One Dimensional Kinematics - Chapter Outline

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One Dimensional Kinematics - Chapter Outline
Lesson 1 Describing Motion with Words Lesson 2
Describing Motion with Diagrams Lesson 3
Describing Motion with Displacement vs. Time
Graphs Lesson 4 Describing Motion with Velocity
vs. Time Graphs Lesson 5 Free Fall and the
Acceleration of Gravity Lesson 6 Describing
Motion with Equations

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1.Describing Motion with Words

1. Know the Language of Kinematics
2. Differentiate Scalars and Vectors
3. Understand Distance and Displacement
4. Be able to calculate Speed and Velocity
5. Be able to calculate Acceleration

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Introduction to the Language of Kinematics

• Mechanics - the study of the motion of objects.
• Kinematics is the science of describing the
  motion of objects using words, diagrams, numbers,
  graphs, and equations. Kinematics is a branch of
  mechanics.

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Scalars and Vectors

• Scalars are quantities that are fully described
  by a magnitude (or numerical value) alone.
• Vectors are quantities that are fully described
  by both a magnitude and a direction.

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Distance and Displacement

• Distance is a scalar quantity that refers to "how
  much ground an object has covered" during its
  motion.
• Displacement is a vector quantity that refers to
  "how far out of place an object is" it is the
  object's overall change in position. Displacement
  has a direction.
• Example consider the motion depicted in the
  diagram below. A physics teacher walks 4 meters
  East, 2 meters South, 4 meters West, and finally
  2 meters North.

the physics teacher has walked a total distance
of 12 meters, her displacement is 0 meters.
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example

• Use the diagram to determine the resulting
  displacement and the distance traveled by the
  skier during these three minutes.

The skier covers a distance of (180 m 140 m
100 m) 420 m and has a displacement of 140 m,
rightward.
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example

• What is the coach's resulting displacement and
  distance of travel?

The coach covers a distance of (35 yds 20 yds
40 yds) 95 yards and has a displacement of 55
yards, left.
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Check Your Understanding

• 1. What is the displacement of the cross-country
  team if they begin at the school, run 10 miles
  and finish back at the school?
• 2. What is the distance and the displacement of
  the race car drivers in the Indy 500?

The displacement of the runners is 0 miles.
The displacement of the cars is somewhere near 0
miles since they virtually finish where they
started. Yet the successful cars have covered a
distance of 500 miles.                            
                                                 
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Velocity vs. Speed

• VELOCITY
• change in DISPLACEMENT occurring over time
• Includes both MAGNITUDE and DIRECTION
• VECTOR
• SPEED
• change in DISTANCE occurring over time
• Includes ONLY MAGNITUDE
• SCALAR

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Average Velocity/Speed Equations

• As an object moves, it often undergoes changes in
  speed. The average speed during the course of a
  motion is often computed using the following
  formula

• In contrast, the average velocity is often
  computed using this formula

d is total displacement

• The direction of velocity is the same as the
  direction of motion.

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example

• Jim gets on his bike and rides 300 meters west in
  60 seconds.
• What is his average velocity?
• What is his average speed?

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example

• Sally gets up one morning and decides to take a
  three mile walk. She completes the first mile in
  8.0 minutes, the second mile in 8.5 minutes, and
  the third mile in 9.0 minutes. What is her
  average speed?

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example

• In a drill during basketball practice, a player
  runs the length of the 30.-meter court and back.
  The player does this three times in 60. seconds.
  What is the average speed of the player during
  the drill?

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example

• The physics teacher walks 4 meters East, 2 meters
  South, 4 meters West, and finally 2 meters North.
  The entire motion lasted for 24 seconds.
  Determine the average speed and the average
  velocity.

her average speed was 0.50 m/s and her average
velocity of 0 m/s.
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example

• Use the diagram to determine the average speed
  and the average velocity of the skier during
  these three minutes.

The skier has an average speed of (420 m) / (3
min) 140 m/min and an average velocity of (140
m, right) / (3 min) 46.7 m/min, right
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example

• What is the coach's average speed and average
  velocity?

