Chapter 2
Motion in One Dimension
Dynamics
? The branch of physics involving the
motion of an object and the
relationship between that motion
and other physics concepts
? Kinematics is a part of dynamics
? In kinematics,you are interested in
the description of motion
? Not concerned with the cause of the
motion
Quantities in Motion
? Any motion involves three
concepts
? Displacement
? Velocity
? Acceleration
? These concepts can be used to
study objects in motion
Brief History of Motion
? Sumaria and Egypt
? Mainly motion of heavenly bodies
? Greeks
? Also to understand the motion of
heavenly bodies
? Systematic and detailed studies
? Geocentric model
,Modern” Ideas of Motion
? Copernicus
? Developed the heliocentric system
? Galileo
? Made astronomical observations with
a telescope
? Experimental evidence for description
of motion
? Quantitative study of motion
2.1 displacement
Position
? Defined in terms
of a frame of
reference
? One dimensional,
so generally the
x- or y-axis
? Defines a starting
point for the
motion
Displacement
? Defined as the change in position
?
? f stands for final and i stands for initial
? May be represented as ?y if vertical
? Units are meters (m) in SI,
centimeters (cm) in cgs or feet (ft) in
US Customary
fix x x? ? ?
Displacements
Vector and Scalar
Quantities
? Vector quantities need both
magnitude (size) and direction to
completely describe them
? Generally denoted by boldfaced type
and an arrow over the letter
? + or – sign is sufficient for this
chapter
? Scalar quantities are completely
described by magnitude only
Displacement Isn’t
Distance
? The displacement of an object is
not the same as the distance it
travels
? Example,Throw a ball straight up
and then catch it at the same point
you released it
? The distance is twice the height
? The displacement is zero
2.2 velocity
Speed
? The average speed of an object is
defined as the total distance traveled
divided by the total time elapsed
? Speed is a scalar quantity
?
?
t ota l di s t a n c e
Av er a ge s pee d
t ota l t i m e
d
v
t
Speed,cont
? Average speed totally ignores any
variations in the object’s actual
motion during the trip
? The total distance and the total
time are all that is important
? SI units are m/s
Velocity
? It takes time for an object to
undergo a displacement
? The average velocity is rate at
which the displacement occurs
? generally use a time interval,so
let ti = 0
????
??
fi
a v e r a g e
fi
xxxv
t t t
Velocity continued
? Direction will be the same as the
direction of the displacement (time
interval is always positive)
? + or - is sufficient
? Units of velocity are m/s (SI),
cm/s (cgs) or ft/s (US Cust.)
? Other units may be given in a
problem,but generally will need to be
converted to these
Speed vs,Velocity
? Cars on both paths have the same average
velocity since they had the same displacement
in the same time interval
? The car on the blue path will have a greater
average speed since the distance it traveled is
larger
Graphical Interpretation of
Velocity
? Velocity can be determined from a
position-time graph
? Average velocity equals the slope
of the line joining the initial and
final positions
? An object moving with a constant
velocity will have a graph that is a
straight line
Average Velocity,Constant
? The straight line
indicates constant
velocity
? The slope of the
line is the value
of the average
velocity
Average Velocity,Non
Constant
? The motion is
non-constant
velocity
? The average
velocity is the
slope of the blue
line joining two
points
Instantaneous Velocity
? The limit of the average velocity as the
time interval becomes infinitesimally
short,or as the time interval
approaches zero
? The instantaneous velocity indicates
what is happening at every point of
time
??
??
?
lim
0t
xv
t
Instantaneous Velocity on
a Graph
? The slope of the line tangent to the
position-vs.-time graph is defined
to be the instantaneous velocity at
that time
? The instantaneous speed is defined as
the magnitude of the instantaneous
velocity
Uniform Velocity
? Uniform velocity is constant
velocity
? The instantaneous velocities are
always the same
? All the instantaneous velocities will
also equal the average velocity
2.3 Acceleration
? Changing velocity (non-uniform)
means an acceleration is present
? Acceleration is the rate of change
of the velocity
? Units are m/s2 (SI),cm/s2 (cgs),
and ft/s2 (US Cust)
????
