AIHL 5.13.1 Kinematics | Free Mathematics Applications & Interpretation (AI) Video | RevisionDojo
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AIHL 5.13.1 Kinematics Learn AIHL 5.13.1 Kinematics in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 5.13—Kinematic problems.
About this video Learn AIHL 5.13.1 Kinematics in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 5.13—Kinematic 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 .
Displacement is defined as the change in position with direction, while distance is the total path length without direction.Velocity includes direction, whereas speed does not. The relationship between these concepts is illustrated through examples involving driving a car and analyzing motion graphs.
Additionally, the video explains how to calculate displacement and velocity using integrals and derivatives, emphasizing the importance of understanding the differences between these terms and their applications in physics.
Video transcript 00:00 Hi everybody. So in this
00:02 lesson we're going to look
00:03 at kinematics, which is the
00:04 study of the motion of
00:07 objects. So we're going to
00:08 look at how objects move,
00:10
which includes displacement, velocity, acceleration,
00:14 distance traveled and speed. So
00:18 firstly, let me bring your
00:21 attention to some of these
00:23 terms. Displacement, velocity, acceleration, what
00:25 are they? Okay, well,
00:28 displacement, imagine, okay, displacement is,
00:33 there's a strong connection between
00:35 displacement and distance, but be
00:38 careful, there is a big
00:40 difference between the two. Imagine
00:43 you lived, imagine you lived
00:46 here, and your friend lives,
00:50 let's say your friend lives
00:56 So the south. So you
01:01 live here, your friend is
01:02 here, and this is two
01:03 kilometers. Now, if you walk
01:06 to your friend's house straight
01:08 down here, you will travel
01:10 a distance of two kilometers,
01:11 and your displacement will be
01:14 two kilometers south. So displacement
01:17 has direction in it as
01:19 well, whereas distance does not.
01:25 your friend's house, but you
01:28 don't know, let's say via
01:30 the shop or via something
01:31 that's on over here. And
01:33 you might even want to
01:34 go and see someone else
01:36 that's there and then you
01:37 come back. Or it might
01:40 even be that this is
01:41 the road to get there.
01:43 There isn't a direct straight
01:44 road from your house to
01:45 his house. Now, if this
01:47 is, let's say this is
01:48 four and this is one
01:55 way to your friend's house,
01:57 you will travel a distance
01:59 of 10 kilometers, but your
02:02 displacement will still be 2
02:04 kilometers south. And actually, if
02:07 you walk all the way
02:08 home, if you do this,
02:09 go to his husband and
02:10 walk home, your displacement is
02:11 zero, because you are zero
02:14 kilometers or meters or whatever
02:16 from where you left, from
02:18 your starting point.
02:20 That's the difference between displacement
02:22 and distance traveled. Fine. Velocity
02:26 and speed, they are also
02:28 very closely linked, but one
02:30 of them has direction and
02:32 one of them doesn't. So
02:33 let's say you are, I'm
02:35 going to actually use this
02:36 analogy for my example here.
02:38 Imagine we're driving in a
02:40 car and we're going, let's
02:42 say, 100 kilometers an hour
02:48 velocity is 100 kilometers per
02:50 hour and your speed is
02:55 your speed is 100 kilometers
02:59 per hour. But if you
03:02 then start reversing or just
03:06 going backwards or just going
03:08 the other direction, let's say
03:09 you start going this way,
03:11 it as reversing, but certainly
03:15 you can just turn the
03:15 car around but you're
03:16 driving the other direction, this
03:18 is then negative. So your
03:21 velocity is now negative. Let's
03:23 say you are reversing, so
03:25 you're going a lot slower,
03:26 let's say negative 20 kilometers
03:28 per hour. Your velocity is
03:31 now negative 20, but your
03:33 speed is still 20. When
03:35 you're reversing in a car,
03:36 you don't say to someone,
03:38 oh my speed, I'm going
03:39 negative 20 kilometers per hour,
03:41 you're just saying 20 kilometers
03:42 per hour in reverse.