The coach has an average speed of (95 yd) / (10
min) 9.5 yd/min and an average velocity of (55
yd, left) / (10 min) 5.5 yd/min, left
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Average Speed vs. Instantaneous Speed
Average speed is a measure of the distance
traveled in a given period of time Suppose that
during your trip to school, you traveled a
distance of 5 miles and the trip lasted 0.2 hours
(12 minutes). The average speed of your car could
be determined as
Instantaneous Speed - the speed at any given
instant in time. For example, your speedometer
tells the instantaneous speed. During your trip,
there may have been times that you were stopped
and other times that your speedometer was reading
50 miles per hour. Yet, on average, you were
moving with a speed of 25 miles per hour.
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Constant speed vs. changing speed
An object with a changing speed would be moving a
different distance each second. The data tables
below depict objects with constant and changing
speed.
Constant speed the object will cover the same
distance every regular interval of time
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In conclusion

• Speed and velocity are kinematics quantities that
  have distinctly different definitions. Speed,
  being a _______quantity, is the rate at which an
  object covers ___________. The average speed is
  the _____________ (a scalar quantity) per time
  ratio. Speed is ignorant of direction. On the
  other hand, velocity is a _________quantity it
  is direction-aware. Velocity is the rate at which
  the position changes. The average velocity is the
  ______________ or position change (a vector
  quantity) per time ratio.

scalar
distance
distance
vector
displacement
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Acceleration

• Definition how fast the velocity is changing -
  change in VELOCITY over TIME
• Change in speed (speed up or slow down)
• Change in direction
• Equation
• VECTOR
• measured in velocity unit / time unit (m/s2,
  mi/hr, etc.)

? (delta) means change ?v vf - vi
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Anytime an object's velocity is changing, the
object is said to be accelerating it has an
acceleration
In a car, there are three controls that can
create acceleration gas petal, brake, steering
wheel.
According to the data, the velocity is changing
over the course of time. In fact, the velocity is
changing by a constant amount - 10 m/s - in each
second of time. This is a case of constant
acceleration.
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The Meaning of Constant Acceleration

• constant acceleration is when an accelerating
  object will change its velocity by the same
  amount each second.

Acceleration is constant
Acceleration is changing
Do not confuse constant velocity with constant
acceleration
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Example
Monty the Monkey accelerates uniformly from rest
to a velocity of 9 m/s in a time span of 3
seconds. Calculate Monty's acceleration.
A child riding a bicycle at 15 meters per second
accelerates at -3.0 meters per second2 for 4.0
seconds. What is the childs speed at the end of
this 4.0-second interval?
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The Direction of the Acceleration Vector

• The direction of the acceleration vector depends
  on whether the object is speeding up or slowing
  down or changing directions.
• If an object is speeding up, then its
  acceleration is in the same direction of its
  motion.
• If an object is slowing down, then its
  acceleration is in the opposite direction of its
  motion.
• If an object is traveling east initially (vi),
  then goes west (vf), its acceleration is west
  (same as vf)

Positive accelerations don't necessarily indicate
an object speeding up, and negative accelerations
don't necessarily indicate an object slowing down.
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Example

• The instant before a batter hits a 0.14-kilogram
  baseball, the velocity of the ball is 45 meters
  per second west. The instant after the batter
  hits the ball, the balls velocity is 35 meters
  per second east. The bat and ball are in contact
  for 1.0 102 second. Determine the magnitude
  and direction of the average acceleration of the
  baseball while it is in contact with the bat.

vi 45 m/s W vf 35 m/s E t 2.0x10-2 s If
East is Then vi -45 m/s, vf 35 m/s
East
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2. Describing Motion with Diagrams

• Know how to interpret motion using Ticker Tape
  Diagrams
• Know how to interpret motion using Vector
  Diagrams

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Ticker Tape Diagrams

• A common way of analyzing the motion of objects
  in physics labs is to perform a ticker tape
  analysis. A long tape is attached to a moving
  object and threaded through a device that places
  a tick upon the tape at regular intervals of time
  - say every 0.10 second. As the object moves, it
  drags the tape through the "ticker," thus leaving
  a trail of dots. The trail of dots provides a
  history of the object's motion and therefore a
  representation of the object's motion.

