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00:00:04,080 --> 00:00:07,460
So suppose we have a
point mass particle

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00:00:07,460 --> 00:00:12,560
of mass m moving with
a velocity vector,

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00:00:12,560 --> 00:00:15,540
v. We can introduce
a quantity we call

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00:00:15,540 --> 00:00:17,120
the momentum of that particle.

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00:00:21,690 --> 00:00:24,400
I'll label it with the symbol p.

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00:00:24,400 --> 00:00:28,810
And it's equal to the product
of the mass times the velocity.

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00:00:28,810 --> 00:00:30,960
This is something you've
undoubtedly seen before.

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00:00:30,960 --> 00:00:32,558
Now, let's think
about the dimensions

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00:00:32,558 --> 00:00:33,600
of momentum for a moment.

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00:00:33,600 --> 00:00:37,010
So dimensionally,
momentum has units

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00:00:37,010 --> 00:00:38,980
of mass times the velocity.

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00:00:38,980 --> 00:00:45,130
And so that's in SI units, the
units of mass are kilograms

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00:00:45,130 --> 00:00:52,490
and then the units of velocity
are meters per second.

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00:00:52,490 --> 00:00:54,800
You can also express
momentum dimensionally

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00:00:54,800 --> 00:00:58,920
as a product of a
force times time.

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00:00:58,920 --> 00:01:02,060
And so, again in SI units,
units force are the Newton

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00:01:02,060 --> 00:01:04,670
and the units of
time is the second.

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00:01:04,670 --> 00:01:07,210
So these are the SI units for
momentum, two different ways

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00:01:07,210 --> 00:01:10,970
of writing the same dimensions.

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00:01:10,970 --> 00:01:20,330
Now, we've seen that
for a single particle,

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we can write Newton's
second law as the force is

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equal to the mass
times the acceleration.

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00:01:29,550 --> 00:01:31,190
Or equivalently,
we can write that

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00:01:31,190 --> 00:01:34,630
as the mass times
the time derivative

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00:01:34,630 --> 00:01:37,970
of the acceleration, dv/dt.

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00:01:37,970 --> 00:01:51,350
Now, if m is a constant,
then I can rewrite this as F

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00:01:51,350 --> 00:02:00,960
is equal to the time derivative
of the mass times the velocity,

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00:02:00,960 --> 00:02:05,750
or equivalently as the time
derivative of the momentum,

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since mv is just equal to p.

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00:02:08,759 --> 00:02:10,600
So I actually want
to stress this much--

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I'm going to put it
in a box because it's

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00:02:12,308 --> 00:02:15,010
very important that I can
write Newton's second law,

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00:02:15,010 --> 00:02:17,520
instead of F equals ma,
as F equals the time

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00:02:17,520 --> 00:02:20,084
derivative of the momentum.

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00:02:20,084 --> 00:02:21,500
And this is
absolutely where we'll

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00:02:21,500 --> 00:02:23,600
see the momentum
becomes very useful.

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00:02:29,400 --> 00:02:32,760
Because it turns out that this
form of Newton's second law

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00:02:32,760 --> 00:02:37,230
is actually the most
general form of the equation

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00:02:37,230 --> 00:02:40,340
because it's applicable not
just to a single point mass,

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00:02:40,340 --> 00:02:42,410
but also to a more
complicated system.

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00:02:42,410 --> 00:02:44,540
A system consisting
of many masses

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00:02:44,540 --> 00:02:46,430
or a system where
the masses changing

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00:02:46,430 --> 00:02:48,550
or the masses flowing,
as in a fluid.

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In all of those cases, this
form of Newton's second law

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00:02:52,079 --> 00:02:52,940
is correct.

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00:02:52,940 --> 00:02:55,710
F equals ma, which is
probably more familiar to you,

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00:02:55,710 --> 00:02:58,030
is actually a special case
of this law for the case

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00:02:58,030 --> 00:02:59,780
of a single point mass.

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00:02:59,780 --> 00:03:02,230
So this is where we'll
see that momentum is quite

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00:03:02,230 --> 00:03:04,650
a useful concept, especially
as we start considering

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00:03:04,650 --> 00:03:08,840
more complicated systems, as
we'll get to a little later

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00:03:08,840 --> 00:03:10,490
in the course.

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00:03:10,490 --> 00:03:12,950
What I want to do now though,
is to take a closer look

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at this equation, force is
equal to the time derivative

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00:03:17,100 --> 00:03:17,950
of the momentum.

