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Kinematics - Analyzing
motion under
One Dimensional Motion
What is Kinematics?
• There are two aspects to any motion
– The movement itself
– What causes the motion or what changes the
motion, which requires that forces be considered
• Mechanics –study of motion and its causes
• Two parts to mechanics
– Kinematics- deals with the concepts that are
needed to describe motion, w/out references to
forces
– Dynamics- deals with the effect that forces have on
motion
YOU deserve a speeding ticket!
I am the LAW around here and
the LAW says that the speed limit
is 55 miles per hour!
Here is the scenario!
You wake up late and have 20 minutes to get to school and you
especially do not want to be late for physics! You decide to travel
at 65 mph in a 55 mph zone. Unfortunately, Chief Genova pulls
you over. You see, the LAW (that’s me!) states that you must pay
$10 for every mile you are over the speed limit. Therefore, you
MUST pay $100 to cover the fine. Being a law abiding citizen you
agree, and pay the fine. But as you begin to leave you hear Chief
Genova say……..
BUT WAIT!
Motion Speed
Over
Fine
On road 10 mph $100
Earth’s
Rotation
1000 mph
Our city council just recently passed a
law 3 months ago that requires us to fine
you for being on the planet Earth as it
rotates on it’s axis. Since Earth rotates on
its axis at 1000 mph you must pay ….
But as you
begin to
leave you
hear Chief
Genova
say……..
$10,000
BUT WAIT!
Motion Speed
Over
Fine
On road 10 mph $100
Earth’s
Rotation
1000 mph $10,000
Earth’s
Revolving
66,621
mph
Our city council just last month passed a
law that requires us to fine you for being
on the planet Earth as it revolves around
our Sun at a speed of 66,621 mph. Thus
your fine is…..
But as you
begin to
leave you
hear Chief
Genova
say……..
$666,210
BUT WAIT!
Motion Speed
Over
Fine
On road 10 mph $100
Earth’s
Rotation
1000 mph $10,000
Earth’s
Revolving
66,621
mph
$666,210
To Vega 44,041
mph
Our city council just last week passed a
law that requires us to fine you for being
on the planet Earth which moves towards
Vega in the constellation Lyra at a speed
of 44,041 mph. Thus your fine is…..
But as you
begin to
leave you
hear Chief
Genova
say……..
$440,410
BUT WAIT!
Motion Speed
Over
Fine
On road 10 mph $100
Earth’s
Rotation
1000 mph $10,000
Earth’s
Revolving
66,621
mph
$666,210
To Vega 44,041
mph
$440,410
Milky Way 558,900
mph
Our city council just yesterday passed a
law that requires us to fine you for being
on the planet Earth in the Milky Way
which rotates at a speed of 558,900 mph.
Thus your fine is…..
Let me total
your fine!
$5,589,000
You owe……(somehow)
Your total fine is:
$6,705,720
Now be a law abiding citizen and PAY UP!
The bottom line….Motion is
RELATIVE
• It depends completely on how you want to look at the moving object.
• Example: You are sitting in an airplane which is moving at a speed of 100 km/h and there is a fly sitting on your head.
(a) What is your speed relative to the ground?
(b) What is your speed relative to the seat you're sitting it?
(c) What is the speed of the fly relative to you?
• To describe motion of an objet, must specify the location of the object at all times= establish a frame of reference
100 km/hr
0 km/hr
0 km/hr
Scalars and Vectors
• Vectors are quantities that have both a
direction and a magnitude (size).
– Ex. 2 km, 30ο north of east
• Examples of Vectors used in Physics
– Displacement
– Velocity
– Acceleration
– Force
• Scalars are quantities that have only a
magnitude(size) are called.
Scalar Example Magnitude
Speed 20 m/s
Distance 10 m
Age 15 years
Heat 1000 calories
Displacement
• Displacement (x or y) = "Change in position" is a “vector”
• Not necessarily the total distance traveled. Displacement &
distance are entirely different concepts.
– Displacement is relative to an axis, Distance how far travel w/out a
direction
o "x" displacement means moving horizontally either right or left.
o "y" displacement means moving vertically either up or down.
o Word change is expressed using the Greek letter DELTA ( Δ ).
o Find the change ALWAYS subtract FINAL - INITIAL position
o It is therefore expressed as either Δx = xf - xi or Δy = yf – yi
o SI unit = meter, can be + or – in value
o + moving along line in same direction
o - moving along line in opposite direction
Example
• Suppose a person moves in a straight line from the
lockers(at a position x = 1.0 m) toward the physics
lab(at a position x = 9.0 m) , as shown below. What is
their displacement?