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Examples of ticker tape diagram
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Example

• Oil drips at 0.4 seconds intervals from a car
  that has an oil leak.  Which pattern best
  represents the spacing of oil drops as the car
  accelerates uniformly from rest?
•     .    .    .    .    .    .    .
•     .    .         .                 .
•     .  .  .  .      .      .      .
•     .      . .    .   . .      . .    .

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Example

• A spark timer is used to record the position of a
  lab cart accelerating uniformly from rest. Each
  0.10 second, the timer marks a dot on a recording
  tape to indicate the position of the cart at that
  instant, as shown.
• The linear measurement between t 0 second to t
  0.30 is 5.4 cm.
• Calculate the average speed of the cart during
  the time interval t 0 second to t 0.30
  second.

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Vector Diagrams

• Vector diagrams are diagrams that depict the
  direction and relative size of a vector quantity
  by a vector arrow.

Constant acceleration
Both velocity and acceleration change
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3. Describing Motion with d-t Graphs

1. Understand The Meaning of Slope for a d-t Graph
2. Know The Meaning of Shape for a d-t Graph
3. Be able to Determine the Slope on a d-t Graph

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The Meaning of Slope for a d-t Graph
Distance or displacement

• The slope of the line on a d-t graph is equal to
  the speed/velocity of the object.

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Slope is speed/velocity in a d-t Graph

• If the object is moving with a velocity of 4
  m/s, then the slope of the line will be ______.
  If the object is moving with a velocity of -8
  m/s, then the slope of the line will be _______.
  If the object has a velocity of 0 m/s, then the
  slope of the line will be ________.

4 m/s
-8 m/s
0 m/s
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The Meaning of Shape for a d-t Graph
As the slope goes, so goes the velocity
Distance or displacement
Distance or displacement
Slope is Constant, positive, velocity is
constant, positive or speed is constant
Slope is increasing, positive, velocity is
increasing, positive or speed is increasing.
There is acceleration
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As the slope goes, so goes the velocity
Slow, Positive, Constant Velocity
Fast, Positive, Constant Velocity
displacement
displacement
Fast, Negative, Constant Velocity
displacement
displacement
Slow, Negative Constant Velocity
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As the slope goes, so goes the velocity
displacement
displacement
Slope is negative, increasing (steeper)
Slope is negative, decreasing (flatter)
velocity is negative, increasing (faster in
negative direction)
velocity is negative, decreasing (slower in
negative direction)
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Check Your Understanding

• Use the principle of slope to describe the motion
  of the objects depicted by the two plots below.

displacement
displacement
velocity is positive, increasing (faster in
positive direction)
velocity is negative, increasing (faster in
negative direction)
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example
displacement
Describe the velocity of the object between 0-5 s
and between 5-10 s.
The velocity is positive constant between 0-5
seconds The velocity is zero between 5-10 seconds
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example
A cart travels with a constant nonzero
acceleration along a straight line. Which graph
best represents the relationship between the
distance the cart travels and time of travel?
A
B
C
D

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example

• Which graph best represents the motion of a block
  accelerating uniformly down an inclined plane?

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example

• The displacement-time graph below represents the
  motion of a cart initially moving forward along a
  straight line. During which interval is the cart
  moving forward at constant speed?

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• Determining the Slope on a d-t Graph

Displacement (m)

1. Pick two points on the line and determine their
   coordinates.
2. Determine the difference in y-coordinates of
   these two points (rise).
3. Determine the difference in x-coordinates for
   these two points (run).

1. Divide the difference in y-coordinates by the
   difference in x-coordinates (rise/run or slope).
2. Make sure all your work has units.

