# Introductory Physics: Graphs of Motion Example

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let’s take a look at an example

involving graphing motion and we’re going to start with this graph it is a

graph of velocity versus time for some object notice that we’ve got the axes

labeled where we’ve got time here and velocity here the object begins at four

meters per second and the velocity declines as time goes on until it

reaches a velocity of zero meters per second after eight seconds have elapsed

and so what we’d like to do is convert this into an acceleration versus

time graph and a position versus time graph. So how do we do that?

The easiest

thing to do is to start with acceleration versus time if we do that we can say all right acceleration versus

time over eight seconds one two three four five six seven eight and to

determine acceleration acceleration is going to be equal to our change in

velocity over our change in time and so our change in velocity was well we went

from four meters per second to zero so our change was a negative four meters

per second and that happened over eight seconds and so what we get is well 4

over 8 is one-half so we get negative 1/2 meters per second squared and that

is our acceleration which means I need to bring down my axis into the negative

so if this is going to be our acceleration in meters per second

squared then if this is one negative one there’s positive 1 negative 1/2 meters

per second squared is going to be right here and because the slope of our

velocity versus time graph is constant the acceleration at all moments is

constant so we’ll have constant acceleration continuing on across all of

these time durations all of these time

intervals so negative 1/2 so there’s our acceleration versus time graph now let’s

take a look at position which we’ll call X and that’ll be in meters divided by or

meters over seconds like that well generally what we’ll see is there’s

gonna be a negative acceleration which means that as time goes on the slope of

our X versus time graph should not be constant why because the slope of X

versus time which is going to be the change in X over change in time Y change

in Y over change in X x-axis well that’s equal to the velocity and notice how

velocity is changing which means if it’s changing to be a smaller number we start

with a higher slope and end with a lower slope and so the way that we can do that

is by as each second goes by we can draw something that looks like this

a higher slope slightly less high slope like this like so so we can either draw something

that looks like this or if because we have a positive velocity we should be

traveling in the positive x-direction so we can start with a high slope and then

end up with a low slope looking like that so this is probably what our graph

is going to look like right here all right but how do we describe this motion

well that means we’re moving in the positive x-direction and as we move in

the positive x-direction we are traveling a shorter displacement as each

second goes on that is we’re making less and less progress as each second goes on

because we’re losing velocity we are d accelerating until finally we get to a

zero acceleration and zero I’m sorry until finally we get to a zero

velocity and at that point the object stops and is at rest a couple other

things that we can do we can determine how far we have traveled during this

during this eight second duration in order to figure that out we can use our

graph or remember that the displacement is equal to velocity times change in

time well we can do that so take a look at our graph over here it turns out that

displacement from a velocity versus time graph can be get can be given by the

area under the curve because velocity times time is going to be equal to our

displacement but the velocity changes so it can’t just be the rectangle here it’s

going to be this triangle so we’ll have to divide the area of what would be the

rectangle by half so the triangle area well that’s going to be equal to the

area of a triangle is equal to one-half base times height so in this case it’ll

be one-half times eight seconds times the height

which is going to be four seconds I mean four meters per second so what we get is

8 times 4 times 1/2 which gives us 16 metres notice how the seconds cancel out

so our displacement total displacement in X is 16 meters that’s how far we

travel as we are decelerating from 4 meters per second to 0 over the course

of 8 seconds