The answer is positive so the person must have
been traveling horizontally to the right.
Example
•Suppose the person turns around and goes back to where
they started? What is their displacement?
The answer is negative so the person must have been traveling
horizontally to the left
What is the DISPLACEMENT for the entire trip?
What is the total DISTANCE for the entire trip?
mxxx initialfinal 00.10.1
m1688
Time Interval
• The time it takes for a given motion to
occur is called the time interval.
– There is a difference between a time and a
time interval.
– Unlike displacements, time intervals are
always positive.
if ttt
Velocity
• Average Velocity is defined as: “The RATE at
which DISPLACEMENT changes”.
• Rate = ANY quantity divided by TIME.
• SI unit for velocity is m/s
• Velocity can be + or -,
depending on its direction
•Average SPEED = the “RATE at which DISTANCE changes”
•Speed – the distance traveled divided by the elapsed time
•SI unit for speed is m/s
•Speed indicate how fast object moves but does not
give a direction t
ds
Example
• A quarterback throws a pass to a defender on the
other team who intercepts the football. Assume the
defender had to run 50 m away from the quarterback
to catch the ball, then 15 m towards the quarterback
before he is tackled. The entire play took 8 seconds.
Let's look at the defender's average velocity:
smss
mm
t
xv /38.4
08
035
Let's look at the defender's speed:
sms
m
t
ds /125.8
8
65
“m/s” is the derived unit
for both speed and
velocity.
Slope – A basic graph model
A basic model for understanding graphs in physics is SLOPE.
Using the model - Look at the formula for velocity.
Who gets to play the role of the slope?
Who gets to play the role of the y-axis or the rise?
Who gets to play the role of the x-axis or the run?
What does all the mean? It means that if your are given a Displacement
vs. Time graph, to find the velocity of an object during specific time
intervals simply find the slope.
t
xv
Run
Riseslope
Velocity
Displacement
Time
Displacement vs. Time graph
What is the velocity of the object from 0 seconds to 3 seconds? The velocity is the slope!
Steeper slope means a faster moving object.
Displacement vs. Time graph
• What is the velocity of the object from 7 seconds to 8
seconds? Once again...find the slope!
A velocity of 0 m/s. What does this mean?
It is simple....the object has simply
stopped moving for 1 second.
Displacement vs. Time graph
• What is the velocity from 8-10 seconds? You must remember!
To find the change it is final - initial.
The answer is negative! It is no surprise,
because the slope is considered to be
negative.
This value could mean several things:
The object could be traveling WEST or
SOUTH. The object is going backwards - this
being the more likely choice!
You should also understand that the slope
does NOT change from 0-3s , 5 to 7s and 8-
10s.
This means that the object has a CONSTANT
VELOCITY or IT IS NOT ACCELERATING.
Example
• It is very important that you are able to look at a graph and explain it's motion
in great detail. These graphs can be very conceptual.
Look at the time interval t = 0 to t = 9
seconds. What does the slope do?
It increases, the velocity is increasing
Look at the time interval t = 9 to t = 11
seconds. What does the slope do?
No slope. The velocity is ZERO.
Look at the time interval t = 11 to t = 15
seconds. What does the slope do?
The slope is constant and positive. The
object is moving forwards at a constant
velocity.
Look at the time interval t = 15 to t = 17
seconds. What does the slope do? The slope is constant and negative.
The object is moving backwards at a
constant velocity.
• Average Acceleration- change in an object’s velocity
over time period in which the change occurs
• The RATE of CHANGE of VELOCITY
• (a) can be +, or – value =vector that shows direction
• When a & v opposite directions = object slows down
or is decelerating
• When a & v same directions = object speeds up or is
accelerating
2///seconds
ondmeters/secsmorssm
t
va
Acceleration
• Example: A Cessna Aircraft goes from 0
m/s to 60 m/s in 13 seconds. Calculate the
aircraft’s acceleration.
13
060
t
va 4.62 m/s2
Example
acceleration
• Example: The Cessna now decides to land and goes from 60 m/s to 0 m/s in 11 s. Calculate the Cessna’s acceleration ?
deceleration
11
600
t
va - 5.45 m/s2
Example
Slopea
Runt
Risev
t
va
Run
RiseSlope
Acceleration
What is the acceleration from t=0s to t=3s?
