Representing
Motion
Chapter 2
Pop Quiz!
• How fast are you moving at this moment?
o A.) 0m/s
o B.) 783 mi/h
o C.) 350m/s
o D.) 30 km/s
Pop Quiz!
• How fast are you moving?
oA.) 0m/s
oB.) 783 mi/h
oC.) 350m/s
oD.) 30 km/s
Speed of the earth’s
rotation around its
axis @ 40o N latitude
Speed of the earth’s
revolution around the
sun
Pop Quiz!
• How fast are you moving?
oA.) 0m/s
oB.) 783 mi/h
oC.) 350m/s
oD.) 30 km/s
Speed of the earth’s
rotation around its
axis @ 40o N latitude
Speed of the earth’s
revolution around the
sun
oIn order to describe motion, you
need a frame of reference!
Frame of Reference
• A system of objects that are not moving with
respect to one another
Relative Motion
• Movement in relation to a frame of reference.
• Different frames of reference will give the same
object different motion.
Coordinate Systems
Coordinate Systems
Coordinate Systems
Distance
• The length of a path between two points
• SI Unit – meter
• Only length of the path is important
o Direction doesn’t matter
o Scalar (magnitude only)
Displacement
• The strait line distance and direction between two
points.
• Vector (magnitude and direction)
Time Intervals
• ∆𝑡 = 𝑡𝑓 − 𝑡𝑖
• Where:
• 𝑡𝑖 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑡𝑖𝑚𝑒
• 𝑡𝑓 = 𝑓𝑖𝑛𝑎𝑙 𝑡𝑖𝑚𝑒
• ∆𝑡 = 𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
o ∆= 𝑐ℎ𝑎𝑛𝑔𝑒
Displacement
• ∆𝑑 = 𝑑𝑓 − 𝑑𝑖
• Where:
• 𝑑𝑖 = 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛
• 𝑑𝑓 = 𝑓𝑖𝑛𝑎𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛
• ∆𝑑 = change in position
Displacement
If displacements are in the same direction, add the
vectors.
If displacements are in different directions, subtract the
vectors.
3
+
3
-
2
2
=5
=1
Vector vs. Scalar
Quantities
• Scalar
o Magnitude only
• Vector
o Magnitude and Direction
Scalar Quantities
Vector Quantities
Distance
Velocity
Displacement
Time
Temperature
Force
Vectors and Motion
A quantity that requires both a magnitude (or size) and a direction can be
represented by a vector. Graphically, we represent a vector by an arrow.
The velocity of this car is 100 m/s (magnitude) to the left (direction).
This boy pushes on his friend with a force of 25 N to the right.
Displacement Vectors
A displacement vector starts at an
object’s initial position and ends at its
final position. It doesn’t matter what
the object did in between these two
positions.
In motion diagrams, the
displacement vectors span
successive particle positions.
Slide 1-29
Exercise
Alice is sliding along a smooth, icy road on her sled when she suddenly
runs headfirst into a large, very soft snowbank that gradually brings her to
a halt. Draw a motion diagram for Alice. Show and label all displacement
vectors.
Slide 1-30
Velocity Vectors
Slide 1-34
Making a Motion Diagram
Slide 1-11
Examples of Motion Diagrams
Slide 1-12
The Particle Model
A simplifying model in which we treat the object as if all its mass were
concentrated at a single point. This model helps us concentrate on the overall
motion of the object.
Slide 1-13
Checking Understanding
Two runners jog along a track. The positions are shown at 1 s time
intervals. Which runner is moving faster?
Slide 1-18
Checking Understanding
Two runners jog along a track. The times at each position are shown. Which
runner is moving faster?
They are both moving at the same speed.
Slide 1-20
Speed
(m)
The car moves 40 m in 1 s. Its speed is
The bike moves 20 m in 1 s. Its speed is
=
40 𝑚
.
1𝑠
=
20 𝑚
1𝑠
.
Slide 1-22
Velocity
Slide 1-23
Speed and Velocity
• Speed vs. Velocity
o Speed – Scalar
o Velocity - Vector
• 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑝𝑒𝑒𝑑 =
𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒
• 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦: 𝑣 =
∆𝑑
∆𝑡
o SI Unit: meter per second (m/s)
=
𝑑𝑓 −𝑑𝑖
𝑡𝑓 −𝑡𝑖
Position vs. Time Graph
•
Give a qualitative description of the motion of the object depicted in
the graph.
Position vs. Time Graph
•
The object remained stationary for 1s. It then moved 40m in the
positive direction for 2s. It remained stationary for 1.5s and moved
40m in the negative direction for 1.5s.
Position vs. Time Graph
•
•
What is the total distance the object traveled?
What is the object’s displacement?
Position vs. Time Graph
•
What is the object’s average velocity between 1s and 3s?
•
𝑣=
∆𝑑
∆𝑡
=
𝑑𝑓 −𝑑𝑖
𝑡𝑓 −𝑡𝑖
=
60𝑚−20𝑚
3𝑠−1𝑠
=
40𝑚
2𝑠
= 20𝑚/𝑠
Position vs. Time Graph
•
What is the object’s average velocity between 1s and 6s?
•
𝑣=
•
What is the object’s average speed between 1s and 6s?
•
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑝𝑒𝑒𝑑 =
∆𝑑
∆𝑡
=
𝑑𝑓 −𝑑𝑖
𝑡𝑓 −𝑡𝑖
=
20𝑚−20𝑚
6𝑠−1𝑠
=
0𝑚
5𝑠
𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒
= 0𝑚/𝑠
=
40𝑚+40𝑚
5𝑠
=
80𝑚
5𝑠
= 16𝑚/𝑠
Position vs. Time Graph
•
The average velocity of an object during the time interval ∆𝑡 is equal
to the slope of the straight line joining the initial and final points on a
graph of the object’s position versus time.
