If you toss a rock at an initial height with an initial velocity
then its height after seconds is given by the
formula where on Earth. (On other celestial
bodies would be different. The minus sign is due to the
convention that the positive direction is up and gravity pulls down.)
You have probably seen this formula before, and chances are you
were simply told that this is the way it is. With Calculus, we can
make perfect sense of this formula. The underlying observation is the
experimentally observed fact that, ignoring air resistance, on Earth a
free falling object increases its speed by 32 feet per second every
second. The velocity of the object is the derivative of height, and the acceleration is the derivative of velocity. So we have to work out that
the formula given above has the right properties.
If is given by the above expression, then
.
The velocity is
and
.
Moreover, the acceleration is
.