Motion Along a Line

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Chapter 2

• Motion Along a Line

2
Motion Along a Line

• Position Displacement
• Speed Velocity
• Acceleration
• Describing motion in 1D
• Free Fall

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Introduction

• Kinematics - Concepts needed to describe motion -
  displacement, velocity acceleration.
• Dynamics - Deals with the effect of forces on
  motion.
• Mechanics - Kinematics Dynamics

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Goals of Chapter 2
Develop an understanding of kinematics that
comprehends the interrelationships among

• physical intuition
• equations
• graphical representations

When we finish this chapter you should be able to
move easily among these different aspects.
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Kinematic Quantities Overview
The words speed and velocity are used
interchangably in everyday conversation but they
have distinct meanings in the physics world.
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Position Displacement
The position (x) of an object describes its
location relative to some origin or other
reference point.
The position of the red ball differs in the two
shown coordinate systems.
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The position of the ball is
The indicates the direction to the right of the
origin.
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x (cm)
1
0
2
?2
?1
The position of the ball is
The ? indicates the direction to the left of the
origin.
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The displacement is the change in an objects
position. It depends only on the beginning and
ending positions.
All ? quantities will have the final value 1st
and the inital value last.
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Example A ball is initially at x 2 cm and is
moved to x -2 cm. What is the displacement of
the ball?
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Example At 3 PM a car is located 20 km south of
its starting point. One hour later its is 96 km
farther south. After two more hours it is 12 km
south of the original starting point.
(a) What is the displacement of the car between 3
PM and 6 PM?
xi 20 km and xf 12 km
Use a coordinate system where north is positive.
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Example continued
(b) What is the displacement of the car from the
starting point to the location at 4 pm?
xi 0 km and xf 96 km
(c) What is the displacement of the car from 4 PM
to 6 PM?
xi 96 km and xf 12 km
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Velocity Rate of Change of Position
Velocity is a vector that measures how fast and
in what direction something moves.
Speed is the magnitude of the velocity. It is a
scalar.
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In 1-dimension the average velocity is
vav is the constant speed and direction that
results in the same displacement in a given time
interval.
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On a graph of position versus time, the average
velocity is represented by the slope of a chord.
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This is represented by the slope of a line
tangent to the curve on the graph of an objects
position versus time.
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The area under a velocity versus time graph
(between the curve and the time axis) gives the
displacement in a given interval of time.
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Example (text problem 2.11) Speedometer readings
are obtained and graphed as a car comes to a stop
along a straight-line path. How far does the car
move between t 0 and t 16 seconds?
Since there is not a reversal of direction, the
area between the curve and the time axis will
represent the distance traveled.
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Example continued
The rectangular portion has an area of Lw
(20 m/s)(4 s) 80 m. The triangular portion
has an area of ½bh ½(8 s) (20 m/s) 80 m.
Thus, the total area is 160 m. This is the
distance traveled by the car.
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The Most Important Graph- V vs T
The values of the curve gives the instantaneous
VELOCITY.
The slope of the curve gives the ACCELERATION.
Negative areas are possible.
Area under the curve gives DISTANCE.
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Acceleration Rate of Change of Velocity
These have interpretations similar to vav and v.
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Example (text problem 2.29) The graph shows
speedometer readings as a car comes to a stop.
What is the magnitude of the acceleration at t
7.0 s?
The slope of the graph at t 7.0 sec is
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Motion Along a Line With Constant Acceleration
For constant acceleration the kinematic equations
are
Also
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A Modified Set of Equations
For constant acceleration the kinematic equations
are
Also
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Visualizing Motion Along a Line with Constant
Acceleration
Motion diagrams for three carts
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Graphs of x, vx, ax for each of the three carts
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Free Fall
A stone is dropped from the edge of a cliff if
air resistance can be ignored, we say the stone
is in free fall. The magnitude of the
acceleration of the stone is afree fall g
9.80 m/s2, this acceleration is always directed
toward the Earth. The velocity of the stone
changes by 9.8 m/s every sec.
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Free Fall
Assumption acceleration due to gravity is g g
9.8 m/s2 10 m/s2
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Example You throw a ball into the air with speed
15.0 m/s how high does the ball rise?
Given viy 15.0 m/s ay ?9.8 m/s2
To calculate the final height, we need to know
the time of flight.
Time of flight from
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Example continued
The ball rises until vfy 0.
The height
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Example (text problem 2.45) A penny is dropped
from the observation deck of the Empire State
Building 369 m above the ground. With what
velocity does it strike the ground? Ignore air
resistance.
Given viy 0 m/s, ay ?9.8 m/s2, ?y ?369 m
Unknown vfy
ay
Use
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Example continued
(downward)
How long does it take for the penny to strike the
ground?
Given viy 0 m/s, ay ?9.8 m/s2, ?y ?369 m
Unknown ?t
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Summary

• Position
• Displacement Versus Distance
• Velocity Versus Speed
• Acceleration
• Instantaneous Values Versus Average Values
• The Kinematic Equations
• Graphical Representations of Motion
• Free Fall