00:00Hi everybody. So in this
00:02lesson we're going to look
00:03at kinematics, which is the
00:04study of the motion of
00:07objects. So we're going to
00:08look at how objects move,
00:10which includes displacement, velocity, acceleration,
00:14distance traveled and speed. So
00:18firstly, let me bring your
00:21attention to some of these
00:23terms. Displacement, velocity, acceleration, what
00:25are they? Okay, well,
00:28displacement, imagine, okay, displacement is,
00:33there's a strong connection between
00:35displacement and distance, but be
00:38careful, there is a big
00:40difference between the two. Imagine
00:43you lived, imagine you lived
00:46here, and your friend lives,
00:50let's say your friend lives
00:53two kilometers,
00:56So the south. So you
01:01live here, your friend is
01:02here, and this is two
01:03kilometers. Now, if you walk
01:06to your friend's house straight
01:08down here, you will travel
01:10a distance of two kilometers,
01:11and your displacement will be
01:14two kilometers south. So displacement
01:17has direction in it as
01:19well, whereas distance does not.
01:22But imagine you
01:24I wanted to go to
01:25your friend's house, but you
01:26wanted to go via, I
01:28don't know, let's say via
01:30the shop or via something
01:31that's on over here. And
01:33you might even want to
01:34go and see someone else
01:36that's there and then you
01:37come back. Or it might
01:40even be that this is
01:41the road to get there.
01:43There isn't a direct straight
01:44road from your house to
01:45his house. Now, if this
01:47is, let's say this is
01:48four and this is one
01:51and this is
01:52Now if you go this
01:55way to your friend's house,
01:57you will travel a distance
01:59of 10 kilometers, but your
02:02displacement will still be 2
02:04kilometers south. And actually, if
02:07you walk all the way
02:08home, if you do this,
02:09go to his husband and
02:10walk home, your displacement is
02:11zero, because you are zero
02:14kilometers or meters or whatever
02:16from where you left, from
02:18your starting point.
02:20That's the difference between displacement
02:22and distance traveled. Fine. Velocity
02:26and speed, they are also
02:28very closely linked, but one
02:30of them has direction and
02:32one of them doesn't. So
02:33let's say you are, I'm
02:35going to actually use this
02:36analogy for my example here.
02:38Imagine we're driving in a
02:40car and we're going, let's
02:42say, 100 kilometers an hour
02:45this way.
02:48velocity is 100 kilometers per
02:50hour and your speed is
02:55your speed is 100 kilometers
02:59per hour. But if you
03:02then start reversing or just
03:06going backwards or just going
03:08the other direction, let's say
03:09you start going this way,
03:11I like to think of
03:11it as reversing, but certainly
03:15you can just turn the
03:15car around but you're
03:16driving the other direction, this
03:18is then negative. So your
03:21velocity is now negative. Let's
03:23say you are reversing, so
03:25you're going a lot slower,
03:26let's say negative 20 kilometers
03:28per hour. Your velocity is
03:31now negative 20, but your
03:33speed is still 20. When
03:35you're reversing in a car,
03:36you don't say to someone,
03:38oh my speed, I'm going
03:39negative 20 kilometers per hour,
03:41you're just saying 20 kilometers
03:42per hour in reverse.
03:44Okay, so that's the difference
03:46between speed and velocity. Acceleration
03:50is just the rate of
03:52change of velocity. So again,
03:56I like the car example
03:57because if you press the
03:58accelerator, you start to accelerate.
04:00Your velocity starts increasing. You
04:02start going faster and faster
04:03and faster. That means you're
04:04accelerating. If you press the
04:06break, you will start to
04:07decelerate. Now I am going
04:09to, instead of, I'm not
04:11going to use the word
04:12decelerate unless I'm explaining something
04:14like this. Instead of using
04:16decelerate, I'm just going to
04:17say my acceleration is negative.
04:21And this leads me nicely
04:23to these rules in the
04:25form of the booklet. So
04:27displacement, we use the letter
04:29S to tell us the
04:31displacement. The last D's V
04:33acceleration is A. Those two
04:34are easy. Why do we
04:35use S and not D?
