Rates of change and the slope of a curve
So what is the derivative, after all?
UBC Calculus Online Course Notes

Velocity of the Falling Ball

For a final interpretation of the derivative, let's return to our earlier discussion of the falling ball. Remember that we looked at a falling ball at various times using a strobe light and that we found pictures like this:

Now to represent our data on a graph, we see the following picture:

Don't be confused by the fact that the ball appears to be moving upwards now---we are simply representing the data in a different way. The quantity $  y  $ measures the distance the ball has fallen downwards. So a positive value of $  y  $ (as seen in the graph above) indicates that the ball is below where it started.

What is velocity?

Our intuitive notion of velocity suggests that this quantity measures how fast the ball is falling. However, we would like to measure this quantitatively so we need to be a little more careful. In the case of our ball, we could compute an average velocity by finding how far the ball moved between stroboscopic snapshots (just by subtracting succesive values of y) and dividing by the time interval between the strobe flashes. That is,

Average velocity = = distance travelled/time taken

Notice that the average velocity we get will depend on the particular data points we use to compute this average. For example, we could use two positions of the ball that show up when the strobe light flashed consequtively.

The time interval between such flashes might be a fraction of a second. This will give us one set of values for the average velocity (and how it changed over the course of the trajectory.)

If we repeat the same experiment, but make the strobe flash twice as often, we will have more pairs of values $ (y,t) $ in the graph, and we will be able to compute the average velocity over the now smaller time intervals.

  1. Sketch what the graph of the average velocity of the ball might look like if we used the position of the ball at succesive strobe flashes to calculate this average velocity at every instant.

  2. Sketch a second graph to show how the situation might change if the strobe flashed twice as fast.

  3. The average velocity you are computing is an average rate. Explain why this average velocity is the slope of the line drawn between the two data points on the graph of $  y  $ versus $  t  $ .

Note: A good solution to this problem does not require any calculations. You are not meant to use derivatives or Calculus as yet! You should only use the strobe picture, your imagination, and a bit of common sense. We are not looking for an accurate picture, only for a real rough sketch.

Finding instantaneous velocity

Remember our discussion of the "two faces of functions"? We will now switch from the world of the scientist who makes a finite number of measurements at discrete points, and the world of the mathematician who calls up "idealized" representations of functions: nice and smooth, prescribed by a simple formula, and thus "easy" to live with.

It is a bit amazing to discover that (if the wind is relatively still on this particular day at the park, and the ball is pretty smooth and other imperfections of the data are neglected) the picture we get is nicely described by the simple function

\[  y = \frac 12 a t^2  \]

Here, $ a $ is just a constant. (Indeed, we will find that this constant is precisely the acceleration of the ball, though we have not yet discussed this concept in detail.) Why should this be true? It was known by Gallileo that this type of graph is a parabola. You might remember his famous experiments on falling objects in which the Leaning Tower of Piza played a prominent role! Among other things, he concluded that the size of the object does not influence the time it takes to fall. (A fact that many first year physics students often forget.)

Now that we have an idealized description of the ball's trajectory, we can discuss various properties of the motion such as its velocity.


What do you think would happen if you really dropped a ball and made the type of measurements we are discussing? Would the distance it dropped actually be related to the time by the above simple mathematician's formula? Why or why not?

To calculate the average velocity, we might take two points on the graph of

\[  y = \frac 12 a t^2,  \]

separated by a time interval $ h $ and calculate the velocity.

We note that

  • At time $ t $ , the ball has fallen by a distance $ 
y(t) = \frac 12 a t^2 $ .
  • A little later, at time $ t+h $ , the ball has fallen by a distance $  y(t+h) = \frac 12 a (t+h)^2  $ .
  • The time between these two data points is $ h $ , which we may think of as the time between strobe flashes.
  • Therefore, the average velocity between these two instants is:

    \[  \mbox{Average Velocity } = \frac{y(t+h)-y(t)}{h}. \]

When we actually use the values of $ y $ at the two time points in this expression and then simplify, we find that

\mbox{Average Velocity } & = & 
 \frac 12 \frac{a (t+h)^2 - a t^2}{h} \\ 
 & = & \frac a2 \frac{t^2 + 2th + h^2 -t^2}{h} \\ 
 & = & \frac a2 \frac{ 2th + h^2}{h} \\ 
 & = & at + \frac a2 h 

To summarize, we have found that the average velocity between time $  t  $ and $  t + h  $ is given by $  at + \frac a2 h  $ . This means that it depends on:

  • the time $  t  $ at which it is calculated. (We knew this already! We have noticed that the ball starts out slow and accelerates to a faster velocity as it drops.)
  • the constant $  a $ . (We hinted that this is the acceleration. On earth, the acceleration due to gravity is $  a=9.8  $ meters per second per second, but a ball falling on the moon would experience a lower gravitational force and would not speed up as quickly.)
  • the time $  h  $ between the points we chose.

  1. How is the average velocity influenced by the constant $ 
a  $ ? If you are on a planet on which $  a $ is twice as large, how does the average velocity of the ball compare to that on Earth.

  2. What would happen to the average velocity we calculate when the time interval $  h  $ between the points (or between the strobe flashes) decreases?

  3. Argue that if we take measurements very close together---that is, have a very small value of $  h  $ ---then the average velocity we get becomes closer and closer to the limiting value of $  at  $ .

    We call this the instantaneous velocity (or simply the velocity) of the ball at time $  t  $ .

    \[ \mbox{The Velocity of the ball at time } t  =  a t 

  4. Draw a graph of the Velocity as a function of time. You may wish to assume that $  a=9.8m/s^2  $ .

  5. Refer to the page on rates of change and slopes and discuss the connection between average velocity and average slope with a friend.

  6. Refer to the same page and discuss the connection between velocity and the slope of a tangent line with a friend. Which tangent line would velocity represent?

  7. On the graph you made in part (4), indicate with a sketch, what is meant by the average velocity and by the Velocity.

  8. Explain why the instantaneous velocity of a falling ball is just another incarnation of the derivative which we have seen by looking at a graph under a microscope and by computing average rates of change.

The kind of problem we have been studying was considered by many thinkers from the earliest days of science. It was Galileo, however, who first described motion in a way similar to our discussion. He found out a lot about the motion of objects before Calulus was ever invented. The Life of Gallileo is interesting. Why not explore a bit on the web and see what else you can find out?

See one of us or drop us an email (David Austin, Leah Keshet) and tell us which parts are unclear. Chances are, others will benefit from your comments as well.