Learn AASL 5.9 Kinematics in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 5.9—Kinematics problems.
Learn AASL 5.9 Kinematics in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 5.9—Kinematics problems.
The video provides an overview of kinematics, focusing on the motion of objects through key concepts such as displacement, velocity, acceleration, distance traveled, and speed.
Key formulas include:
The video also emphasizes the importance of understanding graphs related to these concepts, such as velocity-time graphs, and how to interpret them to determine the motion of a particle.
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
00:38careful, 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.