ZOOM PART 2B | RELATIVE DISPLACEMENT & INTERCEPTION | S6 MATH TERM 1 HOLIDAY | MON 11TH MAY 2026 |

Ajouté : 2026-05-13

[00:00:03]Okay.

[00:00:09]So just doing a review that if they they give you V A B it means the loss of A relative to B meaning B is the observer.

[00:00:19]That's what should be in your mind observer. And how do we expand it? It will be VA minus VB. But the condition is that this should be a vector that should be in vector. So they can give you a question whereby the velocities are in vector form. It is okay. We can just code this formula direct. But if the questions are in terms of magnitude and direction, then the formula it is still the same. But we do something funny. we say VA plusative VB whereby negative VB means reverse velocity of B like that so each time you're going to draw a geometrical diagram remember that there will be something called reversing the velocity reversing the velocity and the rule is still the same as it was in resulting velocity that the velocity they give you must be constant so the bodies is move with constant velocity.

[00:01:21]Let's begin. It says suppose a particle A starts from point X. Okay. With position vector A R. So sorry R. Now in vectors we could just say O A to mean position vector. But here in this in this topic R would not have position vector or a displacement. Position vector or a displacement. Just as V means velocity vector like that.

[00:01:50]So with a position vector this and moves with a constant it should the velocity should always be constant to point Y in time t as shown below. So let's try to show an just an animation. Let's assume.

[00:02:11]Okay, this is the ship.

[00:02:14]Mhm. That the ship. This is just a 3D view. You So we have this one called the origin because all position all vectors they have a origin. That's why they say vector of A is O A.

[00:02:32]Just a moment. Network may go off.

[00:02:34]Hello.

[00:02:37]Yes.

[00:02:43]Okay, let me check.

[00:03:25]Okay, I've been telling you that in vectors we always have an origin. For example, if they say vector of A, it means we [clears throat] shall write as O A. Why? Because we assume the origin is we take origin to be O. So the point if the particle is at point A we in vector form we say O A to show that there's an origin then based on the origin where is that particle like that so this shape it is here but in vector form we shall need to get this distance from here to here and we shall call it now r in our case in the yeah in what we write we wrote we said it was it's a position r a so that's what they mean by r Based on the origin, where are you located? We call it R A.

[00:04:13]Based on the origin, where is A located?

[00:04:17]Mhm.

[00:04:20]So this will be now R A. Not that here now R means like position vector or position or displacement vector in this topic. So when it moves to a new position in that position is now position y. Mhm.

[00:04:41]It has moved with a certain velocity.

[00:04:43]The velocity is constant and it has moved in time t. So that mean that distance covered. We all know that distance is equal to speed time time.

[00:04:53]That is if speed is constant. So that means that if they ask for the displacement from the starting point we shall just say VT that is called displacement but not just any displacement it must be displacement from from what from starting point or from initial point starting point I want you to get used of these words because they will always ask you they can say find the displacement from starting point. What do you do? You just get velocity time.

[00:05:31]Velocity time is distance displacement from starting point. Okay. Another word they use is called displacement from origin. Do you see origin is here? Now what will be the displacement from the origin? Means from this origin to the final point where the particle is. So this distance is called displacement.

[00:05:55]displacement from origin.

[00:06:01]Okay. If they don't want to ask for distance and if they don't want to use the word displacement from origin, they can say find the final position vector.

[00:06:10]Final position vector. Why are they the same? Because we know that in vectors position vector you base on the origin to to write the position vector you base on the origin to write the position where the boat is.

[00:06:30][snorts] So these two words mean the same. When they ask for position vector after time t it is the same as displacement from origin. But be keen there's a second type of displacement which is which you call displacement from starting point.

[00:06:48]Displacement from starting point is just from X to Y. Displacement from origin it is from O to Y. Get you should must be of those two terms. Now what do we write? What symbol do we use for position vector at any time t? We write this but add on a bracket to show us that this is called position vector at any time t.

[00:07:13]at any time t we added this one in bracket. If we write like this and write here six, it means that [snorts] it's another word to mean that position vector after six maybe 6 seconds or after 6 hours depending on the unit of t like that.

[00:07:35]This one we just write a this one here.

[00:07:37]This one just write a then here we just we write vt. Okay, now we know the rules of vector that if I want to move from here to here, that's what we did more in O level vector geometry. It means it is the same as moving from O to X. Okay, then from X to Y that can give you O Y. So that means that O Y O Y is equal to O X + X Y. But what is our o x? O x we know it is r a and what is xy? xy is bt. But remember v is a vector and this this is a vector. So that is where this expression came from.

[00:08:27]If we go back I didn't put some arrows.

[00:08:31]Let me put some put something.

[00:08:59]Okay. So that animation means that now we can understand these words. Let's read them and we see suppose a particle A starts from point X with position vector R A tilda to show a vector and moves the constant velocity VA to point Y in time t as shown below. Now this what they're saying the initial position vector o x is noted by r a. So r is a symbol we shall always use for position vector or displacement vector. The displacement vector xy in time t will be given by tv. Reason distance is equal to speed * time time.