04:36Well, D is good good
04:40good good
04:40confused with distance, so we're
04:43going to go with s.
04:45So the displacement is s.
04:47Now the rate of change
04:48of displacement with respect to
04:51time is what velocity is.
04:54Look, if you are going
04:55100 kilometers per hour, that
04:57means your displacement is changing
04:59by 100 kilometers every hour.
05:03So this is a really
05:04nice, it's actually a really
05:07nice and useful application.
05:08of calculus that we can
05:11actually differentiate, if we differentiate
05:13this displacement, this ds dt,
05:15we get ds dt, this
05:17differentiate displacement with respect to
05:19time, since the change in
05:21displacement with respect to time,
05:25that, sorry, I don't want
05:26to put that there, that
05:28gives us the velocity, the
05:30velocity is the change in
05:32displacement with respect to time.
05:36And then,
05:36what's acceleration? Well, acceleration is
05:40the change in velocity with
05:42respect to times how my
05:43velocity is change. I'm getting
05:44faster and faster by press
05:45the accelerator. So it's dv
05:47dt and it's there in
05:51the formula booklet. Now they
05:52don't give us ds dt
05:53but I guess they assume
05:54we should know that. But
05:56if we're saying v is
05:58ds dt then dv dt
06:01is just the second derivative
06:03of this which is
06:04d2s dt squared. So the
06:09second derivative will give me
06:11the acceleration. Okay, now that
06:14leads me to a little
06:19rhyme that I have to
06:22help us remember this. So
06:23if we have displacement, velocity,
06:30acceleration, to go
06:33this way we differentiate, I'm
06:36just right, diff, and then
06:39to go this way, I
06:42don't know why I'm using
06:42green and red, but I
06:43guess positive and negative, to
06:45go this way we integrate,
06:48because if I, if I'm
06:49displacement to get to velocity,
06:51differentiate, if I'm in velocity
06:53to get to acceleration, differentiate,
06:55but we know to integrate
06:57is the reverse of differentiation,
07:00so if I want to
07:00go from
07:01acceleration to velocity, I integrate
07:03and if I want to
07:04go from velocity to displacement,
07:05I integrate. So here I
07:07have SVA. The best, the
07:13best rhyme that I've got
07:17and I've asked many, many
07:19students to give me a
07:20rhyme is so, sorry, I'm
07:22going to rhyme, but it's
07:23so very awesome. Something to
07:27remember this SVA.
07:29VA so very awesome. So
07:31SVA differentiate to go to
07:33the right and integrate to
07:34go to the left. Okay,
07:37fine. Let us go into
07:39this example. So this is
07:41basically a pass paper question
07:43that I've adapted slightly. I've
07:45actually made it a bit
07:46longer just to include all
07:48the different types of questions
07:50that you could be asked.
07:52No doubt the IB are
07:53able to come up with
07:56a very
07:57very difficult strange questions that
07:58you haven't seen before, but
08:00hopefully if you understand how
08:02to do all of this,
08:04you will be able to
08:05at least give them some,
08:06give every question that you
08:08might get a good attempt.
08:10Obviously, practice the raspberry questions,
08:13once you've finished watching this
08:14video. Okay, so the initial,
08:18sorry, a particle P. There's
08:20a particle P, it's moving
08:22along a straight line, so
08:23that it's velocity V
08:25meter per second after t
08:27seconds is given by this.
08:29So this is a velocity
08:31function and this is a
08:32velocity time graph. The x
08:34-axis is time. The y
08:37-axis is velocity is not
08:40x and y, it's t
08:41and v. So time and
08:43velocity and this tells us
08:45what its velocity is after
08:47a certain amount of time.
08:49Find the initial velocity of
08:51p. Okay.
