SPH3U1
Lesson 05
Kinematics
THREE STEPS TO GRAPHICAL ANALYSIS
LEARNING GOALS
Students will


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.
1.
2.
3.
You can read values off the graph.
You can find slope(s) of the graph (tangent = instantaneous, secant = average).
You can calculate the area between the curve and the x-axis of the graph.
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
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.
1
SPH3U1
Lesson 05
Velocity vs Time
10.0
Rise ( ⃑)
5.0
Run (
Velocity vs Time
15.0
Velocity (m/s [E])
Velocity (m/s [E])
Average
Acceleration
15.0
Kinematics
)
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 ____________
⃑
Velocity is _______________
t
Value of acceleration is ____________
Direction of velocity ____________
⃑
Velocity is _______________
t
Value of acceleration is ____________
Direction of velocity ____________
⃑
Direction of acceleration is ____________
t
Velocity is _____________ (increasing,decreasing)
Direction of velocity above axis ____________
⃑
Direction of velocity below axis ____________
What happened at crossing of axis?
t
_____________________________
Direction of acceleration is ____________
Velocity above axis is ______________
Velocity below axis is ______________
2
SPH3U1
Lesson 05
Kinematics
Direction of velocity ____________
⃑
Direction of acceleration is ____________
t
Velocity is _____________ (increasing,decreasing)
Direction of velocity above axis ____________
⃑
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.
3
SPH3U1
Lesson 05
Kinematics
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.
HOMEWORK
In your textbook, do p26#1, p30 #2-6
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