Motion in one dimension
2.1
Displacement and velocity
2-1 Objectives
1. Describe motion in terms of displacement, time, and
velocity.
2. Calculate the displacement of an object traveling at a known
velocity for a specific time interval.
3. Construct and interpret graphs of position versus time.
Do Now - Notes
Key terms and
ideas
Frame of reference
Displacement
distance
Vector and Scalar
Average velocity
Velocity (instantaneous
velocity)
Velocity in a d. vs. t graph
notes
Question
• How far do you travel to get to school in the
morning?
• How do you compare this distance to the
approximate straight-line distance between
their home and school?
Distance vs. Displacement
xi
xf
• There could be many distances between xf and xi many be
many, distance depends on the path.
• There is only one displacement between xf and xi.
displacement refers to shortest distance between the xf and xi
and direction from xi to xf
Displacement = change in position = final position – initial position
∆x
=
xf
-
∆ denotes change
xi
Scalar vs. Vector
• SCALAR
– A measured quantity that has NO DIRECTION
– Examples
• Distance, Time, Mass, Volume
• VECTOR
– A measured quantity that includes DIRECTION
– SIGN SHOWS DIRECTION
– Example
• Displacement
Example
• A man drives his car 3 miles north, then 4
4 mi
miles east.
East
3 mi
North
Distance
7 mi
Displacement
5 mi
Somewhat
Northeast
What distance did he travel?
What is his displacement from his point of origin?
Example
• Three men leave the same house on foot. The first man walks
30 feet north, then 40 feet west. The second man walks 90 feet
south, then 88 feet north. The third man walks 10 feet east,
then 50 feet west.
• Which man has traveled the greatest distance?
The second man
• Who is farthest from the house?
The first man
• Who is closest to the house?
The second man
The frame of reference
• http://www.physics-chemistry-interactive-flashanimation.com/mechanics_forces_gravitation_energy_in
teractive/frame_of_reference_motion_child_ball_train.h
tm
• The choice of a reference point for the coordinate
system is arbitrary, but once chose, the same point must
be used throughout the problem.
• Text book, p41, figure 2-2, what would the displacement
of the gecko be if the zero end of the meter stick had
been lined up with the gecko’s first position?
Positive and negative displacement
How do you describe the speed of the car?
The speed changes
Average speed
Instantaneous speed
Average Velocity vs. Average Speed
• AVERAGE VELOCITY
– change in DISPLACEMENT occurring over time
– Includes both MAGNITUDE and DIRECTION
• VECTOR
• The direction of the velocity vector is simply the same as
the direction that an object is moving.
• AVERAGE SPEED
– change in DISTANCE occurring over time
– Includes ONLY MAGNITUDE
• SCALAR
Calculate Average Speed and
Average Velocity
• The average speed during the course of a motion is often
computed using the following formula:
Does NOT include DIRECTION!
• In contrast, the average velocity is often computed using this
formula
v  vavg
x x f  xi


t t f  ti
The language of physics: ∆ means change
Example
• Sally gets up one morning and decides to
take a three mile walk. She completes the
first mile in 8.3 minutes, the second mile
in 8.9 minutes, and the third mile in 9.2
minutes.
– What is her average speed during her walk?
vavg = d / t
vavg = 3 mi / (8.3 min + 8.9 min + 9.2 min)
vavg = 0.11 mi / min
Example
• Tom gets on his bike at 12:00 pm and
begins riding west. At 12:30 pm he has
ridden 8 miles.
– What was his average velocity during his ride?
vavg = d / t
vavg = 8 mi / 30 min
vavg = 0.27 mi / min WEST
Example
• During a race on level ground, Andra runs with an
average velocity of 6.02 m/s to the east. What
displacement does Andre cover in 137 s?
(∆t ) vavg =
∆x
∆t
(∆t )
∆x = vavg (∆t ) = (6.02 m/s)(137 s) = 825 m
Answer: 825 m East
Class work
Practice p. 44 #1-6
Interpret velocity in p-t graph
v  vavg
x x f  xi


t t f  ti
What does this remind you of?
Position
Position
Position vs. Time
SLOPE OF A GRAPH!
INCREASING
SLOPE
CONSTANT
CONSTANT
POSITIVE
ZERO
SLOPESLOPE
Time
What is happening in this
graph?
Moving with
Motionless
INCREASING
CONSTANT
Object
positive
velocity
velocity
Graph interpretation of velocity
Alert:
Only use
points on the
line to
calculate
slope.
Average velocity
during 0-55 s
Average velocity
during 11-33 s
Distance vs. Time Graphs
During what time interval was the object NOT MOVING?
Distance vs. Time
18
Distance (m)
2 – 3 seconds
16interval on the graph where
The
the
14distance remains constant!
12
10
8
6
4
2
0
0
1
2
3
4
Time (s)
5
6
7
Displacement vs. Time Graphs
During
During
At
what
what
what
time
distance
time
interval(s)
interval(s)
fromwas
thewas
the
origin
object
thedoes
object
tothe
the
NOT
object
left MOVING?
of stop?
the origin?
Displacement vs. Time
The object’s final position is
at +1 meter (1 meter to the
When
the of
displacement
right
the origin) is
1 – 2 and 4 – negative,
5 secondsthe object has a
positionmeans
to the that
left of the origin
Constant displacement
the object doesn’t move
6
5
Displacement (m)
4
3
2
1
0
-1 0
1
2
3
4
-2
-3
-4
Time (s)
5
6
7
As the slope goes, so goes the velocity
Slow, Positive, Constant Velocity
Fast, Positive, Constant Velocity
Fast, Negative,
Constant Velocity
Slow, Negative
Constant Velocity
example
Determine average velocity
1.
during 0-5 seconds
2.
During 5-10 seconds
The velocity is 5 m/s between 0-5 seconds
The velocity is zero between 5-10 seconds
Displacement-time
graph
Distance-time
graph
• NEVER decreases
• Read graph to find current
total distance
• Subtract points to find
distance traveled between
them
• Average speed = slope or
Δd/Δt
• curve = changing speed
(acceleration or deceleration)
– increasing slope =
increasing speed
– decreasing slope =
decreasing speed



•
Above x-axis = positive displacement from
origin (east, right, up); Below x-axis = negative
displacement from origin (west, left, down)
Read graph to find current position.
Difference between points = displacement
traveled (change in position); Accumulate to
get total distance
velocity = slope or Δd/Δt
positive slope = headed in positive direction
from origin (east, right, up)
negative slope = headed in negative direction
from origin (west, left, down)
curve = changing speed (acceleration or
deceleration)
increasing slope = increasing speed
decreasing slope = decreasing speed
Distance vs. time graph
Displacement vs. time graph
Speed (Instantaneous Speed) and
velocity (Instaneous Velocity)
• Speed, often means instantaneous speed - the speed at any given instant
in time. It is often ref
• Velocity (Instantaneous velocity) - the speed at any given instant in time
with direction at that instant
• The magnitude of Instantaneous velocity is always equal to the
instantaneous speed
• The magnitude of average velocity can be less than the average speed.
Instantaneous velocity
• Text book p. 46 – figure 2-7
• The instantaneous velocity at a given time can
be determined by measuring the slope of the
line that is tangent to that point on the
position versus time graph.
Average speed vs. instantaneous
speed on p-t graph
Position
Position vs. Time
The slope of the secant
line is between points A
and B is the Average
velocity between A & B
B
The slope of the
tangent line is at point
A is the instantaneous
velocity at point A
A
Time
CLASS WORK
• Section Review Worksheet 2-1,
“Displacement and Velocity.” Graph Skills