[00:09:44]Now our speed is now velocity. This one is so why are we using this formula? We using it because velocity is constant. We using it because velocity is constant. If it wasn't constant, we would go back to the equations of motion. We' go back to the equations of motion which says that s is equal to u t + half a t^² like that. But because velocity is constant, acceleration is zero. So we only remain with vt like that.

[00:10:15]That was that line. Then shall go this line. The final position vector o y of the particle is noted by r a t. Now not the difference here is just a A. Another way of writing this can be R A Z to mean that at time tal it means that at any point at any time t at any time t like that. So that this is this is now the position vector after a time t. Position vector after a at time t as you see it here. What is the relationship between these three? It comes from vectors that if I want to get O y I can move from O to X then from X to Y. O Y I move from O to X then from X to Y. But what now? Substitute. We know that O Y is RT which is this. We know that O X is R A and we know that XY is T T A sorry TVA.

[00:11:20]So basically that was the first slide.

[00:11:22]If there's a question, I encourage you to ask him.

[00:11:56]Okay. Is the first slide. Okay.

[00:12:00]Hannah.

[00:12:09]>> Um >> Hannah, are you there?

[00:12:11]>> Yes, I'm here.

[00:12:13]>> Is it understandable the first slide?

[00:12:17]I've just joined so I'm hoping I'll understand as we move on. Okay then Morin.

[00:12:30]>> Yeah it's okay.

[00:12:31]>> Okay [snorts] that's good. So once we understand the formula then we can easily apply it can easily apply it.

[00:12:42]So in that first slide we must have realized that this a relationship between position vector at 10 time t will be equal to initial position vector plus the [clears throat] displacement which is velocity time time initial position vector plus the displacement plus the displacement which is time velocity. So because now they were saying here what if there now two particles what will happen? So suppose there are two particles A and B start simultaneously meaning they start at the same time that is also key in our in this topic from points with position vectors R A and R B those are now the initial positions moving with constant that is the key always all the questions in relative velocity and also resultant velocity velocity must be constant velocity must be constant.

[00:13:44]Mhm. If it is not constant, then you have to go back to the equations of motion. The equation of the path taken by particle A is given by this. I believe by now we know it. Mhm. Equation of the path taken by particle B is given by this. Mhm. Which is also understandable. But what is the expression for relative displacement?

[00:14:07]Narrative displacement.

[00:14:10]Previously we saw what we call relative velocity. Velocity of one as seen by the other. Okay. Narrative displacement means displacement of one as seen by the other. Still in velocity this one it meant what is the velocity of A?

[00:14:29]velocity of A as seen by B or as it appears to be. That means that when it comes to relative still the same thing A means what is the relative displacement of A as seen by B. In short, what is the distance apart between these two bodies?

[00:14:54]So relative displacement here it means the distance apart between two moving bodies at any time t at any time t the formula still remains the same because we said in v is given by va minus vb okay so that means that if it is r a b it will be given by r a minus r b okay what about if it is r at end time t still the same thing a time t will be equal to a at end time t minus b at end time t. That is what this first is saying here that a b is equal to a at time t minus r b at end time t. Okay. Then you remember what we wrote here that this one can be expanded to be this. This can be expanded to be this. Let us substitute this put there this and this put there this like that. What will happen when we rearrange and try to collect like terms?

[00:16:03]Collecting like terms means bring this and this closer also bring this and this closer like that. When you do that closer we shall come up with r a b a minus r b plus t in brackets v a minus vb. And the good thing we know this subtraction is the same as R A B. This subtraction also is the same as V A B.

[00:16:31]So in short we come up with an expression that R A B T is equal to R A B + T VA. It is also connected to this one. If you look at it, this one was just RA A. Now this is RA A B. Mhm. This one was just RA A. This is just this now R A B. This one was T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T and this one was VA. This is now V A B. These have some relationship if you look at them.

[00:17:02]Now this R A is our interest in today's lesson because today's lesson we want to look at the case where there is collision.

[00:17:10]case where there is collision and this one gives us the distance apart at any time t meaning when they collide this distance apart is zero. Let's give us let's first look at this animation just Mhm. [clears throat] Now these are two boats. There is one here ship. Okay.

[00:17:35]Patrol ship A and B.

[00:17:38]For a certain velocity these ships may collide or they may just pass close to each other depending on the velocity with which they are set. Now in this case let's choose a velocities which will make them collide at some point. So when they set off at the same time they will move but at a certain point they will collide. Do you see this? Now what does that mean? It means that in this at this point here R A T will be equal to zero. That is the condition for collision.

[00:18:20]condition for collision. Sometimes they call it collision, other times they call it interception. It is still the same word interception. They intercept or they collide. So the same word. So when the distance apart is zero, we say they will collide. If it is not zero, we shall say they have some distance apart.

[00:18:42]For example, as they move, for example, when this one reached here, this may been here. When they ask AB in that point it means that is this distance apart this one that becomes now our R A T. So r a bt is the distance apart between the two bodies. Distance apart that should be in your mind distance apart. And when they collide the distance is zero. when they are they they can also ask closest distance but that will be seen another day cuz for close distance if you look at this they're still in the same direction but when I alter the speed something happens and what has what have I done when I alter the speed start at the same time something will happen they will not collide but they will pass close to each other there will be a point whereby they will be closest That is why the word closest approach is. But that condition is not for this lesson. For this lesson, we want to see collision. We want to see collision. Okay. So with that we can first go back and read this slide.