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example
Slope -3.0 m/s
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example

• Determine the velocity (i.e., slope) of the
  object as portrayed by the graph below.

The velocity (i.e., slope) is 4 m/s.
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example

• With the given d-t graph of Tom,
• describe his motion during 0-5 s, 5-10 s, 10-12.5
  s.
• What is Toms total displacement in 12.5 seconds?

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example

• The graph represents the relationship between
  distance and time for an object. What is the
  instantaneous speed of the object at  t   5.0
  seconds?

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example

• The graph below represents the displacement of an
  object moving in a straight line as a function of
  time. What was the total distance traveled by the
  object during the 10.0-second time interval?

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4. Describing Motion with v-t Graphs

1. Know The Meaning of Slope for a v-t Graph
2. Be able to describe motion with given the Shape
   for a v-t Graph
3. Be able to Determining the Slope on a v-t Graph
4. Be able to Relate the Shape of d-t graph to the
   shape of v-t graph
5. Know the meaning of the Area on a v-t Graph
6. Be able to determine the Area on a v-t Graph

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The Meaning of Slope for a v-t Graph
The slope of the line on a v-t graph is equal to
the ACCELERATION of the object.
The average velocity of constant acceleration can
be determined by
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Describe motion with given the Shape for a v-t
Graph
The slope of the line on a velocity-time graph
reveals the acceleration of the object.
velocity is constant, Slope is zero, Acceleration
is Zero,
Velocity is Increasing, slope is positive,
constant, acceleration is positive, constant.
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Direction of velocity

• the velocity would be positive whenever the line
  lies in the positive region (above the x-axis) of
  the graph. Similarly, the velocity would be
  negative whenever the line lies in the negative
  region (below the x-axis) of the graph. And
  finally, if a line crosses over the x-axis from
  the positive region to the negative region of the
  graph (or vice versa), then the object has
  changed directions.

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speeding up or slowing down?

• Speeding up means that the magnitude (or
  numerical value) of the velocity is getting
  large.

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Example

• Consider the graph at the right. The object whose
  motion is represented by this graph is ...
  (include all that are true)
• moving in the positive direction.
• moving with a constant velocity.
• moving with a negative velocity.
• slowing down.
• changing directions.
• speeding up.
• moving with a positive acceleration.
• moving with a constant acceleration.

a, d and h
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Example
What is constant? What is zero?
v is constant, positive. Slope is zero, a is
zero.
What is constant? What is zero?
v is constant, negative. Slope is zero, a is
zero.
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What is increasing? What is constant?
What is decreasing? What is constant?
v is increasing, slope is constant, a is
constant. a and v are both positive.
v is decreasing, slope is constant, a is
constant. a is neg. v is pos.
What is increasing? What is constant?
What is decreasing? What is constant?
v is decreasing, slope is constant, a is
constant. a is pos. v is neg.
v is increasing, slope is constant, a is
constant. a and v are both negative.
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Check Your Understanding

• Describe the motion, include the direction of
  motion ( or - direction), the velocity and
  acceleration and any changes in speed (speeding
  up or slowing down) during the various time
  intervals (e.g., intervals A, B, and C).

1.
A direction at a constant speed, zero
acceleration B direction, slowing down,
negative acceleration C direction, a
constant speed slower speed than A, zero
acceleration.
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2.
A direction, slowing down, constant negative
acceleration B zero velocity, zero
acceleration C - direction, speeding up,
constant negative acceleration.
3.
A direction, constant velocity, zero
acceleration B direction, slowing down,
constant negative acceleration C - direction,
speeding up, constant negative acceleration.
Point indicate changing direction
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• Determining the Slope on a v-t Graph

• A method for carrying out the calculation is
• Pick two points on the line and determine their
  coordinates.
• Determine the difference in y-coordinates for
  these two points (rise).
• Determine the difference in x-coordinates for
  these two points (run).
• Divide the difference in y-coordinates by the
  difference in x-coordinates (rise/run or slope).

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Check Your Understanding

• Determine the acceleration (i.e., slope) of the
  object as portrayed by the graph.