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00:03:17,950 --> 00:03:20,480
Whenever we have a
relation involving

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00:03:20,480 --> 00:03:23,990
a derivative like this,
we can always also

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00:03:23,990 --> 00:03:27,156
rewrite it in an equivalent
integral form, which

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00:03:27,156 --> 00:03:29,530
can be very useful and give
us a different way of looking

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00:03:29,530 --> 00:03:30,570
at the same information.

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00:03:30,570 --> 00:03:33,550
So let's take a look at that.

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So if I take this equation
and integrate both sides

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with respect to time,
then I can write

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that as the integral of
F with respect to time

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00:03:48,070 --> 00:03:52,210
is equal to the integral
of the right-hand side,

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00:03:52,210 --> 00:03:56,542
dp/dt with respect to time.

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00:03:56,542 --> 00:03:58,250
Now, let's make this
a definite integral.

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I'll go from time t1 to
time t2 on both sides here.

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00:04:08,420 --> 00:04:12,530
Now, this right-hand
side is just--

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so the integral of dp/dt
with respect to time

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00:04:15,360 --> 00:04:24,280
is just p at time2
minus p at time1.

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00:04:24,280 --> 00:04:29,860
And that is just the change
in the momentum vector

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00:04:29,860 --> 00:04:31,310
going from time1 to time2.

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00:04:34,220 --> 00:04:36,830
Now, this integral on
the left-hand side,

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00:04:36,830 --> 00:04:38,170
we give a special name.

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We call this the impulse.

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00:04:42,870 --> 00:04:47,280
This name, impulse, calls to
mind a short, sharp, shock

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of some sort.

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But it can also
refer to a weak force

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00:04:50,810 --> 00:04:52,590
acting over a long interval.

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00:04:52,590 --> 00:04:55,750
And notice here the
function F, the force F,

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00:04:55,750 --> 00:04:58,014
is in general a
function of time.

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00:04:58,014 --> 00:04:59,930
So this doesn't necessarily
mean a constant f.

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00:04:59,930 --> 00:05:03,160
This could mean a force
that's varying in time.

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00:05:03,160 --> 00:05:05,450
And what this
equation tells us is

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00:05:05,450 --> 00:05:08,170
that the change in the
momentum of the system

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00:05:08,170 --> 00:05:11,440
doesn't depend on the
detailed time dependence of F,

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00:05:11,440 --> 00:05:14,890
but rather just on
the integral of F.

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00:05:14,890 --> 00:05:23,970
And so suppose I were to graph
the force as a function of time

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00:05:23,970 --> 00:05:28,590
going from time0
to a time delta t.

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00:05:28,590 --> 00:05:33,200
And suppose I had some
complicated function that

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00:05:33,200 --> 00:05:34,020
looked like that.

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00:05:36,610 --> 00:05:39,970
The impulse is just the
area under this curve.

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00:05:39,970 --> 00:05:41,440
It's the integral
of this function.

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00:05:45,720 --> 00:05:46,590
That's the impulse.

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00:05:46,590 --> 00:05:49,080
And the change in
the momentum depends

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00:05:49,080 --> 00:05:51,350
only on the area
under this curve

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00:05:51,350 --> 00:05:56,440
and not on the detailed
shape of the curve.

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00:05:56,440 --> 00:06:02,570
So what that means is that I
can define an average force

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00:06:02,570 --> 00:06:09,300
by choosing a
constant force that

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00:06:09,300 --> 00:06:12,600
has the same area as
this example on the left.

102
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So suppose I calculated that.

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00:06:14,750 --> 00:06:18,290
And there is some
constant force here.

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00:06:18,290 --> 00:06:22,660
I'll call this F average.

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00:06:22,660 --> 00:06:24,610
Going over the
same time interval.

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00:06:28,910 --> 00:06:31,730
The average force is
that constant force

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00:06:31,730 --> 00:06:37,380
which has the same area as
the area under my F of t.

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00:06:37,380 --> 00:06:42,440
So in other words, F
average times delta t,

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00:06:42,440 --> 00:06:44,690
which is the area on the
right-hand side here,

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00:06:44,690 --> 00:06:52,690
is equal to the integral
of F of t dt integrated

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00:06:52,690 --> 00:06:57,370
from 0 to delta t,
which is the area

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00:06:57,370 --> 00:06:58,700
under this right-hand curve.

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00:06:58,700 --> 00:07:08,080
And so my average force is
just that integral, F of t dt,

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00:07:08,080 --> 00:07:10,000
divided by delta t.

115
00:07:10,000 --> 00:07:12,830
And this is integrated
from 0 to delta t.

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00:07:12,830 --> 00:07:15,230
So that's my average force.