curvature of x(t)

Page 8: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

Posi

tion

Velo

city

Acc

ele

rati

on

0x

Change in velocity = area under a(t) curve

dtavt

t 1

0

dtvxt

t 1

0

Displacement = area under v(t) curve

0v

vt0 t1

Page 9: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

a

t

Example v(t) from a(t): Draw the velocity vs. time graph that corresponds to the following acceleration vs. time graph. Assume that the velocity at t = 0 is zero.

B C

v

t

v

t

A

v

t

Does your graph look like one of these?

Page 10: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

v

t

NB: a < 0 but object is speeding up.

a

t

NB: a > 0 but object is slowing up.

Page 11: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

Free fall

-Free fall acceleration: g=9.8m/s2

Using the two base equations:

atvv

attvxx

0

2

00 2

Substitute the following into the base equations:

yx

ga

To derive the following equations:

gtvv

gttvyy

0

2

00 2

Page 12: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

Example 1. A particle, a material point, is thrown vertically up. Find the maximum height the particle will reach and the time it will take, if you are given the initial height and the initial velocity.

Given:

g

Vyy

g

Vt

2

20

0max

01

Unknown variables:

? yt

?yy

v

y

max

max

0

0

at

Solution:

g

vyy

g

gv

g

vvyy

g

v tgtv

22

0

20

0max2

200

00max

0110

Answer:

g

vyy

g

vt

2

20

0max

01

0 no! ?

?

?

11

1

vv

y

t

Page 13: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

Example 2. A particle, a material point, is thrown vertically up. Find the velocity with which the particle returns to the point from which it was thrown, and the time this flight will take. The initial height and the initial velocity are given.

Given

12 2tt

?t

?v

yy

v

y

2

2

0

0

0

Solution:

020

02

02

2

20

2

20

2

v v g

vgvv

g

v t

gttv

Answer

02

02

2

vv

g

vt

Compare to example 1:

12 2tt

Page 14: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

Example 3, Two particles, material points, are thrown vertically up. One particle is thrown before the other. Find the time at which both objects are at the same height, and the height at which the objects’ intersection occurs.

Equations used:

Given:

?h

?t

t

v

3

0

0 Unknown variables:

?

?

?

2

1

3

y

y

t21 yy

Too many variables but

Thus, 3 equations and 3 unknowns.

Solution:

2

2

2

20

002

2

01

2

00

ttgttvy

gttvy

gΔΔΔtvyy

g

VtttgttV

gt

gttgt

gttVtV

gttV

ttgttV

gttV

0030300

20

20

03

23

0030

23

30

203

030

23

30

2

2

222

22

g

vt ttgttv

gt

gttgt

gttvtv

gttv

ttgttv

gttv

0030300

20

20

03

23

0030

23

30

203

030

23

30

22

222

22

2

2000

20

2000

2

00000

31

2282

222

g

gv

g

vgtgt

g

vtv

g

vtg

g

vtv

tyh

822

20

2000

3

gt

g

v h;

g

vt t Answer:

Page 15: ACCELLERATION -Acceleration shows how fast velocity changes - Acceleration is the “velocity of velocity”

Free fall (review)

Example1: Ball #1 is thrown vertically upwards with a speed of v0 from the top of a building and hits the ground with speed v1. Ball #2 is thrown vertically downwards from the same place with the same speed v0 and hits the ground with speed v2. Which one of the following three statements is true. Neglect air resistance.A. v1>v2

B. v1=v2

C. v1<v2

D. Depends on which ball is more massiveE. None of the above

Example2: You are throwing a ball straight up in the air. At the highest point, the ball’s

1. velocity and acceleration are zero.

2. velocity is nonzero but its acceleration is zero.

3. acceleration is nonzero, but its velocity is zero.

4. velocity and acceleration are both nonzero.