Physics 30S - Position – Time Graphs: Types of Velocities
There are 3 types of velocities that we will consider.
1. Constant Velocity
2. Average Velocity
3. Instantaneous Velocity
1. Constant Velocity
You are driving down a highway and you notice that it takes equal intervals of time
to travel between hydro poles. This is an indication that you are traveling at
constant velocity. The formula for determining constant velocity is:
v
d
t
v - velocity in m/s, km/h, etc.
d - displacement in m, km, etc.
t - time intervals in s, or h
Note that since displacement is either a positive or negative value, velocity will
have the same directional notation. The similarity in sign for both velocity and
displacement occurs because time interval is always positive.
Practice
Note that when writing the direction of a measurement we can use the actual
direction. It is in a math formula that we need to substitute "+" and "-" for direction
notation.
1) What is the constant velocity of an airplane that flies 602 m E in 2.5 s? Express
your answer in meters per second and in kilometers per hour.
2) The tine on a tuning fork moves 1.0 mm to the right in 4.0 x 10-3s. What is the
constant velocity of the tine in meters per second?
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Physics 30S - Position – Time Graphs: Types of Velocities
3) An electron travels at a constant velocity of 1.3 x 105 m/s. How much time is
required for a displacement of 1.00 m?
4) A spaceship traveled at a uniform velocity of 3.2 x 104 km/h for 2.7 days. What
was the displacement of the spaceship?
5) A motorbike is traveling at 85 km/hr as shown on its speedometer. How many
seconds will it take to cover a 200 m track at this constant rate?
2. Average Velocity ( v )
Average velocity can be determined two ways:
a) by formula:
v
d
t
v - average velocity, d – displacement, t - time
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Physics 30S - Position – Time Graphs: Types of Velocities
This formula would be used when you are asked to determine average velocity in a
word problem.
e.g.1 A bike travel 3.2 km in 45 minutes. What is its average velocity?
e.g.2 A vehicle travels 30 km East in 15 minutes, 45 km West in 20 minutes
then 10 km East in 7.5 minutes. What is its average velocity?
e.g. 3 A orienteering athlete walks 1.2 km East in 25 minutes, then 0.8 km
North in 15 minutes. Determine her average velocity.
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Physics 30S - Position – Time Graphs: Types of Velocities
b) by determining the slope of the line joining two points
In the example graph the slope of the line in time interval A would yield the
average velocity for that time interval.
position (m)
8
B
position (m)
4
A
C
0
F
-4
E
-8
D
-12
0
5
10
15
20
time (s)
v = slope
rise
=
run
6m  0m
=
2s  0s
6m
=
2s
= 3m/s
Determine the average velocity for the
remaining time intervals as well as the
entire trip.
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Physics 30S - Position – Time Graphs: Types of Velocities
When determining the average velocity the following points should be kept in
mind.
a) Positive slope (rises to the right) yields positive velocity and negative slope
(drops to the right) yields negative velocity.
b) Positive velocity results from movement in a positive direction and negative
velocity from movement in the negative direction.
c) A straight-line portion on a graph indicates a constant velocity. This means
that the average and constant velocities are the same value. If average
velocity over more than one straight-line portion of a graph is desired, then
you must construct a straight line from a point on the graph at the beginning
of the time interval to a point on the graph at the end of the time interval. If
d
you use the average velocity formula v  , just make certain that you have
t
determined the total displacement over the time interval.
d) A horizontal line indicates zero slope and therefore zero velocity; that is, the
object is stationary.
3) Instantaneous Velocity (vinst)
There are two possible conditions when vinst is to be calculated.
a) On a straight-line graph: In this case vinst = v because velocity is constant
on any straight-line graph
d
slope A = slope = slope C
slope C
slope B
slope A
t
Any time you are asked for instantaneous velocity, and the graph is a straight line;
simply find the average velocity for the corresponding interval.
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Physics 30S - Position – Time Graphs: Types of Velocities
b) On a curved-line graph: In this case the portion of the graph over which
the vinst is to be calculated is reduced as shown in the diagram below.
d
slope B
slope C
slope A
t
As a shorter and shorter time interval is chosen, shown in the diagram as slopes C,
B, and A the line become progressively shorter and the curve becomes straighter
between those points. If we were to choose a very short time interval so that time
could be considered zero, the length of the curve would also be extremely short.
Since decreasing the length of the time interval caused the portion of the graph to
be straighter it would be reasonable to assume that the very short piece of graph
would, for all purposes, be considered a straight line. It would be impossible to find
the slope of such a short line but we can employ a construction method, which will
lengthen and consequently allow us to calculate the slope of the line under
consideration. The construction involves drawing a tangent line at the point where
we want to find the vinst on the graph. This tangent line represents an extension of
the very short piece of line that was mentioned earlier. The construction of the
tangent line is shown below.
Draw the tangent line that best
approximates the slope of the
curved line at that point.
vinst  slope of tangent line