The acceleration (i.e., slope) is 4 m/s/s.
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Example
Describe motion from 0 4 s, and 4 8 s with
the given diagram.
From 0 s to 4 s slope 0 m/s2 a 0 m/s2 From
4 s to 8 s slope 2 m/s2 a 2 m/s2
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Example

• The velocity-time graph for a two-stage rocket is
  shown below. Determine the acceleration of the
  rocket during the listed time intervals.
• t 0 - 1 second
• t 1 - 4 second
• t 4 - 12 second

40 m/s/s
20 m/s/s
-20 m/s/s
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Relate d-t graph with v-t graph

• In d-t graph, the slope is velocity.
• In v-t graph, the slope is acceleration.
• Motion can be describe by both graphs.
• Example an object is moving with positive,
  constant velocity.

all graphs represent the same motion
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example

• Graph d-t and v-t graphs for an object moving
  with constant, positive acceleration.

all graphs represent same motion
Slope is velocity. Positive acceleration means
increasing velocity in positive direction, which
in turn means increasing slope, in positive
direction.
Slope is acceleration. Positive acceleration
means constant, positive slope.
acceleration is constant, positive.
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example
Which pair of graphs represents the same motion
of an object?
A
B
C
D
   
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example
Which pair of graphs represents the same motion?
A
B
C
D
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example
Which pair of graphs represents the same motion
of an object?
A
B
C
D
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• The meaning of the Area on a v-t Graph

For velocity versus time graphs, the area bound
by the line and the axes represents the
displacement.
A ½ b x ( h1 h2) Or A ½b2xh2 - ½ b1xh1
A b x h A v x t A displacement
A ½ b x h
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Determine the area of displacement

• Determine the displacement (i.e., the area) of
  the object during the first 4 seconds (Practice
  A) and from 3 to 6 seconds (Practice B).

90 m
120 m
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example

• Determine the displacement of the object during
  the first second (Practice A) and during the
  first 3 seconds (Practice B).

5 m
45 m
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example

• Determine the displacement of the object during
  the time interval from 2 to 3 seconds (Practice
  A) and during the first 2 seconds (Practice B).

25 m
40 m
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example

• The graph below represents the velocity of an
  object traveling in a straight line as a function
  of time. Determine the magnitude of the total
  displacement of the object at the end of the
  first 6.0 seconds.

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5. Free Fall and the Acceleration of Gravity

1. Understand the meaning of Free Fall
2. Know The Acceleration of Gravity
3. Be able to Represent Free Fall by Graphs
4. Be able to steer clear from The Big Misconception

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What is Free Fall?

• Free-falling objects do not encounter air
  resistance.

Ticker tape trace for free fall
As an object free falls, its speed in increasing.
The distance traveled during each second also
increases.
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Acceleration of gravity

• The acceleration for any object moving under the
  sole influence of gravity is called acceleration
  of gravity
• The symbol g is used to represent the
  acceleration of gravity.
• g 9.81 m/s2, downward
• The value of the acceleration of gravity (g) is
  different in different gravitational
  environments.
• On the moon, g 1.6 m/s2
• On Mercury, g 3.7 m/s2

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Acceleration is the rate at which an object
changes its velocity. It is the ratio of velocity
change to time between any two points in an
object's path.

• To accelerate at 9.81 m/s/s means to change the
  velocity by 9.81 m/s each second.

Time (s) Velocity (m/s)
0 0
1 - 9.81
2 - 19.62
3 - 29.43
4 - 39.24
5 - 49.05

• If the velocity and time for a free-falling
  object being dropped from a position of rest were
  tabulated, then one would note the following
  pattern.

