U1S6V14 Acceleration.txt

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One of the most important applications
of the second derivative is when we're dealing with motion.
We know if f of t is position as a function of time,
then the derivative f prime of t is velocity.
And that's the rate of change of position.
If we take the derivative of that,
we're going to get f double prime.
We'd be measuring how fast the velocity is changing,
and this rate of change of velocity
is called acceleration.
Let's examine this with our old friend, the pumpkin.
Here f of t is measuring height, and the graph
of f of t well it goes up and then down as we throw,
the pumpkin.
What happens with the graph of velocity?
Well, we start with an initial upward or positive velocity
at time zero.
And then as the pumpkin goes up, the velocity gets slower.
And then the velocity is 0 when the pumpkin reaches the top.
And then the velocity is negative, and then more
negative.

Now acceleration is the derivative of this function.
And we see that the velocity is just
decreasing the entire time.
It goes from positive, to 0, to negative.
So its derivative will always be negative, that's acceleration.
And the minus sign means that the acceleration
is in the downward direction.

What does that mean practically?
Well, in the first part of the trajectory
the velocity, which we'll denote with v, that's upwards.
And we said that the acceleration a is downward.
So that's the opposite direction from the velocity.
And the opposite direction means that the pumpkin
is decelerating, it's slowing down.
In the second part of the trajectory,
the velocity is downward and the acceleration is still downward.
So they're together.
They're pointed in the same direction,
and that means that the pumpkin is gaining more
and more downward velocity.
It's accelerating downward.
And that is our second derivative.