Let's consider a one
dimensional motion that
has a non-uniform acceleration.
What we'd like to
do is explore how
do you differentiate
position functions,
to get velocity functions, to
get acceleration functions.
So what we're going to
consider is a rocket.
So I'm going to choose
a coordinate system y.
And here's my rocket.
And I have a function y of t.
And I'll have a j-hat
direction, but this will
be a one dimensional motion.
Now I want to express while
the rocket is thrusting upwards
and the engine is burning, we
can describe a function y of t
to be equal to 1/2
a constant a naught
minus the gravitational
acceleration, times t squared.
And we're going to have a
separate term here, which
is minus 1 over 30.
And you'll see where
this 30 comes in
as we start to differentiate.
The same constant a naught,
t to the 1/6 over t naught
to the 1/4.
Now in this expression, a
naught is bigger than g.
And also, this is
only true, this
holds for the time
interval 0 less than
or equal to t,
less than t naught.
And at time t equals to t
naught, the engine shuts off.
And at that moment,
our expectation
will be that the y component
of the acceleration
should just be minus g, for
t greater than t naught.
So now let's calculate
the acceleration
as the velocity and the
position as functions of time.
So the velocity--
in each case, we're
going to use the
polynomial rule.
So the y component
of the velocity
is just the derivative of t
squared, which is just 2t.
And so we get a naught
minus g times t.
And when we differentiate
t at the 1/6, the 6 over 30
gives us factor 1 over 5.
So we have minus 1 over 5
times a naught, t to the 1/5
over t naught to the 1/4.
And this is a combination
of a linear term and a term
that is decreasing by
this t to the 1/5 factor.
And finally, we now take the
next derivative, ay of t,
which is d dy dt.
I'll just keep functions of t,
but we don't really need that.
And when we differentiate
here, we get a naught minus g.
Now you see the
5s are canceling,
and we have minus a naught
t to the 1/4 over t naught
to the 1/4.
Now at time t equals t
naught, what do we have?
Well, ay at t equals t naught.
This is just a factor
minus a naught.
Those cancel, and we get minus
g, which is what we expected.
Now this is a
complicated motion.
And let's see if we can
make a graphical analysis
of this motion.
So let's plot y as
a function of t.
Now notice we have a
quadratic term and a factor t
to the 1/6 with the minus sign.
So for small values of t, the
quadratic term will dominate.
But as t gets larger, then
the t to of the 1/5 term
will dominate.
That's t squared.
And let's call t
equal to t naught.
Now we have to be a
little bit careful.
Because when the
engine turns off,
the rocket is still
moving upwards.
So even though it starts
to grow like this,
it will start to still
fall off a little bit,
due to this t to the 1/6 term.
It has a slope that
is always positive.
So we're claiming that our
velocity term is positive.
And then somewhere, if
the engine completely
didn't turn off, this
term would still--
where is the point where the
velocity, the vertical velocity
is 0, because gravity will--
this term will eventually
dominate.
And that, we can see, is going
to occur at some later time,
even though that's not
part of our problem.
Now in fact, if we
want to define, just
to double check that,
where the y of t equals 0,
then we have a
naught minus g over t
equals 1/5 a naught t to the
1/5 over t naught to the 1/4.
And so we have the quadratic--
we have this equation,
a naught minus g times t naught
to the 1/4 equals t to the 1/4.
Or t equals 5 times
a naught minus g,
a quantity bigger than
1, times t naught.
This quantity is larger than 1.
And so we see that
when given this motion,
the place where the
velocity reaches--
the position reaches its
maximum would occur after t
equals t naught.
So this graph looks reasonable.
And that would be the
plot of the position
function of the rocket
as a function of time.
As an exercise, you may want to
plot the velocity as a function
of t 2, to see how that looks.
That would correspond to
making a plot of the slope
of the position function.