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• Representing Free Fall by Graphs

Velocity is increasing in neg. direction. Slope
is increasing in negative direction.
Acceleration is constant, negative, slope is
constant, negative. Velocity starts from zero,
increase in neg. direction.
velocity
time
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Graphs of up and down motion
velocity
Upward velocity is big, positive, decreasing,
slope is constant (a -9.8 m/s/s). Top, velocity
is zero. Slope remains the same (acceleration is
still -9.8 m/s/s) Downward velocity increases
in negative direction at the same constant rate
of -9.81 m/s/s (slope remains the same), reaches
the same speed as it started upward.
time
Upward displacement increases, slope is
positive, decreasing (velocity is positive,
decreasing) Top slope 0 (its velocity is
zero) Downward displacement decreases, its slope
increases in negative direction (velocity is
negative, increasing)
position
time
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The Big Misconception

• Does heavy object falls faster than lighter
  object?
• NO. The acceleration of a free-falling object (on
  earth) is 9.81 m/s2. This value (known as the
  acceleration of gravity) is the same for all
  free-falling objects regardless of
• how long they have been falling,
• whether they were going up, down or go side ways.
• How massive they were.

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6. Describing Motion with Equations

1. Know The Kinematics Equations
2. Be able to apply Kinematics Equations to solve
   Problems
3. Be able to apply Kinematics Equations to solve
   problems in Free Fall
4. Be able to Relate Kinematics Equations and Graphs
   

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Kinematics Equations

• d - displacement t time a acceleration
• - velocity vi - initial velocity vf - final
  velocity

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The strategy for solving problems

1. Identify and list the given information in
   variable form.
2. Identify and list the unknown information in
   variable form.
3. Identify and list the equation that will be used
   to determine unknown information from known
   information.
4. Substitute known values into the equation and use
   appropriate algebraic steps to solve for the
   unknown information.
5. Make sure your answer has proper unit

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Example A

• Ima Hurryin is approaching a stoplight moving
  with a velocity of 30.0 m/s. The light turns
  yellow, and Ima applies the brakes and skids to a
  stop. If Ima's acceleration is -8.00 m/s2, then
  determine the displacement of the car during the
  skidding process. (Note that the direction of the
  velocity and the acceleration vectors are denoted
  by a and a - sign.)

vi vf ?t a d
30.0 m/s 0.00 m/s 15 m/s -8.00 m/s2 ?
3 sig figs.
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Example B

• Ben Rushin is waiting at a stoplight. When it
  finally turns green, Ben accelerated from rest at
  a rate of a 6.00 m/s2 for a time of 4.10 seconds.
  Determine the displacement of Ben's car during
  this time period.

vi vf ?t a d
0.00 m/s 4.10 s 6.00 m/s2 ?
3 sig figs.
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Example C

• A race car starting from rest accelerates
  uniformly at a rate of 4.90 meters per second2.
  What is the cars speed after it has traveled
  200. meters?

vi vf ?t a d
0.00 m/s ? 4.10 s 4.90 m/s2 200. m
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Kinematics Equations and Free Fall

• An object in free fall experiences an
  acceleration of -9.81 m/s2. ( downward
  acceleration.)
• If an object is merely dropped (as opposed to
  being thrown) from an elevated height, then the
  initial velocity of the object is 0 m/s. (vi 0)
• If an object is projected upwards in a perfectly
  vertical direction, then it will slow down as it
  rises upward. The instant at which it reaches the
  peak of its trajectory, its velocity is 0 m/s.
  This value can be used as one of the motion
  parameters in the kinematics equations for
  example, the final velocity (vf) after traveling
  to the peak would be assigned a value of 0 m/s.
• If an object is projected upwards in a perfectly
  vertical direction, then the velocity at which it
  is projected is equal in magnitude and opposite
  in sign to the velocity that it has when it
  returns to the same height. That is, a ball
  projected vertically with an upward velocity of
  30 m/s will have a downward velocity of -30 m/s
  when it returns to the same height.