03:44 Okay, so that's the difference
03:46 between speed and velocity. Acceleration
03:52 change of velocity. So again,
03:56 I like the car example
03:57 because if you press the
03:58 accelerator, you start to accelerate.
04:00 Your velocity starts increasing. You
04:02 start going faster and faster
04:03 and faster. That means you're
04:04 accelerating. If you press the
04:06 break, you will start to
04:07 decelerate. Now I am going
04:09 to, instead of, I'm not
04:11 going to use the word
04:12 decelerate unless I'm explaining something
04:14 like this. Instead of using
04:16 decelerate, I'm just going to
04:17 say my acceleration is negative.
04:21 And this leads me nicely
04:23 to these rules in the
04:25 form of the booklet. So
04:27 displacement, we use the letter
04:31 displacement. The last D's V
04:33 acceleration is A. Those two
04:36 Well, D is good good
04:40 confused with distance, so we're
04:45 So the displacement is s.
04:47 Now the rate of change
04:48 of displacement with respect to
04:51 time is what velocity is.
04:54 Look, if you are going
04:55 100 kilometers per hour, that
04:57 means your displacement is changing
04:59 by 100 kilometers every hour.
05:04 nice, it's actually a really
05:07 nice and useful application.
05:08 of calculus that we can
05:11 actually differentiate, if we differentiate
05:13 this displacement, this ds dt,
05:17 differentiate displacement with respect to
05:19 time, since the change in
05:21 displacement with respect to time,
05:25 that, sorry, I don't want
05:26 to put that there, that
05:28 gives us the velocity, the
05:30 velocity is the change in
05:32 displacement with respect to time.
05:36 what's acceleration? Well, acceleration is
05:40 the change in velocity with
05:42 respect to times how my
05:43 velocity is change. I'm getting
05:44 faster and faster by press
05:45 the accelerator. So it's dv
05:47 dt and it's there in
05:51 the formula booklet. Now they
05:53 but I guess they assume
05:54 we should know that. But
05:56 if we're saying v is
06:01 is just the second derivative
06:04 d2s dt squared. So the
06:09 second derivative will give me
06:11 the acceleration. Okay, now that
06:14 leads me to a little
06:19 rhyme that I have to
06:22 help us remember this. So
06:23 if we have displacement, velocity,
06:33 this way we differentiate, I'm
06:36 just right, diff, and then
06:42 don't know why I'm using
06:42 green and red, but I
06:43 guess positive and negative, to
06:45 go this way we integrate,
06:48 because if I, if I'm
06:49 displacement to get to velocity,
06:51 differentiate, if I'm in velocity
06:53 to get to acceleration, differentiate,
06:55 but we know to integrate
06:57 is the reverse of differentiation,
07:01 acceleration to velocity, I integrate
07:04 go from velocity to displacement,
07:05 I integrate. So here I
07:07 have SVA. The best, the
07:13 best rhyme that I've got
07:17 and I've asked many, many
07:19 students to give me a
07:20 rhyme is so, sorry, I'm
07:22 going to rhyme, but it's
07:23 so very awesome. Something to
07:29 VA so very awesome. So
07:31 SVA differentiate to go to
07:33 the right and integrate to
07:34 go to the left. Okay,
07:37 fine. Let us go into
07:39 this example. So this is
07:41 basically a pass paper question
07:43 that I've adapted slightly. I've
07:45 actually made it a bit
07:46 longer just to include all
07:48 the different types of questions
07:50 that you could be asked.
07:53 able to come up with
07:57 very difficult strange questions that
07:58 you haven't seen before, but
08:00 hopefully if you understand how
08:05 at least give them some,
08:06 give every question that you
08:08 might get a good attempt.