08:53begin. So before we actually
08:57do that question, I want
08:58to look at the graph
08:59and I want you to
09:00think about what's actually happening
09:02to this particle and let's
09:04actually think of it as
09:05a curve. So we've a
09:07curve moving in the curve
09:10is moving in a straight
09:12line. So let's think of
09:13it as it's going forwards
09:14and then it goes straight
09:17into reverse and it goes
09:18backwards. So what's actually happening
09:20at this point here?
09:21Now, the graph, the gradient
09:25of the graph is negative,
09:26or it's decreasing, but that
09:28doesn't mean the curve is
09:30going backwards. At this point
09:31at the very, very beginning,
09:33the curve is going forwards
09:34because its velocity is positive.
09:37So its velocity is positive
09:38here, and it's positive all
09:40the way down to that
09:41point there, which is, that
09:45is a zero, it's where
09:47its velocity is zero. So
09:48it's going forwards,
09:49He's going forwards and he's
09:51slowing down and right at
09:52this point his velocity zero
09:54so he stops instantaneously immediately
09:57goes into reverse and now
09:58he starts going backwards. So
10:00he's actually going backwards all
10:01the way along here because
10:03the velocity is negative even
10:05though there's some turning points
10:06in the road or there's
10:08some turning points in the
10:09time. He is going backwards
10:14all the way to here.
10:15At this point then he
10:17instantaneously
10:17So he's going forward, backwards,
10:27forwards. Now, what is happening
10:28at these kind of turning
10:29points? Well, he is reversing
10:36here, but he's getting faster
10:37and faster and faster and
10:38faster. So he's going backwards,
10:42but he is putting his
10:44foot on the
10:45accelerator. So his acceleration is
10:51negative here, but he is
10:54going faster and faster and
10:56he is decelerating if you
10:59want to think of it
10:59as that word. So he's
11:01decelerating to here. And then
11:04he decides, hang on, I'm
11:05going too fast. I'm going
11:07to slow down a bit,
11:08but I'm still in reverse,
11:09but I'm going to slow
11:10down a bit. Then you
11:12go, I'll speed up a
11:12little bit.
11:13I'm going to slow down
11:14a little bit, but it's
11:15all the time in reverse.
11:17And then straight away, he
11:18goes into first gear and
11:20now he starts to speed
11:21up and he's going forward,
11:22forward, forward, forward. And then
11:23at this point he presses
11:24the break and says hang
11:25on, I'm going too fast,
11:26I'll slow down and that's
11:28the end of it. So
11:29it's only goes from zero
11:30to five seconds. Now, yes,
11:33if it's a car, this
11:36is all happening pretty quickly
11:37for zero to five seconds.
11:38It isn't a car, it's
11:39a particle, but I like
11:40to use the current analogy,
11:41it helps to understand what's
11:43going on. Okay, so hopefully
11:45you have an idea of
11:47what's going on with that
11:48graph. Let's do these questions.
11:51The initial velocity of p,
11:53so initial velocity is when
11:55t is zero at the
11:57start. So I'm trying to
11:58find v of zero. Now
12:00this is a calculator, I
12:04should have put the thing,
12:06a little drawing. This is
12:07a calculator allowed question. So
12:09we
12:09here on our D -zone
12:10calculator. What I'm going to
12:11show you, and how to
12:13do this without a calculator,
12:13because it's straightforward. So V
12:16of zero is two costs
12:18of three times zero, which
12:20is zero minus five sine
12:23of zero minus one. This
12:27is actually cost of zero
12:29is one, so it's two
12:31minus zero minus one, two
12:33minus one equals one meters
12:36per second.
12:38initial velocity is 1 meters
12:39per second. Fine, that's how
12:41to do it without a
12:42calculator. We have a calculator
12:45so I'm going to show
12:47you how to do it
12:47with the calculator as well.
12:50Okay, let's do part B.