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Example A

• Luke Autbeloe drops a pile of roof shingles from
  the top of a roof located 8.52 meters above the
  ground. Determine the time required for the
  shingles to reach the ground.

up is positive, down is negative.
vi vf ?t a d
0.00 m/s ? -9.81m/s2 -8.52m
t 1.32 s
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Example B

• Rex Things throws his mother's crystal vase
  vertically upwards with an initial velocity of
  26.2 m/s. Determine the height to which the vase
  will rise above its initial height.

vi vf ?t a d
26.2 m/s 0.00m/s 13.2m/s -9.81m/s2 ?
d 35.0 m
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Check Your Understanding

• An airplane accelerates down a runway at 3.20
  m/s2 for 32.8 s until is finally lifts off the
  ground. Determine the distance traveled before
  takeoff.
• A car starts from rest and accelerates uniformly
  over a time of 5.21 seconds for a distance of 110
  m. Determine the acceleration of the car.
• Upton Chuck is riding the Giant Drop at Great
  America. If Upton free falls for 2.6 seconds,
  what will be his final velocity and how far will
  he fall?

d 1720 m
a 8.10 m/ s2
vf -25.5 m/s
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• Kinematics Equations and Graphs
• there are now two methods to solve problems
  involving the numerical relationships between
  displacement, velocity, acceleration and time.
• Using equations
• Using graphs the slope of the line on a
  velocity-time graph is equal to the acceleration
  of the object and the area between the line and
  the time axis is equal to the displacement of the
  object.

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Using Kinematics Equations and Graphs to solve
problems

• An object that moves with a constant velocity of
  5.0 m/s for a time period of 5.0 seconds and
  then accelerates to a final velocity of 15 m/s
  over the next 5 seconds. What is the acceleration
  of the object? How far did the object travel in
  total of 10 seconds?
• Using graph

Slope acceleration t 0 5 s a 0 t 5
10 s a 2 m/s2
Area displacement t 0 5 s d 25 m t 5
10 s d 50 m dtotal 75 m
94

• 2. Kinematics equations

vi vf ?t a d1
5.0 m/s 5.0 m/s 5.0 m/s 5.0 s 0 ?
t 0 s - 5 s
vi vf ?t a d2
5.0 m/s 15 m/s 10. m/s 5.0 s ? ?
t 5.0 s 10. s
95
practice

1. A 747 jet, traveling at a velocity of 70. meters
   per second north, touches down on a runway. The
   jet slows to rest at the rate of 2.0 meters per
   second2. Calculate the total distance the jet
   travels on the runway as it is brought to rest.
2. A basketball player jumped straight up to grab a
   rebound. If she was in the air for 0.80 second,
   how high did she jump?
3. A rocket initially at rest on the ground lifts
   off vertically with a constant acceleration of
   2.0 101 meters per second2. How long will it
   take the rocket to reach an altitude of 9.0 103
   meters?

96

• A spark timer is used to record the position of a
  lab cart accelerating uniformly from rest. Each
  0.10 second, the timer marks a dot on a recording
  tape to indicate the position of the cart at that
  instant, as shown.    The linear measurement
  between t 0 second to t 0.30 is 5.4 cm.
  Calculate the magnitude of the acceleration of
  the cart during that time interval.

A
97

• Rennata Gas is driving through town at 25.0 m/s
  and begins to accelerate at a constant rate of
  -1.0 m/s2. Eventually Rennata comes to a complete
  stop.
• Represent Rennata's accelerated motion by
  sketching a velocity-time graph. Use the
  velocity-time graph to determine this distance.
• Use kinematic equations to calculate the distance
  that Rennata travels while decelerating.

vf2 vi2 2ad (0 m/s)2 (25.0 m/s)2 2
(-1.0 m/s2)d 313 m d
Area 313 m
98

• Otto Emissions is driving his car at 25.0 m/s.
  Otto accelerates at 2.0 m/s2 for 5 seconds. Otto
  then maintains a constant velocity for 10.0 more
  seconds.
• Represent the 15 seconds of Otto Emission's
  motion by sketching a velocity-time graph. Use
  the graph to determine the distance that Otto
  traveled during the entire 15 seconds.
• Finally, break the motion into its two segments
  and use kinematic equations to calculate the
  total distance traveled during the entire 15
  seconds.

