Position-Time graphs

WHY GRAPH?

With accuracy Can be understood by everybody in the world 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

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

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

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

GRAPH SET UP

Axis What do the numbers mean

Direction Is direction of motion indicated on the graph?

POSITION-TIME GRPAHS

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 11

Position-time graph Where is the object at the 3 second mark? 12

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 13

Position Time graphs 14

Position Time graphs Graph indicates Positive motion Constant velocity Between fast and slow rate of motion 15

Moving very fast

Moving very slow

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

Moving forward, backward at constant speed

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

Speeding up

22

Graph indicates Non-constant velocity Positive motion Getting faster

24

Graph indicates Negative motion Non-constant velocity Getting faster

Slowing down

Slowing down moving forward

28

Graph indicates Negative motion Non-constant velocity Slowing down

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 30

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 31

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 object’s velocity The amount of tilt will indicate how fast th object goes 32

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 33

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 34

Tangent line Line that touches a graph at only one point 35

Mathematical method to determine slope of tangent lines Determine the rate of change Derivatives in calculus 36

DRAWING AND ANALYZING POSITION TIME GRAPHS

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

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? 39

Position-Time 4 2 5 3 6 1 2

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

Start position

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

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

GIVEN THE GRAPH, DESCRIBE THE MOTION

GIVEN DESCRIPTION OF MOTION, PRODUCE CORRESPONDING GRAPH

Creating a graph from written information Draw a copy on a piece of paper 47

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 48

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 49

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

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

DETERMINE THE VELOCITY FROM A POSITION TIME GRAPH

Position –time graph with number values 53

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? 54

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 55

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 = (y2 –y1) / (x2 – x1) 56

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

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

Calculating the velocity Use (y2-y1) / (x2 – x1) to calculate the slope (velocity) Organization of information Starts with identification of x and y values

Calculate velocity on each section

DETERMINE INSTANTANEOUS VELOCITY FROM POSITION-TIME GRAPH

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.

DETERMINE DISTANCE AND DISPLACEMENT FROM POSITION TIME GRAPH

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? 64

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 = y2 – y1

Example

Find the overall displacement for: The entire graph the first 20 seconds The last 20 seconds

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? 68