Post on 13-Jan-2016
Chapter 3: Acceleration and Accelerated Motion
Unit 3
Accelerated Motion
Chapter 3: Acceleration and Accelerated MotionChapter 3: Acceleration and Accelerated Motion
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 0
1 10
2 20
3 30
4 40
5 50
Velocity/Time Graphvelocity (m/s)
t (seconds)
52 4
0
10
30
50
1 5
20
0
50
3
40
12
12
xx
yyslope
ss
smsmslope
05
/0/50
s
smslope
/10
The slope means acceleration!
210s
m
The slope means something!
Chapter 3: Acceleration and Accelerated Motion
What equation can we get from this graph?
t
v
x
yslope
t
va
The constant acceleration equation!
We can also get the “how fast” equation.
bmxy
if vtav
From Graph:From Algebra:
t
va
t
vva if
if vvta fi vtav
if vtav
Chapter 3: Acceleration and Accelerated Motion
Chapter 3: Acceleration and Accelerated Motion
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 0 10
1 10 10
2 20 10
3 30 10
4 40 10
5 50 10
Chapter 3: Acceleration and Accelerated Motion
acceleration (m/s2)
t (seconds)
52 4
0
5
15
1 5
10
30
15 What would the acceleration/time graph look like?
Horizontal line means constant acceleration.
Let’s look at the area under the ‘curve.’bhArea
)10)(5( 2smsArea
smArea 50
5s
10 m/s2
It is the change in velocity!
Chapter 3: Acceleration and Accelerated MotionChapter 3: Acceleration and Accelerated Motion
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 0 0 10
1 10 10
2 20 10
3 30 10
4 40 10
5 50 10
velocity (m/s)
t (seconds)
52 4
0
10
30
50
1 5
20
0
50
3
40
Area of a triangle:
bhA 21
10 m/s
1 s
)10)(1(21
smsA
How do you find displacement from a velocity/time graph?
Area under the ‘curve.’
m5
5
Chapter 3: Acceleration and Accelerated MotionChapter 3: Acceleration and Accelerated Motion
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 0 0 10
1 10 5 10
2 20 10
3 30 10
4 40 10
5 50 10
velocity (m/s)
t (seconds)
52 4
0
10
30
50
1 5
20
0
50
3
40
Area of a triangle:
bhA 21
20 m/s
2 s
)2)(20(21 sA s
m m20
20
Chapter 3: Acceleration and Accelerated MotionChapter 3: Acceleration and Accelerated Motion
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 0 0 10
1 10 5 10
2 20 20 10
3 30 10
4 40 10
5 50 10
velocity (m/s)
t (seconds)
52 4
0
10
30
50
1 5
20
0
50
3
40
Area of a triangle:
bhA 21
30 m/s
3 s
msA sm 45)3)(30(2
1
45
Chapter 3: Acceleration and Accelerated MotionChapter 3: Acceleration and Accelerated Motion
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 0 0 10
1 10 5 10
2 20 20 10
3 30 45 10
4 40 10
5 50 10
velocity (m/s)
t (seconds)
52 4
0
10
30
50
1 5
20
0
50
3
40
Area of a triangle:
bhA 21
40 m/s
4 s
msA sm 80)4)(40(2
1
80
Chapter 3: Acceleration and Accelerated MotionChapter 3: Acceleration and Accelerated Motion
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 0 0 10
1 10 5 10
2 20 20 10
3 30 45 10
4 40 80 10
5 50 10
velocity (m/s)
t (seconds)
52 4
0
10
30
50
1 5
20
0
50
3
40
Area of a triangle:
