Motion In One Dimension
PLATO AND ARISTOTLE
GALILEO GALILEI
LEANING TOWER OF PISA
Graphing Constant Speed
Distance vs. Time for Toy Car (0-5 sec.)
best-fit line (from TI calculator)
Distance (cm)
1000
d  207.7  t  12.6
800
Constant speed is the slope of the
(best fit) line for a distance vs. time graph.
600
speed 
distance
time
s  208
cm
(3 sig figs)
s
400
Remember, the standard metric
unit for length is the meter!
200
0
cm 10 2 m
m
s  208

 2.08
s
1 cm
s
1.0
2.0
3.0
Time (s)
4.0
5.0
Graphing Average and Instantaneous Speed
600
Average speed is the slope of a secant
line for a distance vs. time graph.
Distance vs. Time for Toy Car (0-0.5 sec.)
distance
time
500  25 mm
savg 
0.5  0.1 s
average speed 
500
savg  1190
Distance (mm)
400
mm
s
3

10 m
1 mm
 1.19
m
s
Instantaneous speed is the slope of a
tangent line for a distance vs. time graph.
300
distance
(as t  0)
time
350  0 mm
approximate slope
s
at t  0.25 s
0.5  0.13 s
mm 103 m
m
s  946

 0.946
s
1 mm
s
inst. speed 
200
(0.5, 350)
100
best-fit quadratic (from TI calculator)
d  2054t 2  26.8t  1.8
(0.13, 0)
0
click for applet
0.1
0.2
0.3
Time (s)
0.4
0.5
tanget line slope at t  0.25 (from calc.)
d  1000t 126.6
Distance, Position and Displacement
Distance (d)
The length of a path traveled by an
object. It is never negative, even if
an object reverses its direction.
One dimensional motion
A
-6
-5
B
-4
-3
-2
x(m)
C
-1
0
+1
+2
+3
+4
+5
+6
1. What is the distance traveled if an object
starts at point C, moves to A, then to B?
d = d1 + d2 = 9 + 3 = 12 m
2. What is the displacement of an object that
starts at point C and moves to point B?
xf – xi = -2 – (+4) = -6 m
Position (x or y)
The location of an object
relative to an origin. It can be
either positive or negative
Displacement (∆x or ∆y)
The change in position
of an object. Also can be
positive or negative.
x  x f  xi
y  y f  yi
3. What is the displacement of an object
that starts at point A, then moves to
point C and then moves to point B?
xf – xi = -2 – (-5) = +3 m
Two dimensional motion
4. What is the distance
traveled and the
displacement of the
person that starts at
point A, then moves
to point B, and ends
at point C?
d = 3 + 4 = 7 m; x = √32 + 42 = 5 m
Distance and Position Graphs
Distance vs. Time
x (m)
positive
d (m)
Position vs. Time
CAR B:
constant
positive velocity
negative
t (s)
CAR C:
constant
negative velocity
t (s)
Distance graphs show how far an object travels. Speed is determined
from the slope of the graph, which cannot be negative.
Position graphs show initial position, displacement, velocity (magnitude
and direction). That’s why position graphs are better!
Remember, all of these graphs show constant speed. (How do you know?)
Average Speed vs. Average Velocity
Average speed is the distance traveled divided by time elapsed.
average speed 
distance traveled
time elapsed
savg 
d
t
Average velocity is displacement divided by time elapsed.
displacement 
average velocity 
time interval
vavg 
x
t
Example: A sprinter runs 100 m in 10 s, jogs 50 m further in 10 s, and then walks back to the finish
line in 20 seconds. What is the sprinter’s average speed and average velocity for the entire time?
x 100 m
m
vavg 

 2.5
t
40 s
s
200
150
150
x (m)
d 200 m
m

5
t
40 s
s
d (m)
savg 
200
100
slope = ave. speed
50
0
10
20
30
40
t (s)
100
50
0
slope = ave. velocity
10
20
30
t (s)
40
Instantaneous Speed and Velocity
Instantaneous speed is the how fast an object moves at an exact
moment in time. Instantaneous velocity has speed and direction.
instananeous speed 
distance
as t approaches zero
time
instananeous velocity 
Honors:
displacement
as t approaches zero
time
s  lim
t0
v  lim
t0
d
t
x dx

t
dt
d (m)
x (m)
Instantaneous speed (or velocity) is found graphically from the slope
of a tangent line at any point on a distance (or position) vs. time graph.
slope of tangent =
instantaneous speed
t (s)
slope of tangent =
instantaneous velocity
sign of slope =
sign of velocity
t (s)
The Physics of Acceleration
“Acceleration is how
quickly how fast changes”
“how fast” means velocity
“how fast changes” means change in velocity
“how quickly” mean how much time elapses
Acceleration is defined as the rate at which an object’s velocity changes.
change in velocity
acceleration 
time
aavg
v

