Chapter 2: Motion along a
Straight Line
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Displacement, Time, Velocity
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One-Dimensional Motion
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The area of physics that we focus on is called mechanics: the
study of the relationships between force, matter and motion
For now we focus on kinematics: the language used to
describe motion
Later we will study dynamics: the relationship between
motion and its causes (forces)
Simplest kind of motion: 1-D motion (along a straight line)
A particle is a model of moving body in absence of effects
such as change of shape and rotation
Velocity and acceleration are physical quantities to describe
the motion of particle
Velocity and acceleration are vectors
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Position and Displacement
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Motion is purely translational, when there is no rotation
involved. Any object that is undergoing purely translational
motion can be described as a point particle (an object with
no size).
The position of a particle is a vector that points from the
origin of a coordinate system to the location of the particle
The displacement of a particle over a given time interval is a
vector that points from its initial position to its final position. It
is the change in position of the particle.
To study the motion, we need coordinate system
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Position and Displacement
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Motion of the “particle” on the dragster can be described in
terms of the change in particle’s position over time interval
Displacement of particle is a vector pointing from P1 to P2
along the x-axis
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Average Velocity
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Average velocity during this time interval is a vector
quantity whose x-component is the change in x divided by
the time interval
x  x2  x1
t  t2  t1
vav x
x2  x1 x


t 2  t1
t
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Average Velocity
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Average velocity is positive when during the time interval
coordinate x increased and particle moved in the positive
direction
If particle moves in negative x-direction during time
interval, average velocity is negative
x  x2  x1  19m  277m  258m
t  t2  t1  25.0s 16.0s  9.0s
vav x  x   258m
 29m / s
t
9.0s
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X-t Graph
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This graph is pictorial way
to represent how particle
position changes in time
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Average velocity depends
only on total displacement
x, not on the details of
what happens during time
interval t
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The average speed of a
particle is scalar quantity
that is equal to the total
distance traveled divided
by the total time elapsed.
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Average Velocity
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Instantaneous Velocity
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Instantaneous velocity of a particle is a vector equal to
the limit of the average velocity as the time interval
approaches zero. It equals the instantaneous rate of
change of position with respect to time.
x dx
v x  lim

t  0 t
dt
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Instantaneous Velocity
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On a graph of position as a function of time for onedimensional motion, the instantaneous velocity at a
point is equal to the slope of the tangent to the curve
at that point.
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Instantaneous Velocity
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Instantaneous Velocity
Concept Question
The graph shows position
versus time for a particle
undergoing 1-D motion.
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At which point(s) is the
velocity vx positive?
At which point(s) is the
velocity negative?
At which point(s) is the
velocity zero?
At which point is speed
the greatest?
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Acceleration
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Acceleration
If the velocity of an object is changing with time, then the
object is undergoing an acceleration.
 Acceleration is a measure of the rate of change of velocity
with respect to time.
 Acceleration is a vector quantity.
 In straight-line motion its only non-zero component is along
the axis along which the motion takes place.
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Average Acceleration
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Average Acceleration over a given time interval is defined
as the change in velocity divided by the change in time.
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In SI units acceleration has units of m/s2.
aav x
v2 x  v1x vx


t 2  t1
t
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Instantaneous Acceleration
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Instantaneous acceleration of an object is obtained by
letting the time interval in the above definition of average
acceleration become very small. Specifically, the
instantaneous acceleration is the limit of the average
acceleration as the time interval approaches zero:
v x dv x
a x  lim

t 0 t
dt
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Acceleration of Graphs
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Acceleration of Graphs
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Acceleration of Graphs
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Constant Acceleration Motion
In the special case of constant acceleration:
 the velocity will be a linear function of time, and
 the position will be a quadratic function of time.
 For this type of motion, the relationships between position,
velocity and acceleration take on the simple forms :
v2 x  v1x
ax 
t 2  t1
vx  v0 x
ax 
t 0
vx  v0 x  ax  t
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Constant Acceleration Motion
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Constant Acceleration Motion
Position of a particle moving with constant acceleration
vav x
x x  x0


t
t 0
vx  v0 x  ax  t
 vav x
x  x0
1
v0 x  a x  t 
2
t
vav x
v0 x  v x

2
1
1
 v0 x  v0 x  a x  t   v0 x  a x  t
2
2
1 2
 x  x0  v0 x t  a x t
2
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Constant Acceleration Motion
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Relationship between position of a particle moving with
constant acceleration, and velocity and acceleration
itself:
vx  v0 x  ax  t 
v x  v0 x
t
ax
 v x  v0 x  1  vx  v0 x 
1 2
x  x0  v0 x t  a x t  x  x0  v0 x 
  a x 

2
 ax  2  ax 
v x2  v02x  2a x ( x  x0 )
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2
Freely Falling Bodies
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Freely Falling Bodies
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Example of motion with
constant acceleration is
acceleration of a body falling
under influence of the earth’s
gravitation
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All bodies at a particular location
fall with the same downward
acceleration, regardless of
their size and weight
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Idealized motion free fall: we
neglect earth rotation, decrease
of acceleration with decreasing
altitude, air effects
Aristotle
384 - 322 B.C.E.
Galileo Galilei
1564 - 1642
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Freely Falling Bodies
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The constant acceleration of a
freely falling body is called
acceleration due to gravity,
g
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Approximate value near
earth’s surface g = 9.8 m/s2 =
980 cm/s2 = 32 ft/s2
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g is the magnitude of a vector,
it is always positive number
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Exact g value varies with
location
Acceleration due to gravity
 Near the sun: 270 m/s2
 Near the moon: 1.6 m/s2
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