[00:19:59]It says if these particles collide after a time t it implies that their relative displacement the relative displacement of A to B at that time is equal to zero. That is the condition for collision. The distance apart is zero. One is here another one is here. The distance apart is zero.

[00:20:23]That's the condition. So for collision we said r a b t this t should be there to show that it's displacement at any time t because if you just write a b it means this is just initial distance apart which is [clears throat] not okay which is not okay. Okay. So it has to be displacement after some time. After some time it is the one which is equal to zero. And the good thing we know that have to expand this. It is expanded as initial relative displacement plus the displacement which is time relative velocity being equal to zero. So that will be the condition for collision which you're going to use in this lesson. Okay. This one is here. You are trying to define what RA means initial displacement vector relative then also VAB what is it like that when I take this this side something happens and that's something which happens is what we're going to use when it comes to geometry though we're going to begin with vector we going to begin with vector but when I rearrange I come up with this and that is a condition for parallel vectors if you look at it one is a scalar multiple of the other. If I say if for example I have have a = 2 BC there it means that these points are colinear or paral colinear or paral is means same lines when I just say par means they can be next to each other but same direction like that. So this condition here is a condition for parallel or colinear vectors. Parallel or colinear vectors. That is why the condition we're going to use when it comes to geometry. But the first for this lesson we're going to begin with vector approach.

[00:22:20]Okay. So there's a question in the in the slide you can ask cuz now from here I'm going to go to the item.

[00:22:28][clears throat] Okay. So the silence means it is better.

[00:22:48]Then let's go to the questions.

[00:22:53]These ones you already seen.

[00:23:00]So question one we sh like I said we're going to begin with vector approach.

[00:23:04]Question one says during a robotic competition at Mccary University. So the items are already in one group.

[00:23:13]Two automated robots P and Q move across a flat flow using programmed motion sensors. Okay. The positions of the robots are measured using vectors XY.

[00:23:27]Now that is called motion in two dimension.

[00:23:31]If it was in three dimension there would have been also Z X Y Z. But this is now motion in two dimension.

[00:23:38]At a certain instant robot P passes through a point whose position vector is this. Okay, that is now R P R P. If you to use the symbol, we know R and continues moving with a constant velocity. This is key in this topic.

[00:23:56]Velocity will always be constant.

[00:24:05]Mhm. at the same instant like this condition because they must begin from the same time. You must time simultaneous time them simultaneously.

[00:24:16]Mhm. Robot passes through a point whose position vector is this. Now this is R2.

[00:24:24]Mhm. While moving the constant ve this now V like that. The good thing they are already in vector form. So we need to just remember the formula. Mhm. The robotic team wants to study the relative motion of the two robots as they move across the floor. Okay.

[00:24:47]Determine the displacement of P relative to Q after time T. That means I want R P relative to Q after time T. That's what they want. The first thing to do is to remember how it expansion at first it must be initial then plus what displacement which is TV pu like that. Then after that you ask yourself how can I get P2 and how can I get V P2 R P will be R P minus R2 VP will be VP minus V2 like that. So the good thing this is vector. So once you know the formula you simply substitute simply substitute. So that's the highlight. So we shall come and see what to do.

[00:25:41]First get your r2 from r p minus r substitute. Like I told you in vector form it is better use column because if you choose to use I J K you must put the two down every I every J and every K in vectors it is advisable to use column form when you're calculating because if you use I J K you must put this tilda symbol on every I every J and every K like that but everything remains the then also get the V P2 which is VP minus V2 the symbol has to be seen substitute and when you come up with this now you're noticing something funny I'm not putting their units do you know why here they also didn't give us the units here I don't have any unit for velocity I don't have any unit for initial position so that is why I'm also not committing myself to a certain unit some students are funny they y that is not okay that is not okay so when they don't put a unit maintain don't put the unit when they put the unit put the unit like that okay now that we have that we can now substitute in our formula which we know bring this substitute bring this substitute so that is now called the general equation of P relative to Q at any time T. P relative to Q at any time T. So if there's a question you ask There's a question I encourage you to ask. So Roman 2 says distance between P and [clears throat] Q after 5 seconds.

[00:27:53]Now that is relative displacement of the two because we said this one means distance apart after end time t distance apart after time t. So this Roman two my work is to simply come and where there is t I put there five I have got the distance apart after five seconds.

[00:28:24]So when t is equal to 5 where there is t I'll now put there five meaning here also this t I'll put there five then I can use a calculator now to come up with this to come up with that and like I've said you don't commit yourself to put a unit because they didn't give it to you in the question they didn't give you the unit in the question okay now because there there's something we know about Distance. Distance is different from displacement. How?

[00:28:57]Distance is a scalar quantity.

[00:29:01]Distance is a scalar quantity and displacement is a vector. So this one is called displacement apart. Displacement apart. But because the question distance, you don't stop there. You go back. You go ahead and get the magnitude.

[00:29:21]go ahead and get the magnitude that is what now what we call distance magnitude is what we call distance because distance is scalar okay so that's what they wanted in that item one here is a question I encourage you to Excuse me, sir.