60/3= 20 m/s2
What is the acceleration from t=3 s to t=5s?
0/2 = 0 m/s2
What is the acceleration from t=8s to t=9s?
0-60/1 = -60 m/s2
Acceleration – Graphical Representation
Kinematic Symbols for
Constant Acceleration
x, y Displacement
t Time
vo Initial Velocity
v Final Velocity
a Acceleration
g Acceleration due to gravity
Kinematic Equations for
Constant Acceleration
Equation #1 v = vo + at
Equation #2 x = ½(vo +v)t
Equation #3 x=vot + 1/2at2
Equation #4 v2 = vo2 + 2ax
Kinematic #1
atvv
atvvt
vv
t
va
o
oo
Kinematic #1
• Example: A boat moves slowly out of a marina (so as to
not leave a wake) with a speed of 1.50 m/s. As soon as it
passes the breakwater, leaving the marina, it throttles up
and accelerates at 2.40 m/s2. How fast is the boat
moving after accelerating for 5 seconds?
What do I know? What do I want?
vo= 1.50 m/s v = ?
a = 2.40 m/s/s
t = 5 s
v
v
atvv o
)5)(40.2()50.1(
13. 5 m/s
Kinematic #3
2
21 attvx ox
x
x
attvx ox
)5)(40.2(2
1)5)(5.1(
21
2
2
37.5 m
Example: A boat moves slowly out of a marina (so as to not
leave a wake) with a speed of 1.50 m/s. As soon as it passes
the breakwater, leaving the marina, it throttles up and
accelerates at 2.40 m/s2. How far did the boat travel during
that time?
What do I know? What do I want?
vo= 1.50 m/s x = ?
a = 2.40 m/s/s
t = 5 s
Kinematic #4
axvv o 222
What do I know? What do I want?
vo= 12 m/s x = ?
a = -3.5 m/s2
V = 0 m/s
x
x
x
axvv o
7144
)5.3(2120
2
2
22
• Example: You are driving through town at 12 m/s when
suddenly a ball rolls out in front of your car. You apply the
brakes and begin decelerating at 3.5 m/s2. How far do you
travel before coming to a complete stop?
20.57 m
Common Problems Students
Have
I don’t know which equation to choose!!!
Equation Missing Variable
x
v
t
atvv o
2
21 attvx ox
axvv o 222
Kinematics for the VERTICAL
Direction
• All kinematics equations can be used to analyze one
dimensional motion in either the X direction OR the y
direction
tvvytvvX
gyvvaxvv
gttvyattvx
gtvvatvv
oyox
oyyox
oyox
oyyo
)(2/1)(2/1
22
21
21
2222
22
Example
• A pitcher throws a fastball with a velocity of 43.5 m/s. It is determined that during the windup and delivery the ball covers a displacement of 2.5 meters. This is from the point behind the body to the point of release. Calculate the acceleration during his throwing motion.
What do I know? What do I want?
vo= 0 m/s a = ?
x = 2.5 m
v = 43.5 m/s
axvv o 222
Which variable is NOT given and
NOT asked for? TIME (t)
2
22
/45.378
)5.2(205.43
sma
a
Example
• How long does it take a car at rest to cross a
35.0 m intersection after the light turns green,
if the acceleration of the car is a constant 2.00
m/s2?
What do I know? What do I want?
vo= 0 m/s t = ?
x = 35 m
a = 2.00 m/s/s st
t
92.5
)2(2
1)0(35 2
Which variable is NOT given and
NOT asked for? Final Velocity
2
21 attvx ox
Final Velocity(v)
Example
• A car accelerates from 12.5 m/s to 25 m/s in
6.0 seconds. What was the acceleration?
What do I
know?
What do I
want?
vo= 12.5 m/s a = ?
v = 25 m/s
t = 6s
atvv o
Which variable is NOT given and
NOT asked for? DISPLACEMENT
2/08.2
)6(5.1225
sma
a
Displacement (x)
Example
• A stone is dropped from the top of a cliff. It is observed to hit the ground 5.78 s later. How high is the cliff?
What do I
know?