08:10 Obviously, practice the raspberry questions,
08:13 once you've finished watching this
08:14 video. Okay, so the initial,
08:18 sorry, a particle P. There's
08:20 a particle P, it's moving
08:22 along a straight line, so
08:23 that it's velocity V
08:25 meter per second after t
08:27 seconds is given by this.
08:29 So this is a velocity
08:31 function and this is a
08:32 velocity time graph. The x
08:34 -axis is time. The y
08:37 -axis is velocity is not
08:43 velocity and this tells us
08:45 what its velocity is after
08:47 a certain amount of time.
08:49 Find the initial velocity of
08:53 begin. So before we actually
08:57 do that question, I want
08:58 to look at the graph
09:00 think about what's actually happening
09:02 to this particle and let's
09:04 actually think of it as
09:07 curve moving in the curve
09:10 is moving in a straight
09:12 line. So let's think of
09:13 it as it's going forwards
09:14 and then it goes straight
09:17 into reverse and it goes
09:18 backwards. So what's actually happening
09:21 Now, the graph, the gradient
09:25 of the graph is negative,
09:26 or it's decreasing, but that
09:28 doesn't mean the curve is
09:30 going backwards. At this point
09:31 at the very, very beginning,
09:33 the curve is going forwards
09:34 because its velocity is positive.
09:37 So its velocity is positive
09:38 here, and it's positive all
09:40 the way down to that
09:41 point there, which is, that
09:45 is a zero, it's where
09:47 its velocity is zero. So
09:48 it's going forwards,
09:49 He's going forwards and he's
09:51 slowing down and right at
09:52 this point his velocity zero
09:54 so he stops instantaneously immediately
09:57 goes into reverse and now
09:58 he starts going backwards. So
10:00 he's actually going backwards all
10:01 the way along here because
10:03 the velocity is negative even
10:05 though there's some turning points
10:06 in the road or there's
10:08 some turning points in the
10:09 time. He is going backwards
10:14 all the way to here.
10:15 At this point then he
10:17 So he's going forward, backwards,
10:27 forwards. Now, what is happening
10:28 at these kind of turning
10:29 points? Well, he is reversing
10:36 here, but he's getting faster
10:37 and faster and faster and
10:38 faster. So he's going backwards,
10:42 but he is putting his
10:45 accelerator. So his acceleration is
10:51 negative here, but he is
10:54 going faster and faster and
10:56 he is decelerating if you
10:59 as that word. So he's
11:01 decelerating to here. And then
11:04 he decides, hang on, I'm
11:05 going too fast. I'm going
11:08 but I'm still in reverse,
11:09 but I'm going to slow
11:10 down a bit. Then you
11:13 I'm going to slow down
11:14 a little bit, but it's
11:15 all the time in reverse.
11:17 And then straight away, he
11:18 goes into first gear and
11:20 now he starts to speed
11:21 up and he's going forward,
11:22 forward, forward, forward. And then
11:23 at this point he presses
11:24 the break and says hang
11:25 on, I'm going too fast,
11:26 I'll slow down and that's
11:29 it's only goes from zero
11:30 to five seconds. Now, yes,
11:36 is all happening pretty quickly
11:37 for zero to five seconds.
11:38 It isn't a car, it's
11:39 a particle, but I like
11:40 to use the current analogy,
11:41 it helps to understand what's
11:43 going on. Okay, so hopefully
11:47 what's going on with that
11:48 graph. Let's do these questions.
11:51 The initial velocity of p,
11:53 so initial velocity is when
11:57 start. So I'm trying to
12:00 this is a calculator, I
12:04 should have put the thing,
12:06 a little drawing. This is
12:07 a calculator allowed question. So
12:10 calculator. What I'm going to
12:11 show you, and how to
12:13 do this without a calculator,
12:13 because it's straightforward. So V
12:16 of zero is two costs
12:18 of three times zero, which
12:20 is zero minus five sine
12:23 of zero minus one. This
12:27 is actually cost of zero
12:31 minus zero minus one, two
12:33 minus one equals one meters
12:38 initial velocity is 1 meters
12:39 per second. Fine, that's how
12:42 calculator. We have a calculator
12:45 so I'm going to show
12:47 with the calculator as well.