??
fi
fi
vvva
t t t
Average Acceleration
? Vector quantity
? When the sign of the velocity and
the acceleration are the same
(either positive or negative),then
the speed is increasing
? When the sign of the velocity and
the acceleration are in the opposite
directions,the speed is decreasing
Instantaneous and
Uniform Acceleration
? The limit of the average
acceleration as the time interval
goes to zero
? When the instantaneous
accelerations are always the same,
the acceleration will be uniform
? The instantaneous accelerations will
all be equal to the average
acceleration
Graphical Interpretation of
Acceleration
? Average acceleration is the slope
of the line connecting the initial
and final velocities on a velocity-
time graph
? Instantaneous acceleration is the
slope of the tangent to the curve
of the velocity-time graph
Average Acceleration
2.4 motion diagrams
? Uniform velocity (shown by red arrows
maintaining the same size)
? Acceleration equals zero
Relationship Between Acceleration and Velocity
Relationship Between
Velocity and Acceleration
? Velocity and acceleration are in the same
direction
? Acceleration is uniform (blue arrows maintain
the same length)
? Velocity is increasing (red arrows are getting
longer)
? Positive velocity and positive acceleration
Relationship Between
Velocity and Acceleration
? Acceleration and velocity are in opposite
directions
? Acceleration is uniform (blue arrows maintain
the same length)
? Velocity is decreasing (red arrows are getting
shorter)
? Velocity is positive and acceleration is
negative
2.5 one-dimensional motion
with constant acceleration
Kinematic Equations
? Used in situations with uniform
acceleration
? ?
??
? ? ? ?
? ? ?
? ? ?
2
22
1
2
1
2
2
o
o
o
o
v v a t
x v t v v t
x v t a t
v v a x
Notes on the equations
t
2
vvtvx fo
a v e r a g e ??
??
?
? ????
? Gives displacement as a function
of velocity and time
? Use when you don’t know and
aren’t asked for the acceleration
Notes on the equations
? Shows velocity as a function of
acceleration and time
? Use when you don’t know and
aren’t asked to find the
displacement
??ov v a t
Graphical Interpretation of
the Equation
Notes on the equations
? Gives displacement as a function
of time,velocity and acceleration
? Use when you don’t know and
aren’t asked to find the final
velocity
2
o at2
1tvx ???
Notes on the equations
? Gives velocity as a function of
acceleration and displacement
? Use when you don’t know and
aren’t asked for the time
? ? ?22 2ov v a x
Problem-Solving Hints
? Read the problem
? Draw a diagram
? Choose a coordinate system,label initial
and final points,indicate a positive direction
for velocities and accelerations
? Label all quantities,be sure all the units
are consistent
? Convert if necessary
? Choose the appropriate kinematic
equation
Problem-Solving Hints,
cont
? Solve for the unknowns
? You may have to solve two equations
for two unknowns
? Check your results
? Estimate and compare
? Check units
Galileo Galilei
? 1564 - 1642
? Galileo formulated
the laws that govern
the motion of objects
in free fall
? Also looked at,
? Inclined planes
? Relative motion
? Thermometers
? Pendulum
2.6 freely falling objects
Free Fall
? All objects moving under the influence
of gravity only are said to be in free fall
? Free fall does not depend on the object’s
original motion
? All objects falling near the earth’s
surface fall with a constant acceleration
? The acceleration is called the
acceleration due to gravity,and
indicated by g
Acceleration due to
Gravity
? Symbolized by g
? g = 9.80 m/s2
? When estimating,use g ??10 m/s2
? g is always directed downward
? toward the center of the earth
? Ignoring air resistance and assuming g
doesn’t vary with altitude over short
vertical distances,free fall is constantly
accelerated motion
Free Fall – an object
dropped
? Initial velocity is
zero
? Let up be positive
? Use the kinematic
equations
? Generally use y
instead of x since
vertical
? Acceleration is g
= -9.80 m/s2
vo= 0
a = g
Free Fall – an object
thrown downward
? a = g = -9.80
m/s2
? Initial velocity ? 0
? With upward
being positive,
initial velocity will
be negative
Free Fall -- object thrown
upward
? Initial velocity is
upward,so positive
? The instantaneous
velocity at the
maximum height is
zero
? a = g = -9.80 m/s2
everywhere in the
motion
v = 0
Thrown upward,cont,
? The motion may be symmetrical
? Then tup = tdown
? Then v = -vo
? The motion may not be
symmetrical
? Break the motion into various parts
? Generally up and down
Non-symmetrical
Free Fall
? Need to divide the
motion into
segments
? Possibilities include
? Upward and
downward portions
? The symmetrical
portion back to the
release point and then
the non-symmetrical
portion
Combination Motions
Motion in One Dimension
Dynamics
? The branch of physics involving the
motion of an object and the
relationship between that motion
and other physics concepts
? Kinematics is a part of dynamics
? In kinematics,you are interested in
the description of motion
? Not concerned with the cause of the
motion
Quantities in Motion
? Any motion involves three
concepts
? Displacement
? Velocity
? Acceleration
? These concepts can be used to
study objects in motion
Brief History of Motion
? Sumaria and Egypt
? Mainly motion of heavenly bodies
? Greeks
? Also to understand the motion of
heavenly bodies
? Systematic and detailed studies
? Geocentric model
,Modern” Ideas of Motion
? Copernicus
? Developed the heliocentric system
? Galileo
? Made astronomical observations with
a telescope
? Experimental evidence for description
of motion
? Quantitative study of motion
2.1 displacement
Position
? Defined in terms
of a frame of
reference
? One dimensional,
so generally the
x- or y-axis
? Defines a starting
point for the
motion
Displacement
? Defined as the change in position
?
