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
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
separated by a
and calculate the velocity.
We note that
When we actually use the values of
at the two time points
in this expression and then simplify, we find that
To summarize, we
have found that the average velocity between time
is given by
. This means that it depends on:
- the time
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
- the constant
. (We hinted that this is the acceleration.
On earth, the acceleration due to gravity is
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
between the points we chose.
- How is the average velocity influenced by the constant
you are on a planet on which
is twice as large,
how does the average velocity of the ball compare to that on Earth.
- What would happen to the average velocity we calculate when the
between the points (or between the strobe
- Argue that if we take measurements very close together---that is,
have a very small value of
the average velocity we get becomes closer and closer to the limiting
We call this the instantaneous velocity (or simply the
velocity) of the ball at time
- Draw a graph of the Velocity as a function of time. You may wish
to assume that
- Refer to the page on rates of change and slopes
and discuss the connection between average velocity and average slope
with a friend.
- 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?
- On the graph you made in part (4), indicate with a sketch, what is
meant by the average velocity and by the Velocity.
- 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?