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Now that we've described
the instantaneous velocity

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of an object, we'd
now like to describe

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what it's acceleration is.

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Philosophy and acceleration
are complicated things.

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One of the reasons
why acceleration

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is such an abstract
quantity is for example,

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if I take this marker
that's in my hand

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and I throw it up in
the air and I catch it,

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you can see with
your eye, velocity.

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You can see the speed.

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You have a sense for it.

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You can see it
change its direction.

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So velocity is something
that as a human being,

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with our eye as a mechanism for
determining motion-- velocity

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is a very basic concept.

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But acceleration is a
much harder concept.

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For instance, when I
take that same object

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and I drop that--
as it falls down,

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its velocity is increasing.

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And so we say it
has an acceleration,

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but is that acceleration
a function of time?

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Or is it constant in time?

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Is it varying in time?

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Is it a polynomial?

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Is it a constant?

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Those are much harder to see.

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Again, let's drop this object.

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So when I drop that object
and I catch it down here,

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our eye does not tell
us intuitively anything

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about the mathematical
nature of that acceleration.

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And that's why acceleration is
a much more abstract quantity.

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In fact, a great question
is the following.

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When I throw this object
up in the air-- let's

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focus at its highest point.

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So I throw it in the
air and it comes down.

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When the object was
at its highest point,

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was its velocity zero?

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Was its acceleration
zero or non-zero?

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A hint to this question
is in Newton's second law,

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F equals ma.

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If there is a force
acting on this object,

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there must be an acceleration.

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And that forces the
gravitational force.

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And when this object was
at the top of its motion

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I didn't turn the earth off.

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So there is still a
gravitational force.

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And so for an object in
free fall-- now free fall

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just as under the-- which
means under the influence

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of the gravitational force.

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We can conclude from
Newton's second law

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that the gravitational
force, ma--

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so this gravitational
force is non-zero.

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And so therefore, the
acceleration is also non-zero.

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So once again, as I
throw this object up

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in the air nobody turned gravity
off, we're not in empty space,

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and near the
surface of the Earth

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the gravitational
force is not zero.

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The acceleration is not zero.

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So throughout this motion
this object is accelerating.

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I choose this example to
show you that acceleration

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is a very abstract concept.

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And now our goal
is to figure out

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how to mathematically
define acceleration

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in terms of our velocity.