Video Transcript
In this video, we will practice
using distance–time graphs to compare the speeds of objects. We see here that those objects can
be runners in a race. And to get started, imagine that
instead of a runner, we have a walker moving along a path. To understand the walker’s motion,
we seek to measure the distance the walker travels and the time the walker takes to
travel those distances. Starting our stopwatch at zero
seconds, we can say this represents the beginning of the walker’s movement. And so at this time, the walker has
traveled zero meters of distance. After one second has passed though,
the walker has covered some ground. At this instant, we’ll say the
walker is one meter from where they began.
Then, say we wait another second
and after two seconds have passed once again measure the distance traveled by the
walker. Say that total distance is now two
meters. And we allow time to keep going
with the walker moving at the same rate. After three seconds, the walker has
covered three meters and after four seconds, a total distance of four meters. The values we recorded here come in
pairs; in each pair, there’s one value of time and one value of distance. It is possible to understand the
walker’s motion by looking at a table of data values like this. But this is often easier to do if
we plot these values on a distance–time graph.
This sort of graph has several
parts to it. First, there are the axes. In general, graphs have a vertical
axis, an axis that goes up and down, and a horizontal axis that goes left and
right. Each axis on a graph needs a
label. The label tells what that axis
shows or represents. A distance–time graph has distance
on the vertical axis and time on the horizontal. Distance and time are
variables. Since we decided to measure the
distance the walker traveled over every second of time passed, the variable that
goes on the horizontal axis is time rather than the other way around.
Even though we now have a distance
and a time axis, we’re not yet able to plot the values that we measured for our
walker on this graph. That’s because currently our graph
is unscaled. That means we haven’t assigned an
orderly progression of time and distance values to these axes. We have measured five pairs of
values called data points, but with unscaled axes, we don’t know where to plot or
locate these values on the graph. So, let’s add scales to our
distance and time axes.
First, notice that all of our time
values have units of seconds and all of our distance values have units of
meters. On our axes labels, we’ll add these
units: seconds for time and meters for distance. This tells us that any time value
that appears on our graph will have units of seconds, and likewise any distance will
have units of meters. Once we’ve added these units to our
labels, we don’t need to write these units again. Let’s first make a scale for our
time axis. And we know that that scale begins
at the origin here. We make a little vertical mark on
the axis, and then we indicate this value corresponds to a time of zero seconds.
Looking back at our data, we see
that the other time values are one, two, three, up to four seconds. We want the scale on our time axis
to extend at least to four seconds. What we’ll do is evenly space out
marks along this axis starting from the origin. The first mark past the origin
corresponds to a time of one second, the next mark to a time of two seconds, and so
on. It’s important that the distance
between each of these marks is the same. We do this to ensure that each
second of time is represented by the same length on the graph.
Now that our time axis has a scale
that covers all of the data we measured, let’s do the same thing for our distance
axis. The scale for distance will also
begin here at the origin. This point corresponds to a
distance of zero meters. The walker’s distance values go
from zero up to four meters, so we want our scale to be able to cover these
distances. We make evenly spaced marks. The first mark corresponds to a
distance of one meter, the second to a distance of two meters, and so on up to four
meters.
Now that our distance and time axes
have scales, we’re able to plot the data points that we measured. As we’ve seen, there are five data
points. The first one is here. At a time of zero seconds, the
walker had moved zero meters. We’re going to locate this pair of
values on our graph. We do this using the scales we’ve
drawn in. First, we find a time of zero
seconds. Looking on the time axis, we see
that occurs at this point in orange. Then, we look for a distance of
zero meters. That occurs at the same point,
which is the origin of our graph. Therefore, this orange dot here
represents this data point of zero seconds of time and zero meters of distance.
What we’re going to do is represent
each of our five data points this way. We move on to the next point where
the walker at one second of time had traveled one meter of distance. To plot this point on our graph, we
first find where one second of time is on the time axis. That time is located right here on
the axis. Next, we look for the distance of
one meter along the distance axis. On the distance scale, that
distance of one meter is here. Notice that these two points we’ve
drawn in don’t lie on top of one another like they did for our first data point.
To find out where our second data
point with a time of one second in a distance of one meter is actually located on
our graph, we sketch a horizontal line out from our vertical axis value of one meter
and then a vertical line from our horizontal axis value of one second. The place where these two lines
intersect is where we plot our data point. This is the only point on our graph
that has a time value of one second along with a distance value of one meter. Now that we know the intersection
point of these two lines, we’ll erase the lines and just leave the data point
itself.
We’re now ready to plot our third
data point. To do this, we first locate a time
value of two seconds on our horizontal axis scale. That’s located here. We then do the same thing for the
distance value of two meters on the distance scale. That distance is right here. Then once more extending a
horizontal line out from our distance value and a vertical line from our time value,
we locate our data point at the intersection of these lines. This is the only point on the graph
that has a time value of two seconds while also having a distance value of two
meters.
