WEBVTT
00:00:04.630 --> 00:00:07.040
Let's consider a one
dimensional motion that
00:00:07.040 --> 00:00:09.910
has a non-uniform acceleration.
00:00:09.910 --> 00:00:11.530
What we'd like to
do is explore how
00:00:11.530 --> 00:00:13.800
do you differentiate
position functions,
00:00:13.800 --> 00:00:17.270
to get velocity functions, to
get acceleration functions.
00:00:17.270 --> 00:00:19.230
So what we're going to
consider is a rocket.
00:00:19.230 --> 00:00:21.970
So I'm going to choose
a coordinate system y.
00:00:21.970 --> 00:00:23.350
And here's my rocket.
00:00:23.350 --> 00:00:26.380
And I have a function y of t.
00:00:26.380 --> 00:00:29.150
And I'll have a j-hat
direction, but this will
00:00:29.150 --> 00:00:32.000
be a one dimensional motion.
00:00:32.000 --> 00:00:36.280
Now I want to express while
the rocket is thrusting upwards
00:00:36.280 --> 00:00:41.100
and the engine is burning, we
can describe a function y of t
00:00:41.100 --> 00:00:45.626
to be equal to 1/2
a constant a naught
00:00:45.626 --> 00:00:49.065
minus the gravitational
acceleration, times t squared.
00:00:49.065 --> 00:00:51.190
And we're going to have a
separate term here, which
00:00:51.190 --> 00:00:52.950
is minus 1 over 30.
00:00:52.950 --> 00:00:54.540
And you'll see where
this 30 comes in
00:00:54.540 --> 00:00:56.210
as we start to differentiate.
00:00:56.210 --> 00:01:01.360
The same constant a naught,
t to the 1/6 over t naught
00:01:01.360 --> 00:01:02.570
to the 1/4.
00:01:02.570 --> 00:01:06.960
Now in this expression, a
naught is bigger than g.
00:01:06.960 --> 00:01:09.580
And also, this is
only true, this
00:01:09.580 --> 00:01:12.870
holds for the time
interval 0 less than
00:01:12.870 --> 00:01:16.190
or equal to t,
less than t naught.
00:01:16.190 --> 00:01:25.680
And at time t equals to t
naught, the engine shuts off.
00:01:25.680 --> 00:01:27.610
And at that moment,
our expectation
00:01:27.610 --> 00:01:30.800
will be that the y component
of the acceleration
00:01:30.800 --> 00:01:37.759
should just be minus g, for
t greater than t naught.
00:01:37.759 --> 00:01:41.910
So now let's calculate
the acceleration
00:01:41.910 --> 00:01:46.030
as the velocity and the
position as functions of time.
00:01:46.030 --> 00:01:48.020
So the velocity--
in each case, we're
00:01:48.020 --> 00:01:49.810
going to use the
polynomial rule.
00:01:49.810 --> 00:01:52.180
So the y component
of the velocity
00:01:52.180 --> 00:01:56.420
is just the derivative of t
squared, which is just 2t.
00:01:56.420 --> 00:02:00.990
And so we get a naught
minus g times t.
00:02:00.990 --> 00:02:05.340
And when we differentiate
t at the 1/6, the 6 over 30
00:02:05.340 --> 00:02:07.090
gives us factor 1 over 5.
00:02:07.090 --> 00:02:13.620
So we have minus 1 over 5
times a naught, t to the 1/5
00:02:13.620 --> 00:02:15.960
over t naught to the 1/4.
00:02:15.960 --> 00:02:19.910
And this is a combination
of a linear term and a term
00:02:19.910 --> 00:02:23.260
that is decreasing by
this t to the 1/5 factor.
00:02:23.260 --> 00:02:27.079
And finally, we now take the
next derivative, ay of t,
00:02:27.079 --> 00:02:31.100
which is d dy dt.
00:02:31.100 --> 00:02:34.160
I'll just keep functions of t,
but we don't really need that.
00:02:34.160 --> 00:02:38.350
And when we differentiate
here, we get a naught minus g.
00:02:38.350 --> 00:02:40.750
Now you see the
5s are canceling,
00:02:40.750 --> 00:02:45.350
and we have minus a naught
t to the 1/4 over t naught
00:02:45.350 --> 00:02:46.840
to the 1/4.