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rise
run
Physics 30S - Position – Time Graphs: Types of Velocities
Practice
Find the instantaneous velocities at points A, B, and C using the graph shown
below.























y
B
C
A




  
   
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   

x
Physics 30S - Position – Time Graphs: Types of Velocities
A) Position-Time Graph For A Rolling Ball
Use the graph below to answer the following questions.
Determine:
a) Δd from 0-4 seconds
f) v from 0-4 seconds
b) Δd from 4-5 seconds
g) v from 6-9 seconds
c) Δd from 6-9 seconds
h) vinst at 10 seconds
d) Δd from 9-14 seconds
i) vinst at 5 seconds
e) Δd from 0-14 seconds
j) v from 0-8 seconds
position vs time
position (m)
10
5
0
-5
0
5
10
time (s)
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Physics 30S - Position – Time Graphs: Types of Velocities
B) Position-Time Graph Using a Curved Line
Use the graph below to answer the following questions.
Determine:
a) v from 0-6 seconds
g) vinst at 18 seconds
b) v from 8-12 seconds
h) d from 0-6 seconds
c) v from 12-16 seconds
i) d from 16-22 seconds
d) v from 16-22 seconds
j) d from 0-22 seconds
e) vinst at 7 seconds
k) v from 6-16 seconds
f) vinst at 14 seconds
15
position (m)
10
5
0
-5
-10
0
2
4
6
8 10 12 14 16 18 20 22 24
time (s)
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Physics 30S - Position – Time Graphs: Types of Velocities
C) Position-Time Graph Reveiw
Use the graph below to answer the following questions.
Determine:
a) the kind of motion from
0-6 seconds
d) v from 6-9 seconds
e) d from 0-6 seconds
b) vinst at 4 seconds
f) v from 10-11 seconds
c) v from 0-6 seconds
g) vinst at 2 seconds
position (m)
position vs time
6
4
2
0
-2
-4
0
2
4
6
time (s)
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8
10
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Physics 30S - Position – Time Graphs: Types of Velocities
D) Position-Time Graph Checkup
Use the following graph to answer the following questions.
Determine:
a) Δd from 0-4 seconds
d) v from 9-16 seconds
b) Δd from 9-16 seconds
e) vinst at 3 seconds
c) v from 0-4 seconds
f) vinst at 10 seconds
position (m)
position vs time
8
6
4
2
0
-2
-4
-6
0
2
4
6
8
time (s)
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10
12
14
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Physics 30S - Position – Time Graphs: Types of Velocities
ANSWERS:
Graph 1
a) +6m
b) 0m
c) -12m
d) +4m
e) -2m
f) 1.5 m/s
g) -4 m/s
h) 0.8 m/s
i) 0 m/s
j) -0.25 m/s
Graph 2
a) 2 m/s
b) -2 m/s
c) 0 m/s
d) 1.3 m/s
e) -7.5 m/s
f) 0 m/s
g) 1.4 m/s
h) 12m
i) 8m
j) -3m
k) -2.3 m/s
Graph 3
a) variable
b) 0.75 m/s
c) 0.67 m/s
d) -1.875 m/s
e) 4m
f) 3.5 m/s
g) 0.4 m/s
Graph 4
a)
b)
c)
d)
e)
f)
g)
-6m
+6m
-1.5 m/s
0 m/s
0.86 m/s
-1.5 m/s
1.57 m/s
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