Moving on to our next data point,
we look on the time axis for a time of three seconds, that’s right here, and then on
the distance axis for a distance of three meters. The place where the horizontal and
vertical lines from these locations meet is where our data point is located. Finally, let’s plot our last data
point on this graph. A time of four seconds is here on
the time axis, and a distance of four meters is here on the distance axis. Horizontal and vertical lines from
these locations cross here. This is the one and only point on
this graph that has a time value of four seconds and a distance value of four
meters.
Now that we’ve plotted all our data
points, notice that we just have these five points on our graph. This information is good if we want
to know the distance that the walker covered after, say, one second or three seconds
had elapsed. But what if we wanted to know how
far the walker had traveled after, say, two and a half seconds? That is, what distance value would
correspond to a time value of 2.5 seconds? Our data don’t directly tell us the
answer, but we can get a good approximation using what are called trend lines.
To make a trend line, we simply
draw a line that fits all of our data points. With this line in place, if we
wanted to know the distance the walker had covered at a time of two and a half
seconds, we would draw a vertical line from that time value up until it meets the
trend line. Then, from that point, we draw a
horizontal line over to the distance axis. The point where this line crosses
the distance axis is a good approximation for how much ground the walker covered
after two and a half seconds had passed. Trend lines on distance and time
graphs are really quite useful. Indeed, the slope of the trend line
actually tells us the speed of our moving object.
To see why this is so, let’s recall
that speed is defined as distance traveled divided by time taken to travel that
distance. Our distance–time graph shows
distance over time, and therefore the slope of our trend line tells us the speed of
our moving object, in this case our walker. We can use this fact to compare
speeds. Say that we had a second walker and
that as this one moved along from zero to four seconds, we recorded corresponding
distances of zero, 0.5, one, 1.5, and two meters. Once again, we have five data
points we can plot on our distance–time graph. Plotting all five at the same time,
they would look like this. And we can connect these five
points with their own trend line.
Now, which of our two walkers is
moving at a greater speed? We can tell by comparing the slopes
of our two trend lines. The steeper the slope, the more
distance a given walker covers in some amount of time. Since the orange trend line is
steeper, we know the first walker is moving more quickly at a higher speed. The steeper the trend line on a
given distance–time graph, the greater the speed. One point worth mentioning is that
the way a trend line looks to our eye depends on the scale over which it’s
plotted.
Let’s clear space on the left side
of our screen. And in this space, we’ll plot the
same data points we have here on a distance–time graph with a different scale. Here, we once again have a vertical
distance axis and a horizontal time axis. We also have time values of zero,
one, two, three, and four seconds marked out on the same scale as we used
before. But notice that now our distance
values are spaced farther apart on this axis. They’re still the same distances as
before, zero up to four meters. But it’s just that on our graph,
the physical distance between these meter markings has increased.
When we plot our two sets of data
and their trend lines on our new distance–time graph, they look like this. The trend lines look steeper than
before, and they are. But these are the same sets of data
on each graph. The only reason for the difference
is that our distance scales between these two graphs have not been drawn the
same. Therefore, considering the scale of
an axis is important whenever we want to compare data between different graphs. At first glance, we might say, for
example, that this line is steeper than this and therefore represents a
faster-moving object. But when we look at the scales
involved, we see that actually they represent the same object. The data points are simply plotted
on unequal scales.
Notice that both of the trend lines
on this graph are straight lines. We’ve seen that the slopes of these
lines represent the respective speeds of those moving objects. When a trend line is straight, as
they are here, that represents an object moving with a constant speed. Knowing all this about
distance–time graphs, let’s look at an example.
Which color line on the graph shows
the greatest speed?
This graph is a distance–time
graph. Distance, time, and speed are all
connected by a mathematical equation. If we take the distance that an
object travels and divide that by the time taken to travel that distance, then that
fraction equals the object’s speed. For a distance–time graph then, the
slope of a line on the graph is equal to the speed of a given object. The steeper the slope, the greater
the speed. Notice that the distance and time
axes on this graph are unscaled; that is, we don’t know the time or distance value
at any given point on the graph. Here though, we’re just looking for
which of our three lines shows the greatest speed, that is, the greatest compared to
the other two.
We can answer this question without
knowing specifically what that greatest speed is. As we said, since objects’ speed on
a distance–time graph is equal to the slope of a line, the steepest of these three
lines, red, blue, and green, corresponds to the greatest speed. All three of the lines begin at
zero at the origin, but it’s the green line that travels the greatest distance in
the least amount of time. Visually, this means it has the
steepest slope. And therefore, it shows the
greatest speed. On this unscaled distance–time
graph, the green line shows the highest speed. As a side note, because these three
lines are all straight, they represent objects each moving with a constant
speed.
Let’s finish our lesson by
reviewing a few key points. In this video, we learned that a
distance–time graph can display distance–time data from a table. A straight line drawn through these
data points is called a trend line. The slope of a trend line on a
distance–time graph indicates objects’ speed. Lastly, a steeper trend line on a
distance–time graph indicates greater speed. This is a summary of distance–time
graphs.
