Understanding Position-Time Graphs: A Comprehensive Guide

good morning today today today today today we're going to discuss

graphing and so let's commence operations

for those of you following along in your notes please turn to this page now

graphs graphs and more graphs a position time graph well what does a position time graph

indicate it indicates the position of an object at a particular instant

in time for all the situations we'll be looking at today

we'll be considering motion in a straight line either the object is moving east or west

or the object is moving north or south in general for these graphs that's the type of motion they can only represent

the slope of a position time graph is the velocity or the speed now why is that

recall that slope is rise over run rise over run the rise is the change in position

final position d2 subtract initial position d1 that's displacement

the run is the change in time final time subtract initial time that's delta t

so the slope of a position time graph is the velocity and so the question is what does a

steeper slope correspond to please write down an explanation now

alright i hope you tried it a steeper slope corresponds to a higher velocity or speed

all right i want to show you a simulation right now so in this simulation we can show a car

moving back and forth so let's start off with a car not moving at all let's place it at a

position of five meters and let's press the start button and see what that looks like

it's not a very interesting graph the car hasn't moved and notice it's a straight line over the

course of time of 4 seconds what would negative 10 meters look like let's see

so notice the car is at a negative position with respect to the reference point

and again not very interesting the graph it's a straight line and the graph is not changing

its position because the car is not moving so when an object is stopped on a

position time graph notice it's just a straight line across and by the way the slope of that line

is zero all right let's make it a bit more interesting let's start at a position of

zero and let's give it a speed of two meters per second i'll reset this

and so on a position time graph that's what constant velocity looks like it's just a straight line with a constant

slope notice after four seconds if it's moving at two

meters per second after four seconds it should be around eight and that's where it sort of looks

like it's at what would negative let's make it negative

four meters per second let's see what that would look like well notice the negative means the car

is moving backwards and it's moving a little faster this time it's moving twice as fast so now

it's not just at 8 meters it's at negative 16 meters

notice the slope of this graph is negative this slope is a positive slope this

would be a negative slope corresponding to the velocity and in fact if you do rise over run

negative 16 divided by four would be your negative four meters per second

how would this graph change if instead of starting at the zero you start at the negative two position so let's

see so you're not starting at the zero here you're starting at the negative two

position let's see and notice for the graph the only thing

that changes is where it starts it's no longer starting at zero it's starting at negative two

but the slope doesn't change because the velocity hasn't changed let's show you a case of where the

object starts at say five meters and it's still negative let's make it negative five

so we'll reset that notice the car has started ahead of the zero at the five meter point but now it's

traveling negative five negative five meters per second and notice where as the object ended up

at well it's ended up at negative 15 meters and again this is a negative slope

let's show you one more example let's make the initial position equal to zero

and we'll go at three meters per second and now we'll go up four meters per second which is a slightly higher

velocity and notice the slope is slightly steeper all right so that's the idea for

constant velocity motion you'll get a straight line

slope doesn't change for constant velocity if it's moving in a positive direction

the slope will be positive if it's moving in a negative direction well then the slope will be negative

let's continue on all right there are times where you may be asked to interpret

a graph such as the case here so example one the following graph represents the

motion of a student in a straight line either east or west in the classroom describe the motion of the student in

each section so here's the graph in greater detail

at time zero they're at their reference position or zero meters two seconds later

they've moved two meters east this e stands for east and then another four seconds later so

now at the six second mark they're at four meters

east and then it seems for a few seconds they don't move they're staying at four meters

east then all of a sudden 10 seconds a person decides to move east again and two seconds later they're at six

meters east and then for whatever reason they decide to turn around and head all the way back

to where they started so in 16 seconds they're back where they started they started at zero meters

and they're back at zero meters and so what would we actually say well for section a here person moved two

meters east in two seconds two meters east in two seconds

b another two meters east gone two meters east but this time in four seconds

and see the person hasn't moved at all they're stopped for over four seconds remember what's the slope of this line

slope is zero that means your velocity is zero section d they've moved another two

meters east in another two seconds two meters east in two seconds

and then for section e person hasn't moved east anymore now those they're moving

back to where they started back to position zero and all together they move six meters

west notice the slope is negative on a position time graph when the slope is positive

they're moving east but when the slope is negative they're moving west

so one of the questions that i like to ask my students is calculate the average speed and velocity

from 0 to 16 seconds because i really want them to understand the difference between speed and

velocity so we know the formula for speed that's distance over time

the time is 16 seconds that's obvious from 0 to 16 seconds it's got to be 16 seconds

and the distance how do we get the distance well notice we label each section of the

graph 2 meters 2 meters 0 2 and 6 and all we do is we add remember

distance direction doesn't matter it doesn't matter

the direction when tabulating distance two plus two plus zero plus two plus six that's twelve meters

and when we do our division it's zero decimal seven five meters per second

now velocity is different remember it's displacement over time so for average velocity it's displacement

not distance displacement the time once again is 16 seconds of course that doesn't change because we're

going from 0 to 16 seconds but the displacement when we determine displacement

direction matters formula looks very similar to what we did for distance we add numbers except here

we're adding negative six zero meters remember what displacement is it's the change in position

has the position changed over 16 seconds let's see what 16 seconds the position is zero

at zero seconds the position is also zero so no the position has not changed therefore the displacement is zero

and so therefore the velocity is zero so keep that in mind displacement is the change in position

here's one way of tabulating or calculating displacement adding all the numbers up

and considering the direction but the other way is just looking

where are you on your graph well it's 16 seconds your final position is zero your initial position is also zero

therefore your displacement is zero all right i want you to try this problem

right now just to make sure you really understand the concept

that's the only way in physics you really understand if you understand something is by trying something

so calculate the average speed from 2 to 14 seconds and then calculate the average velocity

from 2 to 14 seconds so from 2 seconds on the graph

to 14 seconds on the graph go ahead pause the video now okay hopefully you try that out

and so the distance starts here this time 2 plus 0 plus 2 plus 3

because we're stopping at 14 seconds distance is 7 meters the time is 12 seconds and i'll let you do that

division of distance over time average velocity however it's only one meter

two plus zero plus two plus negative three now does that make sense it's one meter

let's see it starts at two meters east it ends at three meters east and the question is

what's the change in position well the change in position is one meter east he's moved or she's moved one meter from

where they started you start at two meters you end at three meters you're basically

one step away from where you started on this trip again i'll let you do the division here of 1 over 12 to figure out

the average velocity hope you enjoyed today's lesson have a great day