Now that we've described the instantaneous velocity
of an object, we'd now like to describe
what it's acceleration is.
Philosophy and acceleration are complicated things.
One of the reasons why acceleration
is such an abstract quantity is for example,
if I take this marker that's in my hand
and I throw it up in the air and I catch it,
you can see with your eye, velocity.
You can see the speed.
You have a sense for it.
You can see it change its direction.
So velocity is something that as a human being,
with our eye as a mechanism for determining motion-- velocity
is a very basic concept.
But acceleration is a much harder concept.
For instance, when I take that same object
and I drop that-- as it falls down,
its velocity is increasing.
And so we say it has an acceleration,
but is that acceleration a function of time?
Or is it constant in time?
Is it varying in time?
Is it a polynomial?
Is it a constant?
Those are much harder to see.
Again, let's drop this object.
So when I drop that object and I catch it down here,
our eye does not tell us intuitively anything
about the mathematical nature of that acceleration.
And that's why acceleration is a much more abstract quantity.
In fact, a great question is the following.
When I throw this object up in the air-- let's
focus at its highest point.
So I throw it in the air and it comes down.
When the object was at its highest point,
was its velocity zero?
Was its acceleration zero or non-zero?
A hint to this question is in Newton's second law,
F equals ma.
If there is a force acting on this object,
there must be an acceleration.
And that forces the gravitational force.
And when this object was at the top of its motion
I didn't turn the earth off.
So there is still a gravitational force.
And so for an object in free fall-- now free fall
just as under the-- which means under the influence
of the gravitational force.
We can conclude from Newton's second law
that the gravitational force, ma--
so this gravitational force is non-zero.
And so therefore, the acceleration is also non-zero.
So once again, as I throw this object up
in the air nobody turned gravity off, we're not in empty space,
and near the surface of the Earth
the gravitational force is not zero.
The acceleration is not zero.
So throughout this motion this object is accelerating.
I choose this example to show you that acceleration
is a very abstract concept.
And now our goal is to figure out
how to mathematically define acceleration
in terms of our velocity.