bhA 21
50 m/s
5 s
msA sm 125)5)(50(2
1 What is happening to the amount of distance increased after each second?
125
Chapter 3: Acceleration and Accelerated Motion
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 1 2 3 4 5 6
Time (s)
Vel
oci
ty (
m/s
)
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 0 0 10
1 10 5 10
2 20 20 10
3 30 45 10
4 40 80 10
5 50 125 10
Chapter 3: Acceleration and Accelerated Motion
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 1 2 3 4 5 6
Time (s)
Vel
oci
ty (
m/s
)
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 1 2 3 4 5 6
Time (s)
Vel
oci
ty (
m/s
)
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 1 2 3 4 5 6
Time (s)
Vel
oci
ty (
m/s
)
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 1 2 3 4 5 6
Time (s)
Vel
oci
ty (
m/s
)
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 1 2 3 4 5 6
Time (s)
Vel
oci
ty (
m/s
)
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 1 2 3 4 5 6
Time (s)
Po
siti
on
(m
)
Shape: Top opening parabola (curvy up)
What is the proportionality?2xy 2Axy
221 )( tax
This is the “How Far” equation! (Starting with zero velocity)
Chapter 3: Acceleration and Accelerated Motion
t (time-sec) t ^2 (time^2)-sec^2) X (position-meters)
0 0
1 5
2 20
3 45
4 80
5 125
014
9
1625
y = 5x
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30
t^2 (s^2)
x (
m)
bmxy 05 2 tx2
21 atx
It checks out!!
Chapter 3: Acceleration and Accelerated Motion
Let’s look at another way to get the “How Far” equation.
From the velocity time graph:velocity (m/s)
t (seconds)52 4
0
10
30
50
1 5
20
0
50
3
40
Area under curve = displacement
xA xbh 2
1
xvt ))((21
xvvt if ))((21
From previous “How Fast” equation:
if vtav
tavv if xatt ))((2
1
xat 2212
21 atx flip
Chapter 3: Acceleration and Accelerated Motion
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 1 2 3 4 5 6
Time (s)
Po
siti
on
(m
)
On a position time graph what is slope equal to?
Velocity
Is the slope constant in this graph?
No
You can use a tangent line to tell you the slope at a given point in time. Let’s try.
Finding the slope at 3 seconds:
Draw a tangent line, which is a straight line that touches the curve at only the desired point.
smss
mmvelocityslope /9.28
5.16
0130
smvelocity /30This is instantaneous velocity. (The velocity at that instant.)
Find Average Velocity
t
xx
t
xv ifavg
sm
avg s
mmv 25
5
0125
Chapter 3: Acceleration and Accelerated Motion
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 1 2 3 4 5 6
Time (s)
Po
siti
on
(m
)
Can we make a motion map of this motion?
You Bet!
x (displacement13020 40 60 80 100 1200 30 70 11010 9050 130
0s 1s 2s 3s 4s 5sv v v v v
What happens to the distance between the dots?
What is happening to the velocity?
Chapter 3: Acceleration and Accelerated Motion
We need to make a change one addition to our “How Far” equation.
What if you saw a velocity/time graph that looked like this?
velocity (m/s)
t (seconds)62 4 6
60
20
40
60
1 5
10
50
3
30
00
What is different about this graph than the previous velocity/time graph?
The velocity at t = 0 is 10 m/s. In other words, the car has an initial velocity of 10 m/s.
Chapter 3: Acceleration and Accelerated Motion
velocity (m/s)
t (seconds)62 4 6
60
20
40
60
1 5
10
50
3
30
00
Let’s see how this affects our “how far” equation.
Again, we need to find displacement. How do we do this?
Area under ‘curve’
Let’s look at the time interval of 0 – 1 sec.
This area is a goofy, irregular shape, so we need to look at this as a rectangle and a triangle together!
Green Area
xAreaWhat equation can I make for the area (displacement)? bhbh2
1
Red Area
)10)(1()10)(1(21
sm
sm ssx
mmmx 15105
Chapter 3: Acceleration and Accelerated Motion
xArea bhbh21
)10)(2()1030)(2(21
sm
sm ssx
mmmx 402020
velocity (m/s)
t (seconds)62 4 6
60
20
40
60
1 5
10
50
3
30
00
Green Area Red Area
Let’s do the same thing for 0 – 2 sec.