12:50 Okay, let's do part B.
12:55 The displacement of P after
12:58 five seconds. So, let me
13:02 just get the calculator. I'm
13:06 I'm actually going to graph,
13:12 I'm going to graph this
13:13 function here. So let's do,
13:15 it's two costs of 3t,
13:21 but I'm going to have
13:21 to use x because this
13:24 it's two costs of 3x,
13:26 close my bracket, my bracket
13:30 gone here, minus five sine
13:39 minus 1. Okay, does that
13:44 look like this? Well, not
13:45 exactly, because the domain is
13:49 a lot bigger. I'm actually
13:50 going to get the zoom
13:51 to work a bit better
13:53 now. Because they've given me
13:56 the graph, I like the
13:57 Windows settings, because I can
13:58 just put the settings to
13:59 what they have, like 0
14:02 let's say negative 10 to
14:04 5 but I'd actually prefer
14:06 more than what they give
14:07 go from negative 1 to
14:12 negative 10 to let's go
14:17 looks. Okay, fine. Now the
14:21 initial let's do that part
14:22 A with the calculator. If
14:26 I like this trace button.
14:29 trace and I can move
14:30 along here, it finds me
14:32 in my minimums, my 0,
14:34 and at this point the
14:35 y -intercept, the y -intercept
14:39 the time, the x -coordinates
14:41 0, and the y -coordinate
14:44 is the velocity, so the
14:46 velocity is 1, and there
14:49 calculator question, you wouldn't even
14:51 have to, I would write
14:57 meters per second, you'd actually
14:58 all the marks. Second one,
14:59 the displacement of p after
15:01 five seconds, right? Now if
15:04 we go to this formula,
15:06 the displacement is the integral
15:10 of the velocity. So remember,
15:16 velocity to displacement, we integrate.
15:18 So it's the integral from
15:26 So the integral from 0
15:29 to 5 of the function
15:46 to integrate it. I don't
15:49 process of getting sine of
15:51 3t over 3 and whatever
15:54 that into the calculator. So
15:56 here I'm going to go
15:58 menu, calculus, numerical integral. I'm
16:02 going to go from zero
16:04 to five and then I'm
16:06 going to go to costs.
16:11 I can actually use t
16:12 here. So I'm going to
16:22 of t. And then minus
16:28 one, just make sure I
16:29 now do dt here, presenter.
16:32 And I get negative eight
16:34 point one four eight one
16:36 six negative eight point one
16:39 four eight one six. That's
16:42 what I said. Maybe point
16:44 one four eight one six.
16:45 Yes. So the displacement of
16:48 p after five seconds is
16:50 point one four eight one
16:52 six meters so the negative
16:54 is important and look the
16:57 meters is important because that's
16:58 where it is and does
16:59 it make sense that he'd
17:00 actually be in a negative
17:02 position? Well if I look
17:04 at this graph I'd say
17:04 yes because he's reversing for
17:07 a lot longer than he's
17:08 going forward so it makes
17:10 goes a little bit forwards
17:13 then a little bit forwards
17:17 actually draw that what's happening
17:18 is he's going a little
17:21 bit forwards that's this bit
17:27 backwards which should be like
17:30 this and then he goes
17:34 a bit more forwards a
17:37 bit more than this but
17:38 not as much forwards as
17:40 certainly not enough to get
17:42 back to the start and
17:44 that's why this display if
17:46 starting place, he starts here,
17:49 goes a little bit forwards,
17:50 a lot back, and then
17:52 a little bit more forwards.
17:54 This is his end point.
