2016-03-13
SCIENCE 1206
MOTION UNIT #6
GRAPHING UNIFORM AND NON-UNIFORM
MOTION
GRAPHING MOTION
In this course we will look at 4 graph types
• Distance-time graph
• Displacement-time graph (position-time
graph)
• Speed-time graph
• Velocity-time graph
In all of these cases, time will be the
independent variable and always be on the
x-axis
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GRAPHING – GENERAL
CONCEPTS
Plotting motion data on a set of x-y axis is a visual way of
describing the motion of the object
• X-axis = horizontal
• Y-axis = vertical
Linear graph – a set of data that forms a straight line (rate of
change is constant between time intervals)
Slope of a line - : on a graph, represents the steepness of the
line; the relationship between the two variables
• Slope = rise/run or
𝑦2−𝑦1
𝑥2−𝑥1
SLOPE OF A LINE
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CALCULATING SLOPE
Ex. Look at the graph below and calculate the slope
- Pick two points on the line
(does not matter which two
- Ex. (0, -3) and (2, 1)
𝑦 −𝑦
- Slope = 2 1
𝑥2−𝑥1
- =
1−(−3)
2−0
- = 4/2
- =2
DISTANCE-TIME
GRAPHS
-
Distance is plotted on the y – axis while time is on the xaxis
-
If the graph is a straight line, the speed is constant. The
slope of a distance-time graph is the speed of the object
-
If the graph is curved, the speed of the object is changing.
If it is curving up (becoming more steep) the object is
speeding up. If it is leveling off (becoming less steep
towards horizontal) the object is slowing down.
-
A horizontal line on a d-t graph means the object is
stopped
-
Average speed =
𝒕𝒐𝒕𝒂𝒍 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆
𝒕𝒐𝒕𝒂𝒍 𝒕𝒊𝒎𝒆
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D-T GRAPHS FOR NON-UNIFORM
MOTION
In this graph,
speed is
increasing (slope
getting steeper)
In this graph, speed is
decreasing (slope of
line decreasing)
D-T GRAPH FOR ZERO
MOTION
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D-T GRAPHS FOR
COMBINATIONS OF MOTION
- You may be asked to describe motion on a graph where
the motion changes (going faster, slower, stopped). Just
break the graph up into sections and analyze each section
separately .
- From 0-4 h, object travelling at
a constant speed of 25 km/h
(slope of the line)
- From 4-6 h object is stopped
(slope is zero)
- From 6-8 h object has a speed
of 50 km/h
- From 8-10 hours object has a
speed of 25 km/h
- Note: average speed is
250km/10 h = 25 km/h
POSITION-TIME GRAPHS
A position-time graph looks very similar to a distance-time graph.
However, position-time graphs show change in direction, but a distancetime graph cannot. Important features of a position-time graph:
a) The reference point:
the origin, zero, or the location of the observer
b) The y-axis label
This will give you the relative change in position (how far and the
direction) of the moving object
c) The slope of the line:
The slope of a position-time graph gives the velocity of the
object:
A positive slope means the object:
is moving in the direction indicated on the y-axis
has a constant positive velocity
identified by the right hand side of the line higher than the left
side
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POSITION-TIME GRAPHS
ii) A negative slope means the object:
is moving in the direction opposite to the direction
indicated on the y-axis
has a constant negative velocity
identified by the right hand side of the line lower
than the left side
d) The direction of movement (relative to the reference point).
North or South; East or West
Up or down; right or left
EXAMPLES OF POSITION-TIME
GRAPHS
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EXAMPLES OF POSITIONTIME GRAPHS
Solutions:
a) The object is travelling at a constant velocity (indicated by straight line).
It is constant velocity in the North direction.
b) The object is travelling at a constant velocity North, towards the
reference point (zero) and then further North beyond the reference point.
c) The object is travelling at a constant velocity South (indicated by
negative slope).
d) The object is travelling at a constant velocity South towards the
reference point, and continues travelling further South beyond the
reference point.
SPEED-TIME (V-T)
GRAPHS
-
Shows the relationship between an objects speed over
time
-
Slope of a v-t graph equals the object’s acceleration
-
- Zero slope = constant speed (no acceleration)
- Positive slope = speeding up
- Negative slope = slowing down
Area under the line: the size of the space between the
graph line and the x-axis; (of a speed–time graph) a way of
calculating distance; (of a velocity– time graph) a way of
calculating displacement
-
Instantaneous speed can be read directly from the graph
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SPEED-TIME GRAPHS
A = positive acceleration
B = negative acceleration
C = no acceleration (constant speed)
D = no movement (zero speed)
SPEED-TIME GRAPHS
AND DISTANCE
-
You can calculate the distance travelled from a v-t graph
by finding the area between the line and the x-axis
-
Area of a rectangle = length x width
Area of a triangle = (base x height)/2
If the shape is not a triangle or rectangle, divide it into
these shapes, find the individual areas and add them
together
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SPEED-TIME GRAPHS
AND DISTANCE EXAMPLE
- From 0-4 s, the distance was :
- (4s x 7 m/s) /2 = 14 m
- From 4-7 s, the distance was:
- 3 s x 7 m/s = 21 m
- From 7-9 s, the distance was:
- (2 s x 7 m/s)/2 = 7 m
- Total distance = 14 m + 21 m
+ 7 m = 42 m
VELOCITY-TIME
GRAPHS
Velocity- time graphs are very similar to speed-time graphs
- slope of a velocity-time graph equals acceleration
- area between graph and x-axis equals displacement
Difference is velocity-time graphs indicate direction
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CONVERTING BETWEEN POSITIONTIME GRAPHS AND VELOCITY-TIME
GRAPHS:
For a velocity with a positive slope (in the direction indicated
on the y axis), The position-time graph and its corresponding
velocity-time graph look like this
CONVERTING BETWEEN POSITIONTIME GRAPHS AND VELOCITY-TIME
GRAPHS:
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SUMMARY OF GRAPHS FOR
UNIFORM ACCELERATION
Comparison of position-time graphs and velocity-time graphs
for uniform and accelerated motion
SUMMARY OF GRAPHS FOR
UNIFORM ACCELERATION
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SUMMARY OF GRAPHS FOR
UNIFORM ACCELERATION
Note: Imagine constructing a series of tangent lines to these
curves.
Their steepness of the slope represents their acceleration:
- steeper tangents = speeding up
- flatter, less steep tangents = slowing down
The slope (positive or negative) indicates direction:
- positive = in direction on the y-axis
- negative = in direction opposite to what is on the y-axis
SUMMARY OF VELOCITY-TIME
GRAPHS FOR ACCELERATED
MOTION
Slope represents acceleration (positive is speeding up,
negative is slowing down)
0 means the object has stopped (0 m/s)
positive portion of graph means motion to right (North or
East), negative portion means motion to the left (South or
West).
Refer also to page 390-393 in text.
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SUMMARY OF VELOCITYTIME GRAPHS FOR
ACCELERATED MOTION
Instantaneous Velocity
Can be read from a velocity-time graph. It is the velocity at
a particular point in time.
Average Velocity
The ratio of total displacement divided by total time.
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