[00:30:08]Yes.

[00:30:10]>> Um when you're finding RP on Roman numerous 3 can we substitute into that one?

[00:30:26]>> Yeah, it is okay. It is okay cuz they mean the same.

[00:30:31]>> They mean the same.

[00:30:32]>> Yeah. What I wanted to emphasize is that in this Roman one you don't stop here because when they when they say vector you must give one vector I 1 I 1 G like that but here it is not yet complete for Roman one this one is the complete one for Roman one like that same applies to this this is not complete for Roman one it is this one which is complete 1 I 1 J 1 K that's how your final answer should be >> thank Thank you.

[00:31:03]>> You're welcome.

[00:31:09]>> Okay, there's a question I encourage you to ask.

[00:32:00]Okay. So that can now we can go to item two. The difference between item one and item two, they're almost the same. The only difference I wanted to bring out is that this one is now motion in three dimension. motion in three dimension but all the procedure will be the same. The procedure will be the same. Let's try. During a satellite tracking simulation at Chamogo University, physics students are studying the motion of particles in three-dimension space using vectors.

[00:32:38]Okay. At a certain instant a particle P starts moving from a point whose position vector is this. Now be keen when it comes to 3D you should be keen.

[00:32:50]It must be I J K. But here there's something funny. There is no I. Even though they didn't put it here you must know that it it should always be there.

[00:33:00]So your work is to come and put there zero I when you're writing 0 I 2 J and 2 K. M okay when it come to this also there's no I but you must put it there so you must write 0 I + J + K like that so big when it comes to 3D at the same time another particle Q starts from the point whose position vector is this now the thing all of them are there there's I there is J and there is K okay but still there are no units not that there are no units and moves with a constant velocity. This all of them are there I, J and K. Okay. The students are required to analyze the relative motion of the two particles as they move through space. Okay.

[00:33:55]Calculate the distance between P and Q after 6 seconds.

[00:34:04]After 6 seconds. So they want they want now here they didn't specify which one to start with isn't cuz it can either be this and this isn't Mhm. So when they don't specify it is you to choose which one you want. Okay. But if you want to to if you want to get a guideline like I told you last time that usually the one observing must be moving slow should be moving slower. If he moves faster a lot of things happen and the car tends to be moving backward. So here it is advisable though is not a rule but is advisable that this one being observed has a higher velocity. So you work first come and see this velocity here magnitude is what 1 2 + 1 2 which is <unk>2. What about this one here? You already 2 + 1 2 + 2 2. So this one is bigger. It means that you advisable to use arp choke and arp.

[00:35:11]But like I've told you it is not a rule that when you don't do it you don't get the max because I didn't specify even when you do this you'll get some negatives but because you're consistent you'll be correct because our aim is to get the relative displacement relative displacement. So for me what I advise is to you to first look for the highest velocity highest speed the magnitude which is bigger and make it the one being observed like I've told you it is not a must it is not a rule it is just for simplicity it is just for simplicity so if you don't do it as long as you get the relative displacement that is okay as long as you get the relative displacement that is okay so me I've chosen to do this because VP Okay, you're not seeing the V. But if you look at this V, this one, the magnitude here is bigger than the magnitude of this.

[00:36:10]That's why I've chosen. But if someone does the other way around, it is okay.

[00:36:14]It is okay.

[00:36:18]So come and formula substitute when t is six, what do you get?

[00:36:25]Substitute for t.

[00:36:28]And remember they said let me first see they must have said distance let yeah they said distance means magnitude.

[00:36:36]So when you stop here it is incomplete because this one is called displacement.

[00:36:44]Displacement but the question wants distance.

[00:36:48]So that means that it will be incomplete. You're on track but it is incomplete. So what do you do to make it complete? Come and get the magnitude.

[00:36:57]come and get the magnitude and that is what they want.

[00:37:11]So there's a question you ask Okay. If there's a question you ask.

[00:38:31]>> Excuse me, teacher.

[00:38:32]>> Yes.

[00:38:35]>> So, could you please explain why we do the magnitude thing?

[00:38:40]>> Magnitude is because this >> when do we do it? When they said distance for example this here in this task there are two words they can use let me pass here here they can say calculate that displacement.

[00:38:58]So if instead of the word distance they put their displacement. We know that displacement is a vector.

[00:39:05]It has it must have magnitude and direction. So we leave your answer in vector form. But because they said distance, distance is not a vector. It is a scalar. And scala means we only want magnitude.

[00:39:21]Scala means we only want magnitude. So basically that is why we went ahead to get the magnitude.

[00:39:31]I hope it is it hope it is better.

[00:39:36]>> Yes teacher. Thank you.

[00:39:38]>> Okay.

[00:39:58]So relative displacement is distance apart at any time t. Relative displacement is distance apart at any time t. Remember that cuz we're going to overuse it when it comes to collision and also when it comes to closest approach.

[00:40:16]For example, this one there be some collision. Let's read and we see it says at exactly noon Mhm. during a marine surveillance exercise on Lake Victoria, two patrol ships A and B, okay, are being monitored by a navigation control center using a coordinate tracking system. Okay. At noon the position vectors and velocity vector of ship A are recorded as this.