17:56 This distance from here to
17:58 here, this is displacement. And
18:03 it's negative because he's to
18:05 the left of where he
18:06 started. So that's his displacement.
18:09 the distance traveled, and look,
18:12 that's this next question. The
18:13 distance traveled is going to
18:14 plus this, plus this. But
18:20 it. Again, in the formula
18:24 book that the distance traveled
18:26 is the same formula, but
18:28 this time with the modulus
18:30 sign. And the reason for
18:32 the modulus sign is because
18:33 that gets rid of the
18:35 negatives and it will make
18:37 this negative bit that we've
18:41 traveled here. It will make
18:42 the positive so it will
18:43 just add this plus this
18:45 plus this. So the distance,
18:47 the distance traveled is the
18:50 integral from zero to five
18:51 of the modulus of two
18:55 costs, three t minus five,
18:58 sine t minus one dt.
19:02 So what working do you
19:04 have to write down? Well,
19:04 that's it. You have to
19:06 write down the integral, you
19:07 have to write down the
19:08 integral, and then you can
19:08 just write down the answer.
19:12 positive and it's definitely going
19:15 .14. Let's find out what
19:17 it is. So menu calculus
19:20 numerical integral. I need 0
19:27 want to put these modular
19:28 signs which are there that
19:33 control C and control
19:41 that in there. Perfect. And
19:43 that's dt. Press enter. That's
19:48 what I was expecting. 19
19:50 .19 .5613. So this is
20:07 Okay, next one, C, D.
20:14 And that's the total distance
20:18 that is traveled. D, how
20:20 many times P changes direction?
20:24 So remember, this is our
20:25 car and we're going to
20:26 reverse. Now look, the common
20:30 incorrect answer here, as you
20:32 can imagine, would be one,
20:35 4, he changes direction 4
20:36 times. No, that is incorrect.
20:40 Because this is just when
20:41 he starts to slow down,
20:43 what he's reversing, reversing, reversing,
20:46 reversing, reversing, reversing, reversing, reversing,
20:47 reversing, reversing, is not changing
20:47 directions. This is where he
20:49 changes direction, and this is
20:51 where he changes direction. So
20:53 he changes direction twice, he
20:55 goes from forwards to backwards,
20:57 and then backwards to forwards.
20:58 So the answer to this
21:00 can just write down 2.
21:02 If you want to indicate
21:03 here, and are you even
21:05 right? Crosses t axis twice
21:12 if you really want to
21:13 show some work, but you
21:14 definitely get full marks for
21:15 that. Okay, e, find the
21:17 acceleration of p after three
21:20 seconds. How do I find
21:21 acceleration? Well, here we are.
21:25 Acceleration is the derivative of
21:28 the velocity. Or here, look,
21:30 I have the velocity to
21:31 at acceleration, I differentiate. So
21:34 I need to differentiate this
21:36 thing and it's after three
21:38 seconds. So I want dV
21:41 dT at three seconds when
21:47 T equals three. Now look,
21:51 again, many students will start
21:52 differentiating this and sobbing in
21:56 non -calculated paper fine, yes,
21:58 you have to do that.
22:00 calculator. So all I need
22:03 my calculator, do menu, calculus,
22:07 numerical derivative at a point.
22:11 three, I want the first
22:13 derivative, and I'm actually going
22:15 to just paste this in
22:17 again, control V. So I
22:19 want the derivative of this,
22:22 the derivative of this, at
22:24 x equals three, press enter,
22:27 Oops, that is something wrong.
22:31 Okay, because I used t
22:34 here and one and the
22:35 derivative is with x. All
22:37 right, let me do that
22:38 again. I need menu. Calculus,
22:41 numerical derivative at a point.
22:43 x. Okay, well actually I
22:45 could just use t here.
22:49 there. The value is three.
22:52 Now do my paste control.
22:59 get 2 .4775, 2 .4775.
23:10 Now what that's telling me
23:11 is it's telling me the
23:12 acceleration when t is 3.
