Position-Time graphs

WHY GRAPH?

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

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

Position-time graph

• Where is the object at the 3 second mark?

• 9 second mark?

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

Position Time graphs

Position Time graphs

• Graph indicates

• Positive motion• Constant velocity• Between fast and

slow rate of motion

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

• Graph indicates

• Non-constant velocity• Positive motion• Getting faster

• Graph indicates

• Negative motion• Non-constant velocity• Getting faster

Slowing down

Slowing down moving forward

• 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

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

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

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

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

Tangent line

• Line that touches a graph at only one point

Mathematical method to determine slope of tangent lines

• Determine the rate of change

• Derivatives in calculus

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?

Position-Time

1 22 3

4 56

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

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

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

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

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?

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

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)

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

Run= change in the time

Rise= change in the position

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?

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:

1) The entire graph

2) the first 20 seconds

3) 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?