[00:40:52]So arrow we said arrow means position vector v is velocity vector that is for a at the same instant. This word is very key because when they don't start at the same time it now changes a lot. We have we have to look for making them start at the same time. we have to look forward to making them start at the same time as we shall see some someday among these items. So shift B has portion vector this and vector this. Now the good thing here they've given us units. Do you see these ones? They've given us units meaning your answer must have units.

[00:41:30]Okay.

[00:41:32]And here we have only I J motion in two dimension. The control officers realized that if both ships continue moving with constant velocities without changing direction or speed Mhm. there could be a danger of collision. They therefore need to determine where the whether the paths of the two ships will intersect and if so identify the exact time and location of collision. Time for collision and also the location. Location is like position is like position position vector like that.

[00:42:16]Okay. Mhm. Task one. Show that that two ships will collide. Show. The key word here is show. Show that. So you don't know yet that they are going to collide, but you you are trying to show that they are they will collide. Show is a key word. You don't know yet, but you are trying to see if they will collide. Show that two ships will collide if they continue with their present speeds.

[00:42:46]Okay. Determine the time at which the collision occurs. Mhm. Find the position vector of the collision point.

[00:42:57]Okay.

[00:42:59]So one, we need to bear in mind that for them to collide the relative displacement we used, did they use letters? Let us see.

[00:43:10]Yeah, they used A and B. So it mean that the relative displacement must be equal to zero at any time T.

[00:43:16]And like I've said, they have not specified in the question which one to use. So if someone uses this condition, it is okay. If another one uses this condition, it is still okay because they both relative.

[00:43:31]They're both relative. But if you want to use the the guideline I gave you, you have to come and see here and here and say which one has the biggest magnitude.

[00:43:41]You realize 15 squared + 10 2 square root that one and also this one of 9 squ + -5^ 2 square root you this one has a bigger magnitude. So it is advisable if you wrote R a B like that is advisable reason this one has a bigger velocity bigger magnet velocity but if but I I emphasize I emphasize that it is not a rule that you must if you don't follow it will not be okay it is not a rule that if you don't follow it will not be okay what the rule is relative displacement equal to zero full stop the rule is relative displacement equal to zero full stop the order it is due to choose by yourself. Okay.

[00:44:36]So we shall begin by saying getting the the initial [clears throat] initial displacement r a minus r b they give it to us give this to us when we subtract 7 5 - 7 -2 2 - 7 [clears throat] like that the units are there because they are given in the question we are putting units because they are given in the question so what does that mean if you don't put units again that is problems. When you don't put units that is again problems also get the velocity units are there because they are given then you remember the condition for collision and see if this side will be equal to this side that side will be equal to that side. So let's try that equal to that. Mhm.

[00:45:34]this using the first part. Okay, this zero is okay just put zero because it is zero but it is it's also a vector it is 0 0 like that so that you equate the i components and also equate [clears throat] the j components that's how they equate vectors that's equate so to be more clear you put 0 here in vector form you put 0 0 in vector form so I've used the upper part if I use the lower part and I get the same value of T it means that T is color if I get different value of T it means that T is variable which we don't want because the condition is that this should be constant should be a scalar should be a scalar like that so now that but when I've done it I've got the same value of T meaning T is a constant t is a constant now because T is color and this one is equal to zero. It implies that the two ships will collide.

[00:46:44]Let me repeat because t is color. How do you know that is color? You do for all component. If there was also k I was supposed to do for k also but yeah this is two dimensions. So you do for i and also do for j. If you get the same value of t that's when you can say t is color.

[00:47:02]If the values are different, T will be variable.

[00:47:06]T will be variable which is not okay.

[00:47:09]Which is not okay. So because you have seen that T's color and this relative pos displacement is zero, then we can conclude that they collide like that. So that was Roman one. If there's a question you ask Excuse me teacher.

[00:47:35]>> Yes.

[00:47:39]>> Um when we were working out we were using a A and B but in the conclusion it's B A >> A. Yeah that is typing mistake. Let's edit that very first. Yeah we have to be consistent. So this is going to be now a b a b and this will also be yeah I think there is no bit there is no Okay, that we are going to reify that.

[00:48:38]If there's any inquiry also ask Okay. So now we can that is showing that's how they show that is how they show show that left hand side is equal to right hand side.

[00:49:29]So that means that our t which is a scalar is now the time taken for collision. But remember in our question they gave us the hour in clock time.

[00:49:41]And this first thing this one also is very key known meaning that you should also give your time basing on this reference from here how many minutes past and what is the time currently at the point of collision that is what they want they gave us the time in clock time I don't know if it's understandable so now that mean that we shall come and say after that on the noon we add on the 20 minutes to give you 12:20 now it becomes P.M. it becomes P.M.

[00:50:19]So that's the way of using this because of the other reference I've told you the noon the noon. So when they give you time in clock time you also give your answer of time in clock time. When they give time in clock time you also give the answer in clock time like that.

[00:50:40]Then they also ask for what? Let me see.

[00:50:44]They ask for the position vector. Mhm.

