in this video we learned to visualize position and velocity for an object moving in one dimension and
we established the meaning of slope for a position time graph and the meaning of area for a velocity
time graph so we begin with an example and in this example we're told that a car starts at the origin
and drives 90 miles to the north at a constant rate in two hours after this the car turns around
and drives to the south 90 miles at a constant rate in three hours we want to plot the position
of the car as a function of time and for this we use a position time graph on this graph the
vertical axis tells us the position of the car relative to the origin and the horizontal axis
tells us the time so each ordered pair gives us a moment in time together with a position in space
and this is how we describe the motion of an object moving in a straight line
the natural units for this problem are miles for the position and hours for time and we start
plotting some points on the graph first when the time is equal to zero the car is sitting
at the origin that's an ordered pair 0 0. next we know that when t is equal to 2 hours the car
is 90 miles north of the origin or 90 miles in the positive direction and the scale on this graph is
10 miles for every grid space in the vertical direction so just as an intermediate point 50
miles is right there five grid space is up and 90 miles is nine grid spaces up on the vertical axis
and our grid spacing is one hour per space on the horizontal axis and so a time of two hours
is two grid spaces to the right of the origin so when we get to t equals two hours x is equal to
ninety miles and we get our second point on the graph again that's an ordered pair
with the time coordinate coming first that's two hours and the position coordinate coming second
that's 90 miles finally we know that when t is equal to five hours
so it's three hours later the car is back at the origin it's moved 90 miles back to the
south and it's where it began and that's an ordered pair of five hours comma zero miles
and because these motions occur at a constant rate we can connect these points with straight lines to
obtain the position time graph for the motion of the car next we start to think what does the slope
mean on this graph and there are two different legs of the journey to analyze in the first leg of
the journey as the car moves 90 miles to the north in two hours we can compute the rise for that in
other words the vertical change on the graph and that's plus 90 miles and we compute the run for
that in other words the horizontal change on that leg of the journey and that's plus two hours and
then of course the slope is given by the rise of a run and we're careful to keep the units on this
because it has a physical interpretation that's 90 miles over 2 hours and i end up with a positive
45 miles per hour and i'm going to go ahead and write the plus sign out in front
because that's indicating the direction that the car was moving i get a positive slope on the
position time graph because the car was moving to the north so this is telling us a velocity for the
car in other words a speed and a direction and in fact it's an average velocity on this leg of the
journey if we started this over and i asked what's the average speed of a car moving 90 miles in two
hours of course we would say it's 45 miles per hour and then the plus sign gives the northward
direction here so it's a velocity so as the car works its way back to the original position
i'm going to call that a rise of negative 90. so i started at 90 and ended at zero
and the run for that was plus three hours and again we're careful to keep units on the rise
in the run because they matter and i end up with negative 30 miles per hour and again when we think
about a car moving 90 miles in three hours of course we think that's an average speed of 30
miles per hour and then the minus sign there gives you the direction so this is a velocity
and this gives us our first major conclusion of the video and this is worth remembering so
i'll put it in a box the slope between two points on an xt graph gives us the average
velocity next we consider a velocity time graph for the same example a velocity time graph tells
us the velocity of an object as a function of time for some object moving in one dimension well we
know the car moves positive 45 miles an hour for two hours and then negative 30 miles per hour for
three hours so we plot these constant velocity segments as horizontal lines on the vt graph
like this i'm calling each vertical grid space 10 miles an hour here so 45 is right about here
and that's constant for two hours in other words we stay at exactly the same velocity value
as t goes from zero to two and that gives me a horizontal line then the car turns around and
takes a negative velocity value that's negative 30 miles per hour that starts at t equals two
and goes all the way to t equals five and it might be nice to connect those with a dotted line
next we're asked to interpret the signed area bounded by the graph in other words the area
bounded with respect to the t axis where it's considered negative if the function is below
the axis and positive if it's above all right these areas are just rectangles and i'll just
label them as a1 and a2 for the different parts of the journey and then we'll calculate the area a1
and you just take the width of this thing and multiply by the height but we're going to keep
the units on it so the width for a1 is just 2 hours and i multiply that by the height which
was 45 miles per hour and the hours are going to cancel out of those units and i get 90 miles
but what's the physical interpretation of that that's the actual displacement of the car
during that first leg of the journey it moved 90 miles to the north we do the same thing for a2
that area has a width of 3 hours and then we'll take care to keep the minus sign on the velocity
here so the height of that rectangle is 30 miles per hour but it takes a minus sign because it's a
negative velocity or i could say because the graph is below the t axis the hours cancel and i end up
with a negative 90 miles and again i'm looking at the displacement of the car on that time interval
we went 90 miles to the south and this gives us our second big conclusion of this video
and this is a big deal so it also gets a box the signed area bounded by a velocity time graph
that gives you the displacement for an object if you enjoyed this video or at least found it useful
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