12:55The displacement of P after
12:58five seconds. So, let me
13:02just get the calculator. I'm
13:05going to
13:06I'm actually going to graph,
13:12I'm going to graph this
13:13function here. So let's do,
13:15it's two costs of 3t,
13:18so two costs of 3t,
13:21but I'm going to have
13:21to use x because this
13:22is my xy graph. So
13:24it's two costs of 3x,
13:26close my bracket, my bracket
13:30gone here, minus five sine
13:33of x,
13:34minus 5 sine of x
13:39minus 1. Okay, does that
13:44look like this? Well, not
13:45exactly, because the domain is
13:49a lot bigger. I'm actually
13:50going to get the zoom
13:51to work a bit better
13:53now. Because they've given me
13:56the graph, I like the
13:57Windows settings, because I can
13:58just put the settings to
13:59what they have, like 0
14:00to 5.
14:02let's say negative 10 to
14:045 but I'd actually prefer
14:05to go a little bit
14:06more than what they give
14:07me. So I'm going to
14:07go from negative 1 to
14:096 and I'll go from
14:12negative 10 to let's go
14:16to 8. See how that
14:17looks. Okay, fine. Now the
14:21initial let's do that part
14:22A with the calculator. If
14:25I go to trace, so
14:26I like this trace button.
14:28If I go to menu
14:29trace and I can move
14:30along here, it finds me
14:32in my minimums, my 0,
14:34and at this point the
14:35y -intercept, the y -intercept
14:37is 0, 1, 0 is
14:39the time, the x -coordinates
14:40the time, when t is
14:410, and the y -coordinate
14:44is the velocity, so the
14:46velocity is 1, and there
14:47you have it. In a
14:49calculator question, you wouldn't even
14:51have to, I would write
14:54v of 0, but if
14:56you just put down 1
14:57meters per second, you'd actually
14:58get all
14:58all the marks. Second one,
14:59the displacement of p after
15:01five seconds, right? Now if
15:04we go to this formula,
15:06the displacement is the integral
15:10of the velocity. So remember,
15:13I said to get from
15:16velocity to displacement, we integrate.
15:18So it's the integral from
15:20t1 to t2. t1 will
15:23be zero and t2 will
15:26So the integral from 0
15:29to 5 of the function
15:32which is 2 cos 3t
15:37minus 5 sin t minus
15:401 dt and here, look
15:43if this is a paper
15:442 you do not have
15:46to integrate it. I don't
15:48have to go to the
15:49process of getting sine of
15:513t over 3 and whatever
15:53it is I can just
15:54type
15:54that into the calculator. So
15:56here I'm going to go
15:58menu, calculus, numerical integral. I'm
16:02going to go from zero
16:04to five and then I'm
16:06going to go to costs.
16:11I can actually use t
16:12here. So I'm going to
16:13go 3t, 2 costs of
16:153t minus five sine.
16:22of t. And then minus
16:28one, just make sure I
16:29now do dt here, presenter.
16:32And I get negative eight
16:34point one four eight one
16:36six negative eight point one
16:39four eight one six. That's
16:42what I said. Maybe point
16:44one four eight one six.
16:45Yes. So the displacement of
16:48p after five seconds is
16:49negative eight
16:50point one four eight one
16:52six meters so the negative
16:54is important and look the
16:57meters is important because that's
16:58where it is and does
16:59it make sense that he'd
17:00actually be in a negative
17:02position? Well if I look
17:04at this graph I'd say
17:04yes because he's reversing for
17:07a lot longer than he's
17:08going forward so it makes
17:09sense to me that he
17:10goes a little bit forwards
17:11then a lot back and
17:13then a little bit forwards
17:14again so yeah if I
17:17actually draw that what's happening
17:18is he's going a little
17:21bit forwards that's this bit
17:25then he goes a lot
17:27backwards which should be like
17:30this and then he goes
17:34a bit more forwards a
17:37bit more than this but
17:38not as much forwards as
17:40certainly not enough to get
17:42back to the start and
17:44that's why this display if
17:46this was his
17:46starting place, he starts here,
17:49goes a little bit forwards,
17:50a lot back, and then
17:52a little bit more forwards.
17:54This is his end point.
17:56This distance from here to
17:58here, this is displacement. And
18:03it's negative because he's to
18:05the left of where he
18:06started. So that's his displacement.