[00:50:46]position vector you choose n you can because they they when they meet when a comes when this is a comes and also this is b and comes ar a and ar b are all at the same point so whether I use ar a or I use ar b it is you'll get the same answer you'll get the same answer because at the same point they are at the same point like Mhm. And because I said position vector, yours is in vector form.

[00:51:22]Okay. So, unless there's anything to ask, that was item three.

[00:52:31]Okay, let me just double check.

[00:52:37]Okay, Nimalin, is the solution. Okay.

[00:52:43]>> Yes, teacher.

[00:52:47]Napoleon I want this is the solution okay?

[00:52:56]Yes teacher.

[00:52:59]>> Then we shall now go to Esther Lucy. Is the solution okay?

[00:53:12]>> Yes, are we there?

[00:53:21]>> Yes, teacher.

[00:53:23]>> Okay. [clears throat] Okay, that is it. That is it.

[00:53:30]So that has been vector. That has been vector vector. But not it's not much that they always give you questions in vector form.

[00:53:43]Sometimes they can give magnitude and direction. Now when they give magnitude and direction, it is you to choose what you prefer. If you if you are stuck to vector form, it is still okay because they will not restrict you that use geometry method or use vector. So they give you this question like this. Do you see this? There's magnitude there speed. Okay, let's first read it and you see it says during a navo security exercise on Lake Victoria a patrol ship is moving due north at a constant speed of this. So in short they've given you that this one velocity is 12 and you know north like that. So if you want to use the method of vector it mean that you should come and change this velocity in vector form and write zero zero means zero horizontal everything is entirely vertical like that. Okay, let's go to this one. At the same time. This word is very key because we I want you to always remember that they must begin at the same time.

[00:54:54]When they begin at different times, you must look for a way of making them begin at the same time. An example, let me give an example. They can say one begins at 12 noon. Okay? Another begins at 1 p.m. So what do you do? The first thing to do is to get the position vector of this one which began at 12:00 noon when it is 100 p.m. So in short after 1 hour where will it be? So now we have 1 p.m.

[00:55:21]and 1:00 p.m. Then they can begin moving. So make always it must begin you must begin timing at the same time. You must begin timing at the same time. So this word is very key.

[00:55:36]Another destroyer traveling at a speed traveling at this traveling at this speed is located 60 30 km due east of the patrol ship. Okay. Now here the we have given us magnitude but know this the direction what do we do? We just if you want to use vector form we can say v maybe vb be magnitude like that cuz when you just write this one we expect a vector but because we don't have the direction we just use magnitude like that I'm just giving you an option of if you want to use vector approach but for this solution we're going to use the geometry we're going to use the geometry the command the command center receives information that the destroyer must immediately intercept the moving ship. Okay, since both vessels continue moving with constant, this is also key. Velocities in this topic must be constant. The navigation offices must determine the exact direction of the destroyer.

[00:56:49]the exact direction the destroyer should take should follow in order to meet the ship successfully.

[00:56:59]Okay.

[00:57:01]The officers also need to calculate how fast the destroyer approaches the ship and estimate and estimate the time required for the interception.

[00:57:17]How fast the destroyer meets that is now [clears throat] relative velocity. If you remember velocity means the rest of this as seen by the other. The rest of this as seen by the other. Okay.

[00:57:33]Assuming that neither vessels change their speeds or direction. Okay.

[00:57:38]Determine the course. Course means direction. Course means direction.

[00:57:43]Course which the destroyer should take.

[00:57:46]Mhm. Two, find the velocity of the destroyer relative to the ship. Okay.

[00:57:55]Calculate the time taken for the destroyer to reach the ship.

[00:58:02]Uhhuh. Now here we are going to begin.

[00:58:08]Be attentive. One, we going to first draw the just as we did for resultant val. I think you remember when we doing when we are doing velocity we first drew individual velocities then we went to the vector diagram. So here the the first thing to do is to know what to reverse.

[00:58:29]Know what to reverse. What are you going to reverse? Because you want this. If you look at this question here, if you look at this, it is V destroyer relative to shape. And in geometry, we're going to write V D plus VS. That negative VS means reverse VS. Reverse VS. That is why you're seeing me reversing this one to give me negative VS. Okay, that is one point. The next point, show the initial positions.

[00:59:05]Initial positions. They told us something here.

[00:59:11]They told us when we go back the question, they told us about location 30 km. What is it? At the same time the destroyer is is located 30 kilometers due east of the patrol ship. Destroyer which is D is due east. Where is east? Let me see.

[00:59:32]East is this side. So this is the destroyer. This is the ship like that.

[00:59:37]So locate the initial positions.

[00:59:43]So we shall come here and locate the initial positions here.

[00:59:49]Mhm. Now why are we putting this? It is because of the other condition we saw.

[00:59:54]Let's first go back to the that slide of introduction.

[01:00:00]The introduction here. Yeah. Now if you look at this introduction, this was the condition for vector approach. Okay. But for for geometry, we take this this side and we come up with this. What does that mean? It means that they are parallel vectors. Though in this case, I'm now going to use the word colinear. Coline colinear. So this one show that the this and this must lie on the same line.