23:14 Now here's when t is
23:15 3. What is the acceleration
23:20 here? Where the acceleration is
23:22 the rate of change of
23:23 velocity, which is actually the
23:25 gradient. It's the gradient of
23:29 the curve at that point.
23:31 And does that look like
23:34 it could be a gradient
23:36 of 2 .4775? Well, yeah,
23:38 it's positive. These axes aren't
23:42 actually to the same scale.
23:44 So that's absolutely fine. Yes,
23:45 you can see he's starting
23:47 to accelerate. So the acceleration
23:49 of P after three seconds,
23:50 yes, that is the acceleration
23:51 of P after three seconds.
23:54 He is accelerating. Even though
23:56 he's slowing down, and this
23:57 is the thing I talked
23:58 to you at the start,
24:00 the car is slowing down
24:02 because he's in reverse and
24:04 he's slowing down, but technically
24:05 he is accelerating. Okay, final
24:09 question. Find the maximum speed
24:12 of P. So I've thrown
24:14 in this question to make
24:16 sure you guys understand the
24:19 velocity and speed. So let's
24:25 graph. Where is the maximum?
24:26 Where is the maximum velocity?
24:29 This is velocity and this
24:30 is time. Where is the
24:31 maximum velocity? Well, it's certainly
24:33 here. It's at that point
24:36 somewhere over here. That would
24:37 be the maximum velocity. But
24:39 remember this maximum speed, like
24:42 speed, it doesn't matter if
24:44 we're going forwards or backwards.
24:47 This guy at this point,
24:54 his speed is like 5.
24:56 something, maybe 6, but no,
24:58 probably 5. something. Whereas this
25:00 speed at this point is,
25:04 it's going to be more
25:08 to find, well, we'll check
25:10 both on the graph, but
25:12 I'm pretty confident from looking
25:14 at the graph that his
25:14 maximum speed is going to
25:16 here. Okay, let's get the
25:18 graph. So it's either going
25:23 menu analyze graph. I'm going
25:28 to get my minimum lower
25:30 bound upper bound. So here
25:33 it's negative seven point set
25:36 negative seven point four seven
25:39 one. And this guy, I'll
25:48 note, I am looking at
25:49 the y coordinate because that's
25:51 the velocity. This is the
25:52 time, that's the velocity. Time,
25:55 velocity. His velocity is obviously
26:00 more than his velocity, but
26:02 his speed is bigger than
26:04 his speed. So the maximum
26:22 7 .471 meters per second
26:31 and look that little bit
26:33 of drawing that I've done
26:34 on the graph is certainly
26:36 enough working. You could also
26:40 say happens if you really
26:42 want to just say happens
26:51 1 .16 okay that's it
26:54 so obviously I've gone through
26:56 a lot there kinematics is
26:58 quite a big topic and
26:59 I've taught it all to
27:01 you in one lesson and
27:03 if you understand everything I've
27:04 done there fantastic you're in
27:08 shape, I would actually advise
27:11 do that question I've just
27:12 done again yourself and then
27:15 watch the video after just
27:16 to make sure you know
27:18 what you're doing and that
27:19 you've got it correct. The
27:23 key points to note are
27:26 obviously you know what displacement,
27:27 velocity, acceleration are. This SVA
27:30 thing differentiates to get from
27:34 displacement to velocity and velocity
27:36 to acceleration and then integrate
27:38 to go backwards. And then
27:41 when you are given a
27:42 velocity time graph to make
27:44 sure you understand what's actually
27:47 happening is the particle going
27:49 forwards or backwards. Also, be
27:53 careful. Sometimes they don't give
27:54 you a velocity time graph.
27:58 displacement time graph and it's
28:00 even possible they can give
28:01 you an acceleration time graph.
28:02 So don't assume that it's
28:04 read the question and make
28:06 sure you follow exactly what