00:02:46.840 --> 00:02:52.050
Now at time t equals t
naught, what do we have?
00:02:52.050 --> 00:02:56.660
Well, ay at t equals t naught.
00:02:56.660 --> 00:02:58.460
This is just a factor
minus a naught.
00:02:58.460 --> 00:03:03.730
Those cancel, and we get minus
g, which is what we expected.
00:03:03.730 --> 00:03:05.860
Now this is a
complicated motion.
00:03:05.860 --> 00:03:08.960
And let's see if we can
make a graphical analysis
00:03:08.960 --> 00:03:09.810
of this motion.
00:03:09.810 --> 00:03:14.580
So let's plot y as
a function of t.
00:03:14.580 --> 00:03:17.720
Now notice we have a
quadratic term and a factor t
00:03:17.720 --> 00:03:19.990
to the 1/6 with the minus sign.
00:03:19.990 --> 00:03:27.280
So for small values of t, the
quadratic term will dominate.
00:03:27.280 --> 00:03:33.579
But as t gets larger, then
the t to of the 1/5 term
00:03:33.579 --> 00:03:34.340
will dominate.
00:03:34.340 --> 00:03:36.670
That's t squared.
00:03:36.670 --> 00:03:40.270
And let's call t
equal to t naught.
00:03:40.270 --> 00:03:42.710
Now we have to be a
little bit careful.
00:03:42.710 --> 00:03:44.790
Because when the
engine turns off,
00:03:44.790 --> 00:03:47.070
the rocket is still
moving upwards.
00:03:47.070 --> 00:03:50.390
So even though it starts
to grow like this,
00:03:50.390 --> 00:03:55.430
it will start to still
fall off a little bit,
00:03:55.430 --> 00:03:57.770
due to this t to the 1/6 term.
00:03:57.770 --> 00:04:04.020
It has a slope that
is always positive.
00:04:04.020 --> 00:04:07.450
So we're claiming that our
velocity term is positive.
00:04:07.450 --> 00:04:11.740
And then somewhere, if
the engine completely
00:04:11.740 --> 00:04:17.200
didn't turn off, this
term would still--
00:04:17.200 --> 00:04:20.180
where is the point where the
velocity, the vertical velocity
00:04:20.180 --> 00:04:23.590
is 0, because gravity will--
this term will eventually
00:04:23.590 --> 00:04:24.870
dominate.
00:04:24.870 --> 00:04:29.260
And that, we can see, is going
to occur at some later time,
00:04:29.260 --> 00:04:34.050
even though that's not
part of our problem.
00:04:34.050 --> 00:04:36.530
Now in fact, if we
want to define, just
00:04:36.530 --> 00:04:42.159
to double check that,
where the y of t equals 0,
00:04:42.159 --> 00:04:45.690
then we have a
naught minus g over t
00:04:45.690 --> 00:04:51.360
equals 1/5 a naught t to the
1/5 over t naught to the 1/4.
00:04:51.360 --> 00:04:53.970
And so we have the quadratic--
we have this equation,
00:04:53.970 --> 00:05:02.130
a naught minus g times t naught
to the 1/4 equals t to the 1/4.
00:05:02.130 --> 00:05:07.010
Or t equals 5 times
a naught minus g,
00:05:07.010 --> 00:05:10.440
a quantity bigger than
1, times t naught.
00:05:10.440 --> 00:05:13.310
This quantity is larger than 1.
00:05:13.310 --> 00:05:18.720
And so we see that
when given this motion,
00:05:18.720 --> 00:05:21.465
the place where the
velocity reaches--
00:05:21.465 --> 00:05:25.085
the position reaches its
maximum would occur after t
00:05:25.085 --> 00:05:26.120
equals t naught.
00:05:26.120 --> 00:05:30.370
So this graph looks reasonable.
00:05:30.370 --> 00:05:35.830
And that would be the
plot of the position
00:05:35.830 --> 00:05:39.040
function of the rocket
as a function of time.
00:05:39.040 --> 00:05:43.880
As an exercise, you may want to
plot the velocity as a function
00:05:43.880 --> 00:05:47.280
of t 2, to see how that looks.
00:05:47.280 --> 00:05:50.540
That would correspond to
making a plot of the slope
00:05:50.540 --> 00:05:52.695
of the position function.