[01:00:31]The relative the initial displacement and also the relative velocity. This one means initial displacement. How far are they apart? Isn't then this one means relative velocity. They are on the same line. That is why you are seeing me doing this. Where are we now?

[01:00:52]I encourage you to follow. I encourage you to follow.

[01:00:57]That is why you're seeing me that after put after locating the initial positions, I'm going to put my relative velocity on that line of the initial position. And just a moment network may go off. Hello.

[01:01:12]Yes.

[01:01:14]I'm wearing blowing out suit.

[01:01:33]Okay, clear back. [clears throat] I was saying they the there are steps you need to follow. One, know what to reverse. So, we have reversed it. Then two, locate the initial positions. If you can, I encourage you to write them somewhere. One, know what to reverse.

[01:01:52]Okay, reverse the known direction.

[01:01:54]Reverse known direction. Reverse velocity with non direction.

[01:02:09]That is one. Two. Locate initial positions.

[01:02:14]Locate initial positions. That was this and this. Then three, put relative velocity on that initial positions. Put relative velocity. Put relative [clears throat] velocity.

[01:02:36]Relative velocity on on the line on the line joining joining the initial positions, not the steps. Mhm. We said one we first know what to reverse then two locate the initial positions and three put the relative velocity on the line joining the initial velocity. Now the direction how do you know the direction? We always know that the one being observed is one which is moving. That is why it is this one is the one being observed. D if you look at this when I say v ds it means s is the observer is here is the one observing this one moving. That is why the position is the one moving to the observe to the observe. Mhm. To the observer. That is it. After putting this then you come and put you put now here it is logic here it is logic just as we did also for crossing a river because we know that this one I don't know if I've finished writing these ones I want to remove them but before that let me just first explain do you see this one it is moving down isn't now here when I'm here there are two ways I can choose direction I can choose it to go this side I I can choose it to go downward.

[01:04:11]Okay. But when I choose it to go downward, do you know what this one will do? It will make me just go down and I will never reach here.

[01:04:20]Let me repeat. Do you see this reverse velocity? When I'm here and I go this side, the reverse velocity will just make me go down and I can never reach point S. Okay. But when I go this side up, this reverse velocity can make me come back to where I want.

[01:04:38]Therefore, the best option to use is this one when it is going up. So that this reverse velocity makes me to reach where this one is that is now up logic now cuz there are two options going this side or this side. But which one is better? You look at what you have reversed. Will it enable you to reach where you want or not? So this one down couldn't enable me to reach. So I leave it. But this one up could enable me to reach. That's why I took the one up. Then I went here like that. If there's a question on that diagram, I encourage you to ask cuz it is diagram which determines everything.

[01:05:31]>> Um, excuse me teacher.

[01:05:33]>> Okay.

[01:05:35]U my question is about the direction in which the 36 the speed of D is >> so you wanted to first to go down >> uh no not not that exactly >> okay what did you want >> why why do we assume that it's um like in the direction of west and maybe not east >> and not this side Yeah.

[01:06:05]>> Okay. I'm going to answer you. But there you can aim to reach here S. That's why that of D S means that it means that when S is stationary, D will move. But for me to move from D to S, this I cannot set my boat here because when I set this side and I add on this, it will just make me go elsewhere. I can't reach S. That is not okay. If I go down this one, adding it here, it just make me go elsewhere. That is not okay.

[01:06:40]If I choose this quadrant still, it make me go down when I add on this. So, it's not okay. The only option which will make me achieve what I want is when I go this side. So, that adding on this one, it makes me fall back to where I want.

[01:06:56]That is the same logic we used under crossing a river. If you can try to relate. If you can try to relate. So now you can come back and now ask again or it is better.

[01:07:10]>> Yes, it's better.

[01:07:12]>> Okay, that's good. So here after going through the other three steps, next is now logic looking at what you know. How can I set my destroyer ship to intercept? I can't set it here because this adding on this it will take me this side. That is not okay. I can't say it here because adding on this will make me take this side. Not okay. I can't go this side. It make me go down. So the only true option is this one that this so that this reverse velocity will enable me to reach where I want. Okay.

[01:07:49]And like we said under result velocity after drawing your diagram you must look for a way of getting one angle inside.

[01:07:58]And the good thing here the angle is already there 90. The angle is already there. Meaning we can proceed with the calculations.

[01:08:08]After drawing your diagram you must look for a way of getting an angle inside the triangle. If you don't get the angle, you cannot you cannot go proceed to the calculations because either cosine rule or sign rule or so they will still need an angle. They will still need an angle inside. So now that we have got an angle inside we can proceed with what they want. Let's begin. When [snorts] they said determine the course means direction. Mhm. I now question they gave us in terms of compass if you remember if you look if you recall let's first go back and I show you here do you see this east there's also this north okay meaning you're going to give your direction in terms of compass in terms of compass so when I know this when I know yeah when I know this I can get this angle so I would say it is north west north that angle west like that. So how do I get theta then because this is a right angle I'm going to use soa says this angle is opposite and this one is hypotenuse therefore I'm going to use sign I'll use sign when I use sign I'll come up with the value of theta as 19.47 47 angles at two decimal places.

[01:09:42]Okay. So I used bearing it is not pro is not advisable. So we have to change to to compass direction.