Look for the pattern:2
21
21 atbh
?bh tvi
time initial velocity
Therefore the ‘How Far’ equation becomes:
tvatx i 221
Chapter 3: Acceleration and Accelerated Motion
Now use that equation to find the position of the object at each second.
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 10 0 10
1 20 15 10
2 30 40 10
3 40 10
4 50 10
5 60 10
This comes from slope, which is the same as the first v/t graph.
221 )3)(10()3)(10( ssx s
msm At 3 sec. m75
221 )4)(10()4)(10( ssx s
msm
221 )4)(10()4)(10( ssx s
msm
m120
m175
75120175
From previous pages
At 4 sec.
At 5 sec.
Chapter 3: Acceleration and Accelerated Motion
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 10 0 10
1 20 15 10
2 30 40 10
3 40 75 10
4 50 120 10
5 60 175 10
Let’s make a position/time graph for this motion.
0102030405060708090
100110120130140150160170180
0 1 2 3 4 5
Time (s)
po
siti
on
(m
)
Notice the shape: top opening parabola (curvy up)
How can the position time graph go through (0,0) and the velocity time graph didn’t?
The car can have an initial velocity at t=0, at the ref. point.
Chapter 3: Acceleration and Accelerated Motion
0s 1s 2s 3s 4sv v v v v
x (displacement
18020 40 60 80 100 120 140 160 18010 50 90 130 17030 1100 15070
Let’s make a motion map for this motion also.
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 10 0 10
1 20 15 10
2 30 40 10
3 40 75 10
4 50 120 10
5 60 175 10
5s
Chapter 3: Acceleration and Accelerated Motion
Let’s take the case of an object slowing down…(Negative acceleration)
t (time-seconds)
v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 50
1 40
2 30
3 20
4 10
5 0
velocity (m/s)
t (seconds)
52 4
0
10
30
50
1 5
20
0
50
3
40
Calculate the slope:
ssa s
msm
05
500
sm10
What does slope of a v/t graph mean again???
Oh yeah….. Acceleration!
What does negative acceleration mean?
Chapter 3: Acceleration and Accelerated Motion
What does negative acceleration mean?
It can mean slowing down, but that’s not a complete picture.
It most accurately means that the object is accelerating in the negative direction.
Ex: If your put your car in reverse at the stop sign (reference pt.) and put your foot on the gas pedal, you would be speeding up in the backwards direction. This would also be negative acceleration.
Velocity Acceleration Motion
Positive Positive
Positive Negative
Negative Negative
Negative Positive
Speeding up, forward
Slowing down forwardSpeeding up, backward
Slowing down, backward
Chapter 3: Acceleration and Accelerated Motion
acceleration (m/s2)
0 t (seconds)
52 4
-15-15
-5
10
0 3
15
5
1
-10
5
15
t (time-seconds) v (velocity-m/s) x (position-m) a (acceleration-m/s2)
0 50 -10
1 40 -10
2 30 -10
3 20 -10
4 10 -10
5 0 -10
Chapter 3: Acceleration and Accelerated Motionvelocity (m/s)
t (seconds)52 4
0
10
30
50
1 5
20
0
50
3
40Let’s make the position/time graph:
x (position-meters)
time (t-seconds)
52 40
10
30
50
70
90
110
0 3
130
40
80
120
5
60
1
100
20
Find x at t=1
bhbhx 21
)/40)(1()/10)(1(21 smssmsx m45
221 attvx i
Let’s use the “how far” equation.
221 )2)(10()2)(/50( 2 sssmx
sm m80
221 )3)(10()3)(/50( 2 sssmx
sm m105
221 )4)(10()4)(/50( 2 sssmx
sm m120
221 )5)(10()5)(/50( 2 sssmx
sm m125
Chapter 3: Acceleration and Accelerated Motion
Chapter 3: Acceleration and Accelerated Motion