Displacement, Velocity, and
Acceleration and Their Graphical
Representations
How can we describe motion?
•
“motion” is the state of an object whose
position
is
changing
•
We quantify motion by describing how
far
an object
moves in a particular amount of
time
•
Distance/Displacement
: measures how far something
moves
•
Speed/Veolcity
: measures how fast something is
moving
•
Acceleration
: measures the rate at which an object’s
velocity is changing
Important Distinctions!
•
Distance
–
The
total length of a
path
followed by a
moving obeject
•
Average Speed
–
The ratio of the
distance travelled to
the time required to do
so
•
Displacement
–
The net
change in
position
of a moving
object
•
Average Velocity
–
The ratio of the
object’s displacement
to the elapsed time
required for that
displacement
Graphical Description of Motion
•
We can understand important aspects of an
object’s motion by studying graphs of its
position, velocity, and acceleration as a function
of time.
•
Position
versus
time
:
x
vs.
t
•
Velocity
versus
time
:
v
vs
t
•
Acceleration
versus
time
:
a
vs
t
Constant position
(not moving!)
Position (m)
Time (s)
In each case, the object’s displacement is
0 since its’ final position is the same as its
initial position!
Constant speed / velocity
Position (m)
Time (s)
v = 1 m/s
Constant velocity (v = 1 m/s)
constant
t
x
v
x (m)
t (s)
2
2
4
4
8
8
slope of (x,t) graph!
Constant velocity (starting at x=0)
x (m)
Position vs time
Velocity vs time
The slope of a (v,t) graph is 0 if the velocity is constant!
v (m/s)
v = 1m/s starting at x = 4 m
x (m)
t (s)
Slope is the same, but
position is shifted by 4 m.
Constant velocity (starting at x=4 m)
x (m)
Position vs time
Velocity vs time
Initial position does not change the slope (velocity is
the same)!
v (m/s)
Constant (uniform) acceleration
v (m)
Uniform acceleration is
defined as:
constant
t
v
a
Velocity changes by a fixed
amount over equal time
intervals
a =1 m/s
2
Acceleration is the slope of a v,t graph!
x (m)
Time (s)
Position graph from uniform
acceleration
v (m)
Velocity is higher over each interval, so distance traveled is greater!
x (m)
Time (s)
Velocity graph from position graph
v (m)
Velocity AT 2 s is the
slope of the tangent line
at 2 s
6 s
8 s
slope = 6
slope = 8
Summary: Graphical description of
motion
•
The slope of an (x,t) graph is the velocity
–
Constant velocity: constant slope
–
Constant acceleration: changing slope
•
The slope of a (v,t) graph is the acceleration
–
Constant acceleration: constant slope
•
If something is constant (position, velocity,
acceleration), the slope of its graph is 0.
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