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Position-Time graphs

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WHY GRAPH?

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What is the best way to describe motion of an object to somebody that did not witness it? With accuracy Can be understood by everybody in the world

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Describe my motion Write a description of my motion on a piece of paper

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A better way Represent the motion in the universal language: Math Draw a graph

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Reasons Universal language Convey more information visually Less time to draw than write

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GRAPH SET UP

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Axis What do the numbers mean

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Direction Is direction of motion indicated on the graph?

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POSITION-TIME GRPAHS

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Position-time graphs Position value is recorded as the vertical (y) component Time value is recorded as the horizontal The point (4,-8) means you are at -8 units from the central reference point at the 4 second mark

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Position-time graph Where is the object at the 3 second mark? 9 second mark?

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What information does the graph tell you about motion Shape of the line Straight-vs-curved Tilt of the line Flat-vs-slanted Tilted upward-vs-tilted downward Vertical lines Placement of the line Start position Postive-vs negative territory

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Position Time graphs

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Graph indicates Positive motion Constant velocity Between fast and slow rate of motion

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Moving very fast

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Moving very slow

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At rest How does a graph indicate that the object does not move

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Moving forward, backward at constant speed

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Which line shows no motion? Which line shows fastest rate of motion?

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Speeding up

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Graph indicates Non-constant velocity Positive motion Getting faster

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Graph indicates Negative motion Non-constant velocity Getting faster

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Slowing down

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Slowing down moving forward

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Graph indicates Negative motion Non-constant velocity Slowing down

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Forward, backward, or stopped If the graph is horizontal, then no motion has occurred. The position (vertical) value did not change over time If the final position is more positive than the initial position, it moved forward If the final position is less positive than initial position, it moved backward

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Is the object moving at a constant rate? Constant velocity means the rate of motion does not change over time Graphs show constant velocity by creating a straight line. Angle (tilt) of line does not matter

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How fast is it going? Constant velocity can occur in any direction Being stopped gives you a constant velocity of zero (0). The tilt of a straight line will indicate forward, or backward motion –Slope of the line is a measure of the objects velocity The amount of tilt will indicate how fast th object goes

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What if it is not a straight line? Then the velocity is not constant If the velocity changes over time the car accelerates Any change in velocity indicates acceleration On a P-T graph, acceleration is indicated by a curved line

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It is getting faster or slower? Tangent lines –Pick 2 point along section of graph –Draw tangent lines –If the slope of line increases, then object is getting faster Divide section up into equal 2 blocks of time. –Compare the displacement in each –If amount of displacement increases, it is getting faster

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Tangent line Line that touches a graph at only one point

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Mathematical method to determine slope of tangent lines Determine the rate of change Derivatives in calculus

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DRAWING AND ANALYZING POSITION TIME GRAPHS

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Draw the graph Draw a graph that would represent the following motion: –Positive Motion –Non-Constant Velocity –Slowing Down

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Information from graphs For each section with the same type of motion, you should be able to determine: Is the object moving forward, backward, or stopped Is the motion constant or not? If constant, is the rate of motion fast or slow? If not constant, is the object getting faster or slower?

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Position-Time

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Position time graph Each point on the graph indicates the position of the object at a certain time Shows both distance and displacement Y-axis indicates position X-axis indicates time

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Start position

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Most graphs are made from a combination of different types of motions

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What you should be able to tell me about the graph Whether object is moving or not Which direction it moves Whether motion is constant or not Whether object speeds up or slows

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GIVEN THE GRAPH, DESCRIBE THE MOTION

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GIVEN DESCRIPTION OF MOTION, PRODUCE CORRESPONDING GRAPH

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Creating a graph from written information Draw a copy on a piece of paper

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In the next graph… You will create a graph that represents the following motion Section 1- starts at the -2 meter position and moves with slow positive constant velocity Section 2- moves with fast negative constant velocity

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Information for next graph… Section 3- moves with positive non- constant velocity and is getting faster Section 4- moves with a constant velocity of zero Section 5 – moves with a negative non- constant velocity and is slowing down

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Example #1 Starting from a position of (-3). Object speeds up, moving forward to the origin Object maintains constant velocity moving forward, reaches (4) Object slows down, moving forward, reaches (6) Object Stops for several seconds Object speeds up moving backwards

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Assumption If the question does not specify times, assume that the displacement of interest is over the entire graph

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DETERMINE THE VELOCITY FROM A POSITION TIME GRAPH

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Position –time graph with number values

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What is the velocity of the car during the first 1.5 second? Is it constant Is it relatively fast or slow? How do you find its actual value?

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To answer the initial problem (3 – 0)(m) / (1.5 – 0)(s) = 3 m/s Find the rest of the constant velocity values shown on the graph

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Calculation of a constant velocity Slope of the line = steepness To determine slope, find the rise over run Rise = change in the y values between initial and final points Run change in the x values V = (y 2 –y 1 ) / (x 2 – x 1 )

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Slope of the line Constant velocity is demonstrated by a slanted straight line on a P-T graph The steepness indicates how fast the object moves To measure the steepness of a line, calculate the slope

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How to calculate the slope Run= change in the time Rise= change in the position Slope = Rise / Run

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Calculating the velocity Use (y 2 -y 1 ) / (x 2 – x 1 ) to calculate the slope (velocity) Organization of information Starts with identification of x and y values

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Calculate velocity on each section

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DETERMINE INSTANTANEOUS VELOCITY FROM POSITION-TIME GRAPH

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Calculating the instantaneous velocity value For any section on a graph with constant velocity, all points in time within that section have the same velocity value.

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DETERMINE DISTANCE AND DISPLACEMENT FROM POSITION TIME GRAPH

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Overall Displacement Look at the difference between final position and start position Look at nothing else!!! What is the overall displacement of the above graph?

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Determine overall displacement On a p-t graph: Displacement = the difference between the start position and final position for the portion of the graph you are interested in D = y 2 – y 1

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Example

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Find the overall displacement for: 1)The entire graph 2)the first 20 seconds 3)The last 20 seconds

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Overall Distance Measure the change in position between each hill and valley of the graph Then add up the changes There are no subtractions!! What is the total distance of the given graphs?

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