? f stands for final and i stands for initial
? May be represented as ?y if vertical
? Units are meters (m) in SI,
centimeters (cm) in cgs or feet (ft) in
US Customary
fix x x? ? ?
Displacements
Vector and Scalar
Quantities
? Vector quantities need both
magnitude (size) and direction to
completely describe them
? Generally denoted by boldfaced type
and an arrow over the letter
? + or – sign is sufficient for this
chapter
? Scalar quantities are completely
described by magnitude only
Displacement Isn’t
Distance
? The displacement of an object is
not the same as the distance it
travels
? Example,Throw a ball straight up
and then catch it at the same point
you released it
? The distance is twice the height
? The displacement is zero
2.2 velocity
Speed
? The average speed of an object is
defined as the total distance traveled
divided by the total time elapsed
? Speed is a scalar quantity
?
?
t ota l di s t a n c e
Av er a ge s pee d
t ota l t i m e
d
v
t
Speed,cont
? Average speed totally ignores any
variations in the object’s actual
motion during the trip
? The total distance and the total
time are all that is important
? SI units are m/s
Velocity
? It takes time for an object to
undergo a displacement
? The average velocity is rate at
which the displacement occurs
? generally use a time interval,so
let ti = 0
????
??
fi
a v e r a g e
fi
xxxv
t t t
Velocity continued
? Direction will be the same as the
direction of the displacement (time
interval is always positive)
? + or - is sufficient
? Units of velocity are m/s (SI),
cm/s (cgs) or ft/s (US Cust.)
? Other units may be given in a
problem,but generally will need to be
converted to these
Speed vs,Velocity
? Cars on both paths have the same average
velocity since they had the same displacement
in the same time interval
? The car on the blue path will have a greater
average speed since the distance it traveled is
larger
Graphical Interpretation of
Velocity
? Velocity can be determined from a
position-time graph
? Average velocity equals the slope
of the line joining the initial and
final positions
? An object moving with a constant
velocity will have a graph that is a
straight line
Average Velocity,Constant
? The straight line
indicates constant
velocity
? The slope of the
line is the value
of the average
velocity
Average Velocity,Non
Constant
? The motion is
non-constant
velocity
? The average
velocity is the
slope of the blue
line joining two
points
Instantaneous Velocity
? The limit of the average velocity as the
time interval becomes infinitesimally
short,or as the time interval
approaches zero
? The instantaneous velocity indicates
what is happening at every point of
time
??
??
?
lim
0t
xv
t
Instantaneous Velocity on
a Graph
? The slope of the line tangent to the
position-vs.-time graph is defined
to be the instantaneous velocity at
that time
? The instantaneous speed is defined as
the magnitude of the instantaneous
velocity
Uniform Velocity
? Uniform velocity is constant
velocity
? The instantaneous velocities are
always the same
? All the instantaneous velocities will
also equal the average velocity
2.3 Acceleration
? Changing velocity (non-uniform)
means an acceleration is present
? Acceleration is the rate of change
of the velocity
? Units are m/s2 (SI),cm/s2 (cgs),
and ft/s2 (US Cust)
????