Position vs. Time Graphs – The meaning of Shape
Constant Velocity
Positive Velocity
Constant Velocity
Slow, Rightward (+)
Constant Velocity
Slow, Leftward (+)
Negative (-) Velocity
Slow to Fast
Changing Velocity
Positive Velocity
Constant Velocity
Fast, Rightward (+)
Constant Velocity
Fast, Leftward (+)
Leftward (-) Velocity
Fast to Slow
Check Your Understanding
• Use the principle of slope to describe the motion of the
objects depicted by the two plots below. In your
description, be sure to include such information as the
direction of the velocity vector (i.e., positive or negative),
whether there is a constant velocity or an acceleration,
and whether the object is moving slow, fast, from slow to
fast or from fast to slow. Be complete in your description.
Position vs. Time Graphs – The meaning of Slope
• The slope of the line on a position versus
time graph is equal to the velocity of
the object.
• 𝑆𝑙𝑜𝑝𝑒 =
∆𝑦
∆𝑥
=
𝑦2 −𝑦1
𝑥2 −𝑥1
=
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
• To determine the slope:
 Pick two points on the line and determine their
coordinates.
 Determine the difference in y-coordinates of these two
points (rise).
 Determine the difference in x-coordinates for these two
points (run).
 Divide the difference in y-coordinates by the difference
in x-coordinates (rise/run or slope).
• Check Your Understanding: Determine the
velocity (i.e., slope) of the object as portrayed by the graph
below.
Tangent
Lines
x
t
On a position vs. time graph:
SLOPE
VELOCITY
SLOPE
SPEED
Positive
Positive
Steep
Fast
Negative
Negative
Gentle
Slow
Zero
Zero
Flat
Zero
Determining the Area on a v-t Graph
•
•
For velocity vs. time graphs, the area bounded by the line and the axes
represents the distance traveled.
The diagram shows three different velocity-time graphs; the shaded regions
between the line and the axes represent the distance traveled during the
stated time interval.
The shaded area is representative of the distance traveled by the
object during the time interval from 0 seconds to 6 seconds. This
representation of the distance traveled takes on the shape of a
rectangle whose area can be calculated using the appropriate
equation.
The shaded area is representative of the distance traveled by the
object during the time interval from 0 seconds to 4 seconds. This
representation of the distance traveled takes on the shape of a
triangle whose area can be calculated using the appropriate
equation.
The shaded area is representative of the distance traveled by the
object during the time interval from 2 seconds to 5 seconds. This
representation of the distance traveled takes on the shape of a
trapezoid whose area can be calculated using the appropriate
equation.
•
The method used to find the area under a line
on a velocity-time graph depends on whether
the section bounded by the line and the axes is
a rectangle, a triangle or a trapezoid. Area
formulae for each shape are given below.
Describing Motion with Velocity vs. Time Graphs - Slope
Check Your Understanding
• The velocity-time graph for a two-stage rocket is shown below. Use
the graph and your understanding of slope calculations to
determine the acceleration of the rocket during the listed time
intervals.
o a. t = 0 - 1 second
o b. t = 1 - 4 second
o c. t = 4 - 12 second
Practice Problem
• Calculate the speed of a dog running through a
field if he is covering 23.7 meters in 54 seconds.
Practice Problem
• Which object has a greater velocity, a ball rolling
down a 3.4 meter hill in six seconds or a fish
swimming upstream and covering 5.4 meters in 0.4
minutes?
Practice Problem
• Calculate the velocity of a mountain climber if that
climber is moving northeast at a pace of 1.6 km in
1.4 hours? Give your answer in the SI unit for
velocity.
Practice Problem
• If a cyclist in the Tour de France traveled southwest
a distance of 12,250 meters in one hour, what would
the velocity of the cyclist be?
Instantaneous Velocity
• The speed and direction of an object at a
particular moment.
• The instantaneous velocity 𝑣 is the limit of the
average velocity as the time interval ∆𝑡 becomes
infinitesimally small
∆𝑑
∆𝑡→0 ∆𝑡
• 𝑣 = lim
Instantaneous Velocity
Positions of a Car at
Specific Instants of
Time
t(s)
d(m)
Calculated Values of the Time Intervals, Displacements,
and Average Velocities for the Car
1.00
5.00
Time Interval (s)
∆t (s)
∆d (m)
𝑣 (m/s)
1.01
5.47
1.00 to 3.00
2.00
47.5
23.8
1.10
9.67
1.00 to 2.00
1.00
29.7
29.7
1.20
14.3
1.00 to 1.50
0.50
21.3
42.6
1.50
26.3
1.00 to 1.20
0.20
9.30
46.5
1.00 to 1.10
0.10
4.67
46.7
2.00
34.7
1.00 to 1.01
0.01
0.470
47.0
3.00
52.5
Instantaneous Velocity
Motion of Car
60
50
d (m)
40
30
20
10
0
0
0.5
1
1.5
2
t (s)
2.5
3
3.5
Equation for Motion for
Average Velocity
• 𝑑 = 𝑣𝑡 + 𝑑𝑖
• Look familiar????
• 𝑦 = 𝑚𝑥 + 𝑏
Comparison of Straight Lines
with Position-Time Graphs
General
Motion
Variable
Variable
y
d
m
𝑣
x
t
b
di