18:08If I wanted to get
18:09the distance traveled, and look,
18:12that's this next question. The
18:13distance traveled is going to
18:14be this.
18:14plus this, plus this. But
18:18let me show you a
18:19nice neat way to do
18:20it. Again, in the formula
18:24book that the distance traveled
18:26is the same formula, but
18:28this time with the modulus
18:30sign. And the reason for
18:32the modulus sign is because
18:33that gets rid of the
18:35negatives and it will make
18:37this negative bit that we've
18:41traveled here. It will make
18:42that
18:42the positive so it will
18:43just add this plus this
18:45plus this. So the distance,
18:47the distance traveled is the
18:50integral from zero to five
18:51of the modulus of two
18:55costs, three t minus five,
18:58sine t minus one dt.
19:02So what working do you
19:04have to write down? Well,
19:04that's it. You have to
19:06write down the integral, you
19:07have to write down the
19:08integral, and then you can
19:08just write down the answer.
19:10What is this?
19:11So it's going to be
19:12positive and it's definitely going
19:14to be more than 8
19:15.14. Let's find out what
19:17it is. So menu calculus
19:20numerical integral. I need 0
19:24to 5 and here I
19:27want to put these modular
19:28signs which are there that
19:31and I want to copy
19:33control C and control
19:39V, I want to copy
19:41that in there. Perfect. And
19:43that's dt. Press enter. That's
19:48what I was expecting. 19
19:50.19 .5613. So this is
19:57equal to 19 .5611.
20:07Okay, next one, C, D.
20:14And that's the total distance
20:18that is traveled. D, how
20:20many times P changes direction?
20:24So remember, this is our
20:25car and we're going to
20:26reverse. Now look, the common
20:30incorrect answer here, as you
20:32can imagine, would be one,
20:34two, three,
20:354, he changes direction 4
20:36times. No, that is incorrect.
20:40Because this is just when
20:41he starts to slow down,
20:43what he's reversing, reversing, reversing,
20:46reversing, reversing, reversing, reversing, reversing,
20:47reversing, reversing, is not changing
20:47directions. This is where he
20:49changes direction, and this is
20:51where he changes direction. So
20:53he changes direction twice, he
20:55goes from forwards to backwards,
20:57and then backwards to forwards.
20:58So the answer to this
20:59is just 2. And I
21:00can just write down 2.
21:02If you want to indicate
21:03here, and are you even
21:05right? Crosses t axis twice
21:12if you really want to
21:13show some work, but you
21:14definitely get full marks for
21:15that. Okay, e, find the
21:17acceleration of p after three
21:20seconds. How do I find
21:21acceleration? Well, here we are.
21:25Acceleration is the derivative of
21:28the velocity. Or here, look,
21:30I have the velocity to
21:31get
21:31at acceleration, I differentiate. So
21:34I need to differentiate this
21:36thing and it's after three
21:38seconds. So I want dV
21:41dT at three seconds when
21:47T equals three. Now look,
21:51again, many students will start
21:52differentiating this and sobbing in
21:55T. If it was a
21:56non -calculated paper fine, yes,
21:58you have to do that.
21:59But this is with a
22:00calculator. So all I need
22:02to do is get out
22:03my calculator, do menu, calculus,
22:07numerical derivative at a point.
22:09I want x, I want
22:11three, I want the first
22:13derivative, and I'm actually going
22:15to just paste this in
22:17again, control V. So I
22:19want the derivative of this,
22:22the derivative of this, at
22:24x equals three, press enter,
22:27Oops, that is something wrong.
22:31Okay, because I used t
22:34here and one and the
22:35derivative is with x. All
22:37right, let me do that
22:38again. I need menu. Calculus,
22:41numerical derivative at a point.
22:43x. Okay, well actually I
22:45could just use t here.
22:48I could just use t
22:49there. The value is three.
22:52Now do my paste control.
22:55V press enter and I
22:59get 2 .4775, 2 .4775.
23:10Now what that's telling me
23:11is it's telling me the
23:12acceleration when t is 3.
