SPH4C1
Lesson 02
Motion and its Applications
NON-UNIFORM MOTION
LEARNING GOALS
Students will:




Determine instantaneous and average velocities from position-time graphs.
Describe in words motions represented by position-time graphs.
Learn how to read values, find slopes and calculate areas on graphs.
Learn what these values mean on both position-time and velocity-time graphs.
INSTANTANEOUS VELOCITY
The moment-to-moment measure of an object’s velocity is called its instantaneous velocity. A
vehicle speedometer tells you the instantaneous velocity.
Uniform motion or constant velocity is motion at a constant speed in a straight line. It is the
simplest type of motion that an object can undergo, except for being at rest. In contrast, motion
with non-uniform velocity is motion that is not at a constant speed or not in a straight line.
Motion with non-uniform velocity may also be called accelerated motion.
DETERMINING INSTANTANEOUS VELOCITY FROM A POSITION-TIME GRAPH
If the velocity changes every moment during the motion of an object, no portion of the positiontime graph is a straight line. Remember that each point on the curve represents the position of
the object at each instant in time. To determine the velocity of an object at any instant,
physicists use tangents. A tangent is a straight line that touches a curve at only one point and
is parallel to the curve at that point. Each tangent has a unique slope that represents the
instantaneous velocity at that instant in time.
A. In each of the position-time graph examples below, state whether the object is speeding
up or slowing down.
B. In each of the examples above (in #3), state whether the object is going forwards or
backwards.
C. In the graph below one line is a Ferrari and the other is a Toyota. Which line is most
likely to be the Ferrari? Explain.
1
SPH4C1
Lesson 02
Motion and its Applications
EXAMPLE 1 INSTANTANEOUS VELOCITY ON AN POSITION-TIME GRAPH
The position-time data for an all-terrain vehicle (ATV) approaching a river are shown on the
following Graph. Find the instantaneous velocities at 2.0 s, 3.0 s, and 5.0 s.
Position vs Time
40.0
Position (m [E])
35.0
30.0
25.0
20.0
15.0
10.0
5.0
0.0
0.0
1.0
2.0
3.0
4.0
5.0
Time (s)
Tangent Line
40.0
Position (m [E])
A tangent line at 2.0 s is drawn and only
touches the point at 2.0 s. The slope of the
tangent line is the instantaneous velocity at
2.0 s.
Position vs Time
30.0
Rise ( ⃑)
⃑
20.0
10.0
Run (
)
0.0
0.0
2.0
3.0
Time (s)
4.0
5.0
At 3.0 s,
Position vs Time
40.0
Position (m [E])
1.0
⃑
Rise ( ⃑)
30.0
Run (
20.0
)
10.0
0.0
0.0
2.0
3.0
Time (s)
4.0
5.0
At 5.0 s, the tangent is a horizontal line.
Position vs Time
40.0
Position (m [E])
1.0
⃑
30.0
20.0
10.0
0.0
0.0
1.0
2.0
3.0
Time (s)
4.0
5.0
2
SPH4C1
Lesson 02
Motion and its Applications
REVIEW: POSITION-TIME GRAPHS
Positions can be read off the graph.
Displacements are a change in position. Read the initial and final position and subtract.
Average velocity over an interval is the slope of the line joining the end-points of the interval
(the secant).
Instantaneous velocity at a point is slope of the tangent at that point.
When the curve is a parabola, the instantaneous velocity at the centre of an interval is equal
to the average velocity of the entire interval.


Position (m [E])
Instantaneous
Velocity
40.0
Average
Velocity
40.0
Position vs Time
30.0
Position (m [E])



Rise ( ⃑)
20.0
Run (
10.0
)
0.0
Position vs Time
30.0
Rise ( ⃑)
20.0
Run (
10.0
)
0.0
0.0
1.0
2.0
3.0
Time (s)
4.0
5.0
0.0
1.0
2.0
3.0
Time (s)
4.0
5.0
PRACTICE
Use the graph below to answer the
following.
A.
Position vs Time for a Skier
Going _____ a Hill
Fill in the title on the graph
(up or down)
45
40
B. Is the skier getting faster or
slower?
Position (m)
C. Determine the average
velocity of the skier…
i)
For the first second
35
30
25
20
15
10
5
ii)
From 1.5 s to 3.5 s
0
0
2
4
6
8
Time (s)
iii)
From 4 s to 5 s.
iv)
Draw a tangent line at 2.0 seconds. Find the slope of this line. What does
this tell you?
3
SPH4C1
Lesson 02
Motion and its Applications
Use the graph at the bottom of the page to answer the following questions.
A. Is the skydiver moving up or down?
___________________________________
B. Is the skydiver getting faster or slower?
C. How far does the skydiver fall in total?
___________________________________
___________________________________
D. How far does the skydiver fall in the first 0.4 seconds?
_________________________________
E. How far does the skydiver fall from 2 s to 2.4 s?
___________________________________
F. Why is your answer from #5 bigger than your answer for #4?
G. Draw tangent lines and find their slope to figure out how fast the skydiver was going at
a. 0.4 s
b. 1.2 s
c. 2.2 s
Position vs Time for a Skydiver in Free Fall
35
30
Position (m)
25
20
15
10
5
0
0
0.4
0.8
1.2
1.6
2
2.4
2.8
Time (s)
4
SPH4C1
Lesson 02
Use the graph below to answer the following questions.
Motion and its Applications
A. Label all areas as either “speeding up”, “slowing down”, “constant speed” or “stopped”.
Get teacher’s signature when done. ___________
Position vs. Time for Isay Late for Class
40
35
Position (m)
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
Time (s)
B. At what time did Isay turn around? ______________
Instantaneous Speed: the speed at ONE POINT IN TIME
A. Label each one as an example of average speed or instantaneous speed:
a) speed from a radar gun
________________________
b) slope of position-time graph
c) total distance divided by total time
d) speed on a speedometer
________________________
________________________
________________________
B. How can you find the instantaneous speed of an object from the position-time graph?
C. Draw tangent lines on Isay’s graph at 3.0 s, 10.0 s and 19.0 s. Show your teacher.
_____
D. Find the instantaneous speed at 3.0 s and 19.0 s. Show your work.
5
SPH4C1
Lesson 02
Motion and its Applications
VELOCITY-TIME GRAPHS
Are graphs of instantaneous velocity plotted against time.
Instantaneous velocities can be read from the graph.
Average acceleration over an interval is determined by the slope of the line joining the two
points (the secant).
Instantaneous acceleration at a point is the slope of the tangent at that point.
When the velocity and the acceleration have the same sign, the object is speeding up (the
line goes away from the time axis).
When the velocity and the acceleration have the opposite sign, the object is slowing down
(the line approaches the time axis).
The displacement during an interval can be found from the area between the curve and the
time axis over that interval.
o Areas above the axis are positive and areas below the axis are negative.
Average velocities are found from the definition of average velocity: displacement over time.





