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Let's consider a one
dimensional motion that

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has a non-uniform acceleration.

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What we'd like to
do is explore how

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do you differentiate
position functions,

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to get velocity functions, to
get acceleration functions.

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So what we're going to
consider is a rocket.

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So I'm going to choose
a coordinate system y.

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And here's my rocket.

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And I have a function y of t.

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And I'll have a j-hat
direction, but this will

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be a one dimensional motion.

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Now I want to express while
the rocket is thrusting upwards

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and the engine is burning, we
can describe a function y of t

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to be equal to 1/2
a constant a naught

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minus the gravitational
acceleration, times t squared.

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And we're going to have a
separate term here, which

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is minus 1 over 30.

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And you'll see where
this 30 comes in

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as we start to differentiate.

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The same constant a naught,
t to the 1/6 over t naught

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to the 1/4.

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Now in this expression, a
naught is bigger than g.

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And also, this is
only true, this

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holds for the time
interval 0 less than

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or equal to t,
less than t naught.

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And at time t equals to t
naught, the engine shuts off.

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And at that moment,
our expectation

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will be that the y component
of the acceleration

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should just be minus g, for
t greater than t naught.

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So now let's calculate
the acceleration

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as the velocity and the
position as functions of time.

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So the velocity--
in each case, we're

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going to use the
polynomial rule.

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So the y component
of the velocity

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is just the derivative of t
squared, which is just 2t.

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And so we get a naught
minus g times t.

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And when we differentiate
t at the 1/6, the 6 over 30

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gives us factor 1 over 5.

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So we have minus 1 over 5
times a naught, t to the 1/5

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over t naught to the 1/4.

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And this is a combination
of a linear term and a term

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that is decreasing by
this t to the 1/5 factor.

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And finally, we now take the
next derivative, ay of t,

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which is d dy dt.

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I'll just keep functions of t,
but we don't really need that.

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And when we differentiate
here, we get a naught minus g.

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Now you see the
5s are canceling,

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and we have minus a naught
t to the 1/4 over t naught

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to the 1/4.

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Now at time t equals t
naught, what do we have?

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Well, ay at t equals t naught.

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This is just a factor
minus a naught.

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Those cancel, and we get minus
g, which is what we expected.

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Now this is a
complicated motion.

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And let's see if we can
make a graphical analysis

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of this motion.

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So let's plot y as
a function of t.

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Now notice we have a
quadratic term and a factor t

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to the 1/6 with the minus sign.

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So for small values of t, the
quadratic term will dominate.

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But as t gets larger, then
the t to of the 1/5 term

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will dominate.

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That's t squared.

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And let's call t
equal to t naught.

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Now we have to be a
little bit careful.

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Because when the
engine turns off,

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the rocket is still
moving upwards.

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So even though it starts
to grow like this,

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it will start to still
fall off a little bit,

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due to this t to the 1/6 term.

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It has a slope that
is always positive.

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So we're claiming that our
velocity term is positive.

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And then somewhere, if
the engine completely

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didn't turn off, this
term would still--

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where is the point where the
velocity, the vertical velocity

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is 0, because gravity will--
this term will eventually

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dominate.

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And that, we can see, is going
to occur at some later time,

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even though that's not
part of our problem.

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Now in fact, if we
want to define, just

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to double check that,
where the y of t equals 0,

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then we have a
naught minus g over t

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equals 1/5 a naught t to the
1/5 over t naught to the 1/4.

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And so we have the quadratic--
we have this equation,

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a naught minus g times t naught
to the 1/4 equals t to the 1/4.

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Or t equals 5 times
a naught minus g,

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a quantity bigger than
1, times t naught.

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This quantity is larger than 1.

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And so we see that
when given this motion,

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the place where the
velocity reaches--

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the position reaches its
maximum would occur after t

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equals t naught.

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So this graph looks reasonable.

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And that would be the
plot of the position

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function of the rocket
as a function of time.

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As an exercise, you may want to
plot the velocity as a function

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of t 2, to see how that looks.

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That would correspond to
making a plot of the slope

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of the position function.