Introductory Physics: Graphs of Motion Example

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

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