GRAPHING WITH TANGENT LINES

Graphing Non-Uniform Motion and Using Tangent Lines to Find The Slope of a Curve at a Particular Moment in Time.

Now let's use some real world data courtesy of NASA.  Checkout the photo of the space shuttle taken from the orbiting International Space Station:

The space shuttle goes really, really fast. After launch, it goes from standing still to way faster than a speeding bullet in less than a minute.

However, the shuttle's velocity is definitely not constant.  Let's graph the following data: (NOTE:  Remember that the slope of a displacement vs time graph is the velocity of the object)

Time (sec) Altitude (meters)
0
0
10
241
20
1,244
30
2,872
40
5,377
50
8,130
60
11,617
70
15,380
80
19,872
90
25,608
100
31,412
110
38,309
120
44,726

We find that graphing that data shows not a straight line, but a curve.  What does that tell us?

Recall that a straight line on a displacement vs time graph indicates uniform motion (constant speed in a single direction).  However, the space shuttle's velocity right after launch is changing very rapidly.  It is in fact, increasing every second which is a great example of acceleration which is very definitely non-uniform motion.

SLOPES

Remember, the slope of a displacement vs time graph is velocity.  Also, the slope of a line is defined to be how much the Y-axis values change in relation to the change in the X-axis values (we often refer to this as the "rise over the run" and more mathematically as Δy/Δx).

However, in the graph above, we aren't working with a straight line but instead we have a curve, so we can't talk about the slope of the curve because the curve is changing all the time.

So, what we do is to draw a straight line that is tangent or our best match to what the curve is doing at a particular instant in time.  Once we have such a straight line, we can get the slope of that line.  And since we know that slope of a line on a displacement vs time graph is velocity, we can determine the velocity of that object at that instant in time.  We call that instantaneous velocity.

Now, let's plot a few such points using that graph begriming at time = 30 seconds.  Notice the red line in the following graph. As close as we can, we try to make it match what the curve is doing at just that point (time=30 seconds).

Keep in mind that although we are making our best approximation, that the line isn't exact.  That is just fine.

Now let's find the slope of that red line.  We can take any two points on that line as reference and in fact we could make that line any length we want to make it easiest for ourselves.  Two points stand out as being easy references, (60 seconds, 10,000 meters) and our current point, (30 seconds, 2500 meters). Notice that we ALWAYS include the units when graphing in physics.

Notice also that we have to estimate the displacement at time = 30 seconds, but again, this is an approximation so that is just fine.

Now that we have two points we can calculate the slope of that line, and since the slope of a line on a displacement vs time graph is velocity, we will have the velocity of the line tangent to the graph at t=30 seconds, or the instantaneous velocity at time = 30 seconds:

The change in displacement is:  10,000 m - 2500 m = 7500 m

The change in time is: 60 sec - 30 sec = 30 sec

So the change in Y / Change in X = 250 m/sec.

Let's try again at time = 60 seconds.

Notice the green line is somewhat steeper (sorry it is kind of faint, my software kind of surprised me there).  Using the same method as above, we choose two points on the new line-- this time at time = 70 (70, 15,000) and time = 40 (40, 5000)

The change in Y is:  15,000 m - 5000 m = 10,000 m

The change in X is: 70 sec - 40 sec = 30 sec

So the change in Y / Change in X = 333 m/sec.

Let's try another at time = 90 seconds.

Using the same method as above, we choose two points on the new (yellow) line-- this time at time = 80 (80, 20,000) and time = 100 (100, 30,000)

The change in Y is:  30,000 m - 20,000 m = 10,000 m

The change in X is: 100 sec - 80 sec = 20 sec

So the change in Y / Change in X = 500 m/sec.