P-T Slope

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Constant VelocityConstant Velocity

• Consider a car moving with a Consider a car moving with a constant, constant, rightward (+) velocityrightward (+) velocity - say of +10 m/s. - say of +10 m/s.

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Graph of Constant VelocityGraph of Constant Velocity

• If the position-time If the position-time data for such a car data for such a car were graphed, were graphed, then the resulting then the resulting graph would look graph would look like the graph at like the graph at the right.the right.

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• Motion described as a constant, positive Motion described as a constant, positive velocity results in a line of constant and velocity results in a line of constant and positive slope when plotted as a position-positive slope when plotted as a position-time graph.time graph.

Constant VelocityConstant Velocity

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The Meaning of Slope of p-t The Meaning of Slope of p-t GraphsGraphs

• The slope of a position vs. time graph The slope of a position vs. time graph reveals pertinent information about an reveals pertinent information about an object's velocity. For example:object's velocity. For example:– a small slope = small velocitya small slope = small velocity

– a negative slope = negative velocitya negative slope = negative velocity

– a constant slope (straight line) = constant a constant slope (straight line) = constant velocityvelocity

– a changing slope (curved line) = changing a changing slope (curved line) = changing velocity (more on this later). velocity (more on this later).

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Slope of a p-t GraphSlope of a p-t Graph

• The actual slope value of any straight The actual slope value of any straight line on a graph is the velocity of the line on a graph is the velocity of the object.object.

• Slope = VelocitySlope = Velocity

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Example 1Example 1

• Consider a car moving with a constant Consider a car moving with a constant velocity of +10 m/s for 5 seconds. The velocity of +10 m/s for 5 seconds. The diagram below depicts such a motion.diagram below depicts such a motion.

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Graph-Example 1Graph-Example 1

• The position-time The position-time graph would look like graph would look like the graph at the right.the graph at the right.

• Note that during the Note that during the first 5 seconds, the first 5 seconds, the line on the graph line on the graph slopes up 10 m for slopes up 10 m for every 1 second along every 1 second along the horizontal (time) the horizontal (time) axis. axis.

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CalculatingCalculating SlopeSlope

Pick two points on the line and determine their Pick two points on the line and determine their coordinates. coordinates.

Determine the difference in y-coordinates of Determine the difference in y-coordinates of these two points (these two points (riserise). ).

Determine the difference in x-coordinates for Determine the difference in x-coordinates for these two points (these two points (runrun). ).

Divide the difference in y-coordinates by the Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).difference in x-coordinates (rise/run or slope).

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• Calculate the slopeCalculate the slope

yy22 – y – y11 = = ddff – d – dii = = riserise

xx22 – x – x11 t tff – t – tii

runrun

50-0m50-0m = = 50m50m = 10 = 10 mm

5-0 s 5s 5-0 s 5s ss

Graph-Example 1Graph-Example 1

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ExampleExample 22

• Now consider a car moving at a constant Now consider a car moving at a constant velocity of +5 m/s for 5 seconds, abruptly velocity of +5 m/s for 5 seconds, abruptly stopping, and then remaining at rest (v = 0 stopping, and then remaining at rest (v = 0 m/s) for 5 seconds.m/s) for 5 seconds.

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Graph-Example 2Graph-Example 2

• slope = df – di

tf – ti

25-0m = 25m = +5 m

5-0 s 5s s

• slope = df – di

tf – ti

25-25m = 0m = 0 m

10-5 s 5s s

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NoteNote

• A slope of zero means that A slope of zero means that the object is at rest (not the object is at rest (not moving).moving).

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SlopeSlope PrinciplePrinciple

• The principle is: The principle is: the the slope of the line on a slope of the line on a position-time graph is position-time graph is equal to the velocity equal to the velocity of the objectof the object. .

• If the object is moving If the object is moving with a velocity of +4 with a velocity of +4 m/s, then the slope of m/s, then the slope of the line will be +4 the line will be +4 m/s. m/s.

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33 TrialsTrials

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Interpretation of p-t GraphsInterpretation of p-t Graphs

• Recall: Recall: – On a p-t graph a negative slope means that On a p-t graph a negative slope means that

the object is moving in the negative direction.the object is moving in the negative direction.– A positive slope means that the object is A positive slope means that the object is

moving in the positive direction.moving in the positive direction.– No slope (zero) means that the object is at No slope (zero) means that the object is at

rest. rest.

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Interpretation of p-t GraphsInterpretation of p-t Graphs• Calculate slope for

the following timeframes:

• 0s – 10s

• 10s – 15s

• 15s – 40s

• 40s – 55s