t
Acceleration has units of meters per second per second,
or m/s/s, or m/s2.
Acceleration is considered as a rate of a rate. Why?
Metric (SI) units
m
s or m
s
s2
Types of Acceleration
v (m/s)
Velocity vs. Time
Constant Acceleration
v (m/s)
t (s)
Constant acceleration is the slope
of a velocity vs. time graph.
(Sound familiar?! Compare to,
but DO NOT confuse with
constant velocity on a position vs.
time graph.)
Velocity vs. Time
Varying Acceleration
t (s)
Average acceleration is the slope of a
secant line for a velocity vs. time graph.
Instantaneous acceleration is the slope of a
tangent line for a velocity vs. time graph.
(Again, compare to, but DO NOT confuse
with average and instantaneous velocity
on a position vs. time graph.)
Velocity and Displacement (Honors)
v (m/s)
Velocity vs. Time
30
20
10
0
2
4
6
8
t (s)
area = displacement
= (.5)(3 s)(30 m/s) + (4 s)(30 m/s)
+ (.5)(1 s)(30 m/s) = 180 m
A velocity graph can be used to
determine the displacement
(change in position) of an object.
The area of the velocity graph
equals the object’s displacement.
For a non-linear velocity graph,
the area can be determined by
adding up infinitely many pieces
each of infinitely small area,
resulting in a finite total area!
This process is now known as
integration, and the function is
called an integral.
An Acceleration Analogy
Compare the graph of wage versus time to a velocity versus time graph.
The slope of the wage graph is “wage change rate”. Slope of the velocity
graph is acceleration. What is the slope for each graph, including units?
In this case the “wage change rate” is constant. The graph is linear because
the rate at which the wage changes is itself unchanging (constant)!
The analogy helps distinguish velocity from acceleration because it is clear
that wage and “wage change rate” (acceleration) are different.
slope = “wage change rate”
= $1//hr/month
slope = acceleration
= 1 m/s/s
An Acceleration Analogy
Earnings, Wage, and “Wage Change Rate”
Can a person have a high wage, but a low
“wage change rate”?
Making good hourly money, but
getting very small raises over time.
Position, Velocity, and Acceleration
Can an object have a high velocity,
but a low acceleration?
Moving fast, but only getting a
little faster over time.
Can a person have a low wage, but a high
“wage change rate”?
Can an object have a low velocity,
but a high acceleration?
Making little per hour, but getting very
large raises quickly over time.
Moving slowly, but getting a lot
faster quickly over time.
Can a person have a positive wage, but a
negative “wage change rate”?
Making money, but getting cuts in
wage over time.
Can a person have zero wage, but still
have “wage change rate”?
Making no money (internship?), but
eventually working for money.
Can an object have a positive velocity,
but a negative acceleration?
Moving forward, but slowing
down over time.
Can an object have zero velocity, but
still have acceleration?
At rest for an instant, but then
immediately beginning to move.
Direction of Velocity and Acceleration
Velocity vs. Time
vi
a
motion
+
0
constant positive vel.
–
0
constant negative vel.
0
+
speeding up from rest
0
–
speeding up from rest
v
v
t
v
v
t
v
+
+
t
t
v
speeding up
t
–
–
speeding up
v
+
–
v
slowing down
t
–
+
t
slowing down
click for applet
t
Kinematic Equations of Motion
Assuming constant acceleration, several equations can be
derived and used to solve motion problems algebraically.
Slope equals acceleration
v v f  vi
a


t
t
Velocity vs. Time
(Constant Acceleration)
v (m/s)
v f  vi  at
Area equals displacement
vf
vi
A
1
2
b1  b2 h

x 
1
2
v  v t
i
f
t
Eliminate final velocity
x  vi t  12 at 2
Eliminate time
v f 2  vi 2  2ax
t (s)
Freefall Acceleration
Aristole wrongly assumed that an object
falls at a rate proportional to its weight.
Galileo assumed all objects freefall (in a
vacuum, no air resistance) at the same rate.
An inclined plane
reduced the effect of
gravity, showing that
the displacement of an
object is proportional
to the square of time.
y : t 2
Since the acceleration
is constant, velocity is
proportional to time.
v: t
click for video
Location
g
Equator
-9.780
Honolulu
-9.789
Denver
-9.796
San Francisco
-9.800
Munich
-9.807
Leningrad
-9.819
North Pole
-9.832
Latitude, altitude, geology affect g.
Kinematic equations
of freefall acceleration:
y  vyit  12 gt 2
vyf  vyi  gt
vyf 2  vyi 2  2gy
y 
1
2
v
yi

 vyf t