Velocity (m/s [E])
Average
Acceleration
15.0
Velocity vs Time
10.0
Rise ( ⃑)
5.0
Run (
Velocity vs Time
15.0
Velocity (m/s [E])



)
0.0
Displacement – area
of coloured shapes
under the graph.
10.0
5.0
0.0
0.0
1.0
2.0
3.0
Time (s)
4.0
5.0
0
1
2
3
Time (s)
4
5
INTERPRETING VELOCIT Y-TIME GRAPHS
Direction of velocity ____________
𝑣⃑[𝑁]
Velocity is _______________
t
Value of acceleration is ____________
Direction of velocity ____________
𝑣⃑[𝑁]
Velocity is _______________
t
Value of acceleration is ____________
Direction of velocity ____________
𝑣⃑[𝑁]
Direction of acceleration is ____________
t
Velocity is _____________ (increasing,decreasing)
6
SPH4C1
Lesson 02
Motion and its Applications
Direction of velocity above axis ____________
𝑣⃑[𝑁]
Direction of velocity below axis ____________
What happened at crossing of axis?
t
_____________________________
Direction of acceleration is ____________
Velocity above axis is ______________
Velocity below axis is ______________
Direction of velocity ____________
𝑣⃑[𝑁]
Direction of acceleration is ____________
t
Velocity is _____________ (increasing,decreasing)
Direction of velocity above axis ____________
𝑣⃑[𝑁]
Direction of velocity below axis ____________
What happened at crossing of axis?
t
_____________________________
Direction of acceleration is ____________
Velocity above axis is ______________
Velocity below axis is ______________
POSITION-TIME TO VELOCITY-TIME GRAPHS
You can use the information in a position-time graph to create a velocity-time graph. Find the
slope of each section on the following position-time graph. The sections are determined by a
change in slope and a change in the sign of the position. Make a table of your results below.
Then use the table of values from your position-time graph to plot a velocity time graph.
7
SPH4C1
Lesson 02
Motion and its Applications
VELOCITY-TIME TO POSITION-TIME GRAPHS
You can use the information in a velocity-time graph to draw a position-time graph. To sketch
the position-time graph, find the area under the velocity-time graph for each section. The
sections are determined by a change in slope or sign of the velocity. Make a table showing the
TOTAL AREA FROM 0 up to each 5 s of time. Then plot the data on the position-time graph.
8
SPH4C1
Lesson 02
Motion and its Applications
PRACTICE POSITION-TIME GRAPHS
The position - time graph for an object in motion is shown below.
Position-Time Graph
10
I
8
A
Position (m [N])
6
B
4
G
2
C
0
-2
H
F
0
1
2
3
4
5
D
E
Time (s)
6
7
8
9
-4
-6
a) Write a short story that could represent the motion of this graph.
b) What is the displacement in the intervals:
i)
AD
ii)
EI
ii)
DH
ii)
FG
c) What is the distance traveled in the intervals:
i)
AE
d) What is the velocity in the intervals:
i)
BC
e) What segments of the trip have a zero velocity? Explain?
f)
What is the average speed in the intervals:
i)
AE
ii)
DH
ii)
DH
iii) AH
g) What is the average velocity in the intervals:
i)
AC
9
SPH4C1
Lesson 02
Motion and its Applications
Position (m [up])
FINDING VELOCITY FROM A CURVED POSITION-TIME GRAPH
Position vs. Time for a Ball Thrown Upward
7
6
5
4
3
2
1
0
0
0.5
1
1.5
2
2.5
Time (s)
1. The ball is on its way up from 0s until _________ and it returns to the ground at
________. On the way up, its speed is _____________________ and you can tell
because the slope is getting ____________________________________. On the way
down, the speed is ____________________ and you can tell because the slope is getting
_____________________. At the maximum height, its speed is _________.
2. Find the average velocity of the ball from 0s until 0.60 s.
3. Find the average velocity of the ball from 1.40 s until 2.2 s.
4. Find the instantaneous velocity of the ball at
a. 0.10 s
b. 1.60 s
10