??
fi
fi
vvva
t t t
Average Acceleration
? Vector quantity
? When the sign of the velocity and
the acceleration are the same
(either positive or negative),then
the speed is increasing
? When the sign of the velocity and
the acceleration are in the opposite
directions,the speed is decreasing
Instantaneous and
Uniform Acceleration
? The limit of the average
acceleration as the time interval
goes to zero
? When the instantaneous
accelerations are always the same,
the acceleration will be uniform
? The instantaneous accelerations will
all be equal to the average
acceleration
Graphical Interpretation of
Acceleration
? Average acceleration is the slope
of the line connecting the initial
and final velocities on a velocity-
time graph
? Instantaneous acceleration is the
slope of the tangent to the curve
of the velocity-time graph
Average Acceleration
2.4 motion diagrams
? Uniform velocity (shown by red arrows
maintaining the same size)
? Acceleration equals zero
Relationship Between Acceleration and Velocity
Relationship Between
Velocity and Acceleration
? Velocity and acceleration are in the same
direction
? Acceleration is uniform (blue arrows maintain
the same length)
? Velocity is increasing (red arrows are getting
longer)
? Positive velocity and positive acceleration
Relationship Between
Velocity and Acceleration
? Acceleration and velocity are in opposite
directions
? Acceleration is uniform (blue arrows maintain
the same length)
? Velocity is decreasing (red arrows are getting
shorter)
? Velocity is positive and acceleration is
negative
2.5 one-dimensional motion
with constant acceleration
Kinematic Equations
? Used in situations with uniform
acceleration
? ?
??
? ? ? ?
? ? ?
? ? ?
2
22
1
2
1
2
2
o
o
o
o
v v a t
x v t v v t
x v t a t
v v a x
Notes on the equations
t
2
vvtvx fo
a v e r a g e ??
??
?
? ????
? Gives displacement as a function
of velocity and time
? Use when you don’t know and
aren’t asked for the acceleration
Notes on the equations
? Shows velocity as a function of
acceleration and time
? Use when you don’t know and
aren’t asked to find the
displacement
??ov v a t
Graphical Interpretation of
the Equation
Notes on the equations
? Gives displacement as a function
of time,velocity and acceleration
? Use when you don’t know and
aren’t asked to find the final
velocity
2
o at2
1tvx ???
Notes on the equations
? Gives velocity as a function of
acceleration and displacement
? Use when you don’t know and
aren’t asked for the time
? ? ?22 2ov v a x
Problem-Solving Hints
? Read the problem
? Draw a diagram
? Choose a coordinate system,label initial
and final points,indicate a positive direction
for velocities and accelerations
? Label all quantities,be sure all the units
are consistent
? Convert if necessary
? Choose the appropriate kinematic
equation
Problem-Solving Hints,
cont
? Solve for the unknowns
? You may have to solve two equations
for two unknowns
? Check your results
? Estimate and compare
? Check units
Galileo Galilei
? 1564 - 1642
? Galileo formulated
the laws that govern
the motion of objects
in free fall
? Also looked at,
? Inclined planes
? Relative motion
? Thermometers
? Pendulum
2.6 freely falling objects
Free Fall
? All objects moving under the influence
of gravity only are said to be in free fall
? Free fall does not depend on the object’s
original motion
? All objects falling near the earth’s
surface fall with a constant acceleration
? The acceleration is called the
acceleration due to gravity,and
indicated by g
Acceleration due to
Gravity
? Symbolized by g
? g = 9.80 m/s2
? When estimating,use g ??10 m/s2
? g is always directed downward
? toward the center of the earth
? Ignoring air resistance and assuming g
doesn’t vary with altitude over short
vertical distances,free fall is constantly
accelerated motion
Free Fall – an object
dropped
? Initial velocity is
zero
? Let up be positive
? Use the kinematic
equations
? Generally use y
instead of x since
vertical
? Acceleration is g
= -9.80 m/s2
vo= 0
a = g
Free Fall – an object
thrown downward
? a = g = -9.80
m/s2
? Initial velocity ? 0
? With upward
being positive,
initial velocity will
be negative
Free Fall -- object thrown
upward
? Initial velocity is
upward,so positive
? The instantaneous
velocity at the
maximum height is
zero
? a = g = -9.80 m/s2
everywhere in the
motion
v = 0
Thrown upward,cont,
? The motion may be symmetrical
? Then tup = tdown
? Then v = -vo
? The motion may not be
symmetrical
? Break the motion into various parts
? Generally up and down
Non-symmetrical
Free Fall
? Need to divide the
motion into
segments
? Possibilities include
? Upward and
downward portions
? The symmetrical
portion back to the
release point and then
the non-symmetrical
portion
Combination Motions