For the 1st 5 second d vit 0.5at2 d
(25.0 m/s)(5.0 s) 0.5(2.0 m/s2)(5.0 s)2 d
150 m
last 10 seconds d vit 0.5at2 d (35.0
m/s)(10.0 s) 0.5(0.0 m/s2)(10.0 s)2 d 350
m 0 m d 350 m
Area 500 m
distance 150 m 350 m 500 m
99

• Luke Autbeloe, a human cannonball artist, is shot
  off the edge of a cliff with an initial upward
  velocity of 40.0 m/s. Luke accelerates with a
  constant downward acceleration of -10.0 m/s2 (an
  approximate value of the acceleration of
  gravity).
• Sketch a velocity-time graph for the first 8
  seconds of Luke's motion.
• Use kinematic equations to determine the time
  required for Luke Autbeloe to drop back to the
  original height of the cliff. Indicate this time
  on the graph.

vf vi atup 0 m/s 40 m/s (-10
m/s2)tup tup 4.0 s 2tup 8.0 s
100

• Chuck Wagon travels with a constant velocity of
  0.5 mile/minute for 10 minutes. Chuck then
  decelerates at -.25 mile/min2 for 2 minutes.
• Sketch a velocity-time graph for Chuck Wagon's
  motion. Use the velocity-time graph to determine
  the total distance traveled by Chuck Wagon during
  the 12 minutes of motion.
• Finally, break the motion into its two segments
  and use kinematics equations to determine the
  total distance traveled by Chuck Wagon.

first 10 minutes
d vit 0.5at2 d (0.50 mi/min)(10.0 min)
0.5(0.0 mi/min2)(10.0 min)2 d 5.0 mi
last 2 minutes
Area 5.5 mi
d vit 0.5at2 d (0.50 mi/min)(2.0 min)
0.5(-0.25 m/s2)(2.0 min)2 d 0.5 mi
The total distance 5.5 mi
101

• Vera Side is speeding down the interstate at 45.0
  m/s. Vera looks ahead and observes an accident
  that results in a pileup in the middle of the
  road. By the time Vera slams on the breaks, she
  is 50.0 m from the pileup. She slows down at a
  rate of -10.0 m/s2.
• Construct a velocity-time plot for Vera Side's
  motion. Use the plot to determine the distance
  that Vera would travel prior to reaching a
  complete stop (if she did not collide with the
  pileup).
• Use kinematics equations to determine the
  distance that Vera Side would travel prior to
  reaching a complete stop (if she did not collide
  with the pileup). Will Vera hit the cars in the
  pileup? That is, will Vera travel more than 50.0
  meters?

vf2 vi2 2ad (0 m/s)2 (45.0 m/s)2 2
(-10.0 m/s2)d 101 m d
Area 101 m
102

• Earl E. Bird travels 30.0 m/s for 10.0 seconds.
  He then accelerates at 3.00 m/s2 for 5.00
  seconds.
• Construct a velocity-time graph for Earl E.
  Bird's motion. Use the plot to determine the
  total distance traveled.
• Divide the motion of the Earl E. Bird into the
  two time segments and use kinematics equations to
  calculate the total displacement.

first 10 seconds d vit 0.5at2 d (30.0
m/s)(10.0 s) 0.5(0.0 m/s2)(10.0 s)2 d 300 m
last 5 seconds d vit 0.5at2 d (30.0
m/s)(5.0 s) 0.5(3.0 m/s2)(5.0 s)2 d 150 m
37.5 m d 187.5 m
Area 488 m
The total distance 488 m
103
vocabulary

• Vector
• Scalar
• Distance
• Displacement
• Speed
• Average speed
• Instantaneous speed
• Velocity
• Average velocity
• Instantaneous velocity
• Constant speed
• Acceleration
• Average acceleration
• Constant acceleration

• Slope
• Meaning of slope in p-t graph
• Meaning of slope in v-t graph
• Meaning of area in v-t graph
• Free fall
• Acceleration of gravity
• Factors affect acceleration of gravity
• Kinematics equations