23:14Now here's when t is
23:153. What is the acceleration
23:20here? Where the acceleration is
23:22the rate of change of
23:23the
23:23velocity, which is actually the
23:25gradient. It's the gradient of
23:29the curve at that point.
23:31And does that look like
23:32it might be or that
23:34it could be a gradient
23:36of 2 .4775? Well, yeah,
23:38it's positive. These axes aren't
23:42actually to the same scale.
23:44So that's absolutely fine. Yes,
23:45you can see he's starting
23:47to accelerate. So the acceleration
23:49of P after three seconds,
23:50yes, that is the acceleration
23:51of P after three seconds.
23:54He is accelerating. Even though
23:56he's slowing down, and this
23:57is the thing I talked
23:58to you at the start,
24:00the car is slowing down
24:02because he's in reverse and
24:04he's slowing down, but technically
24:05he is accelerating. Okay, final
24:09question. Find the maximum speed
24:12of P. So I've thrown
24:14in this question to make
24:16sure you guys understand the
24:18difference between
24:19velocity and speed. So let's
24:24just find it on the
24:25graph. Where is the maximum?
24:26Where is the maximum velocity?
24:29This is velocity and this
24:30is time. Where is the
24:31maximum velocity? Well, it's certainly
24:33here. It's at that point
24:36somewhere over here. That would
24:37be the maximum velocity. But
24:39remember this maximum speed, like
24:42speed, it doesn't matter if
24:44we're going forwards or backwards.
24:46Now this
24:47This guy at this point,
24:54his speed is like 5.
24:56something, maybe 6, but no,
24:58probably 5. something. Whereas this
25:00speed at this point is,
25:04it's going to be more
25:06than 5. So I need
25:08to find, well, we'll check
25:10both on the graph, but
25:12I'm pretty confident from looking
25:14at the graph that his
25:14maximum speed is going to
25:15be here.
25:16here. Okay, let's get the
25:18graph. So it's either going
25:20to be this one or
25:21this one. We do our
25:23menu analyze graph. I'm going
25:28to get my minimum lower
25:30bound upper bound. So here
25:33it's negative seven point set
25:36negative seven point four seven
25:39one. And this guy, I'll
25:42do the maximum.
25:44is 5 .47. One. Now
25:48note, I am looking at
25:49the y coordinate because that's
25:51the velocity. This is the
25:52time, that's the velocity. Time,
25:55velocity. His velocity is obviously
26:00more than his velocity, but
26:02his speed is bigger than
26:04his speed. So the maximum
26:06speed of the car is
26:087 .4711.
26:127 .471 max speed is
26:227 .471 meters per second
26:31and look that little bit
26:33of drawing that I've done
26:34on the graph is certainly
26:36enough working. You could also
26:40say happens if you really
26:42want to just say happens
26:43when t equals 1 .16
26:511 .16 okay that's it
26:54so obviously I've gone through
26:56a lot there kinematics is
26:58quite a big topic and
26:59I've taught it all to
27:01you in one lesson and
27:03if you understand everything I've
27:04done there fantastic you're in
27:07you're in very good
27:08shape, I would actually advise
27:10you to even try and
27:11do that question I've just
27:12done again yourself and then
27:15watch the video after just
27:16to make sure you know
27:18what you're doing and that
27:19you've got it correct. The
27:23key points to note are
27:26obviously you know what displacement,
27:27velocity, acceleration are. This SVA
27:30thing differentiates to get from
27:34displacement to velocity and velocity
27:35to accelerate.
27:36to acceleration and then integrate
27:38to go backwards. And then
27:41when you are given a
27:42velocity time graph to make
27:44sure you understand what's actually
27:47happening is the particle going
27:49forwards or backwards. Also, be
27:53careful. Sometimes they don't give
27:54you a velocity time graph.
27:56They can give you a
27:58displacement time graph and it's
28:00even possible they can give
28:01you an acceleration time graph.
28:02So don't assume that it's
28:04velocity.
28:04read the question and make
28:06sure you follow exactly what
28:10they say.