What do I
want?
voy= 0 m/s y = ?
g = -9.8 m/s2
t = 5.78 s
2
21 gttvy oy
Which variable is NOT given and
NOT asked for? Final Velocity
mh
my
y
7.163
7.163
)78.5(9.4)78.5)(0( 2
Final Velocity(v)
The Acceleration of Gravity
• On Earth, the acceleration caused by gravity is about 9.81 m/s2
• Since the acceleration is downward, we write a = -9.81 m/s2
• The quantity 9.81 m/s2 is abbreviated g. Thus, the acceleration of gravity is a = -g
• If an object is in FREE FALL in the VERTICAL
DIRECTION, the acceleration is due to GRAVITY.
• Applies to an object falling freely near the surface of
the Earth (or any other planet) due to gravity,
ignoring air resistance.
2/8.9 smgay •It NEVER ceases to exist •It ALWAYS works DOWN •It is NEVER zero •This is ONLY true in a vacuum (no air) •Acceleration does not depend on mass of object
Free-Fall Acceleration
• Example: A person throws a ball straight
upward into the air.
Q1: What is the Acceleration at the TOP of its
path? -9.8 m/s2
Q2: What is the VELOCITY at the TOP of its path?
ZERO
Acceleration due to Gravity
Q3: What is the magnitude(#value) and direction of the acceleration, HALF way up?
-9.8 m/s2 - ALWAYS DOWNWARD
Q4: What is the magnitude(#value) and direction of the acceleration, HALF way down?
-9.8 m/s2 - ALWAYS DOWNWARD
THE BOTTOM LINE: EVERYTHING will accelerate at -9.8 m/s2 in a VACUUM, that is any situation involving NO AIR.
Acceleration due to Gravity
Graphical Analysis
Let’s Review
smslopeRun
Rise
t
xv /
seconds
meters
VELOCITY is the SLOPE of a
distance, position, or displacement
vs. time graph.
Let’s Review
What is the slope doing?
INCREASING
What is the velocity doing?
INCREASING
Let’s Review
2/seconds
ondmeters/secsmslope
Run
Riseva
Let’s Review
Describe the acceleration
during interval A.
The acceleration or SLOPE
is constant and positive.
Describe the acceleration
during interval B.
The acceleration or SLOPE
is ZERO.
Describe the acceleration
during interval C.
The acceleration or SLOPE is constant and negative.
Let’s Review
What is the acceleration(slope) doing?
The acceleration is INCREASING!
Let’s Review
The slopes help us sketch the motion of other graphs!
NEW MODEL = AREA MODEL
seconds
meters x seconds meters
units of viewofpoint thefromit at Looking
AREAx
HEIGHTv
BASEt
HEIGHTxBASEAREAvtxt
xv
Does all this make sense?
mA
AbhA
50.7
)5.1)(5(
mA
bhA
30
)12)(5(2
1
2
1
mA
AbhA
50.7
)5.1)(5(
1.5
m/s
13.5 m/s
Total displacement = 7.50 + 30 = 37.5 m = Total AREA under the line.
The Area Model
How FAR did this object
travel during interval
A?
mxorArea
AbhA
100
20102
1
2
1
How FAR did this object
travel during interval
B?
mxorArea
AhbA
400
2020
How FAR did this object travel during interval c?
mxorArea
AbhA
400
20402
1
2
1
The Area Model – Acceleration
2seconds
metersseconds
seconds
meters
Areav
Heighta
Baset
HeightxBaseAreatavt
va
The Area Model
v
What is the change in
Velocity during the t=2 to
t=4 interval?
smvA
AhbA
/6
32
In summary
Graph Slope Area
x vs. t Velocity N/A
v vs. t Acceleration Displacement
a vs. t N/A Velocity
Slope Graph Area
x vs. t
v vs. t
a vs. t
t (s) t (s)
x (m) v (m/s)
area = x
t (s)
a (m/s2)
area = v
Comparing & Sketching graphs
One of the more difficult applications of graphs in physics is when given
a certain type of graph and asked to draw a different type of graph
t (s)
x (m)
t (s)
v (m/s)
List 2 adjectives to describe the SLOPE or VELOCITY
1.
2.
The slope is CONSTANT
The slope is POSITIVE
How could you translate what
the SLOPE is doing on the
graph ABOVE to the Y axis on
the graph to the right?
Example
t (s)
x (m)
1st line
•
•
2nd line
•
3rd line
•
•
The slope is constant
The slope is constant
The slope is “+”
The slope is “-”
The slope is “0”
Example – Graph Matching
t (s)
v (m/s)
t (s)
a (m/s/s)
t (s)
a (m/s/s)
t (s)
a (m/s/s)
What is the SLOPE(a) doing?
The slope is increasing
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