[01:09:52]If I send 90 90 - 19 - 19.47 it is 73 70.53.

[01:10:06]Let's change it right away.

[01:10:10]We want north. Uh-huh. 70.53° degrees west. That will be the best way because the in the question it was also in compass direction. So you shall say that if you just write now direction is what is north in brackets we put there 90 - 19.47 west which gives us what? North. Mhm.

[01:10:53]70.53° degrees west. Yeah, that will be better now. That will be better. Or you can even just write there the word course.

[01:11:07]What? What was the question?

[01:11:11]Can just say course.

[01:11:16]Yeah, like that.

[01:11:20]There we shall be good to go. So, and this is a question you ask.

[01:11:27]That was direction. Yeah. This is a question on direction you ask.

[01:11:32]>> Yes. Hannah, >> I'd like to clarify something.

[01:11:37]>> Mhm.

[01:11:40]>> Um, you said that that's where we do the reverse.

[01:11:44]>> Oh, first I've not understood you before.

[01:11:50]>> Or geometry. That's when we >> geometry. Yeah. Because geometry we in we use plus then. So this time it was V SD V D minus V S. But we can't draw this minus in geometry easily. So what we do?

[01:12:05]We say V D plus V S like that. So it is true.

[01:12:15]>> Thank you.

[01:12:16]>> Okay. If there's any inquir on Roman one, I encourage you to ask.

[01:12:24]>> Um, excuse me, teacher.

[01:12:29]>> I've not got why we get 70.53.

[01:12:33]>> Why you get in the c I think gave us that you first there on I think you first subract and see unless I didn't press it well but as if it is correct.

[01:12:45]You know like why we we got 19.47 That's the angle of theta. Then went ahead to find 70.53 because in direction in compass we either begin from north or from south.

[01:13:02]Now here because it's in this quadrant we begin from north and reach here. What is that extra angle that alpha? We know that alpha plus theta will be 90. So what what will be alpha be? It will be 90 minus theta. I know. Is that better now?

[01:13:22]>> It's but Yes.

[01:13:24]>> Pardon? Yes.

[01:13:27]>> Yes, it's better.

[01:13:29]>> Okay.

[01:13:34]The question you ask we're about to finalize. So that was Roman one. Roman one they wanted course and I've told you if they give you in if in the scenario they give direction in terms of compass it is advisable you give you direction in terms of compass that's why you saw me changing at first I had given my direction in terms of bearing but in the scenario it was not in terms of bearing so I had to change and write in terms of compass okay now next they velocity of the destroyer relative to the ship V D relative to ship. It is this one here.

[01:14:18]So still there are two ways. If I use Pythograph it will be correct because by Pythagoras I can say V ds is 36^ 2 minus 12 squ everything under square root it is still okay. Another one can say I really know theta I can use so means adjacent over hypotenuse. So I can say cos 19.47 is equal to what is equal to adjacent which is vds over over what over 36 I'll still get the same answer. I'll still get the same answer.

[01:15:04]Okay. So let's just write it here.

[01:15:14]Yeah, that is it. And because it is velocity and velocity is a is a vector quantity, you also give the direction.

[01:15:21]Velocity is a vector quantity. You also give the direction.

[01:15:26]Okay. Us the question in Roman 2. to ask.

[01:15:42]Okay, then we can now proceed to the last Roman. Last Roman says, capture the time taken for the destroyer to reach the battleship. Uh the same thing I told you last time when they ask for in if it is resultant you use distance along the resultant and resultant velocity. Okay but now this we are now under relative velocity. So we are going to use distance along the relative velocity which is ds [clears throat] this one.

[01:16:14]Okay, now let's go back and and it was given in the question in the question it was this.

[01:16:22]Okay, that's good. So, we know the distance.

[01:16:31]We know the distance ds is 30. And we also know the velocity we have got. Okay, it was on the next slide. Interesting.

[01:16:45]Yeah. Distance is here. Yeah. No, this is this is speed and the distance is third. So my work is to come and divide and that will be my answer. That would be my So if there's a question you ask.

[01:17:00]So here we are not giving the answer in 12 hour clock because they didn't give us the time in our clock in clock time.

[01:17:07]We just use after these hours and that is all. That's how we conclude.

[01:17:13]Okay. Okay. So if there's a question you ask then two I encourage those with who have their pass papers for end of term one to inbox me for guys I already have they did more but for other schools I encourage you to send your passers and see if we can get some questions from there to benefit the rest of us.

[01:17:40]Okay.

[01:17:44]So there's a question on this item I encourage you to ask because the remaining items I it will be you to practice them. The remaining items it will be you to practice. So tomorrow we do now the other part called closest approach. What if they don't collide?

[01:18:03]What happens if they don't collide? They must at least have a point where they are closest to each other. That is what we shall see tomorrow. So I encourage you to ask if there's a question on collision.

[01:18:30]Mhm. Let me double check.

[01:18:33]So patience or model are you now ready?

[01:18:36]Are you cover that now with collision interception?

[01:18:42]>> Yes, teacher.

[01:18:44]>> Okay. Mhm. Nadong Bonji is interception now. Okay.

[01:18:54]>> Yes, teacher. Thank you.

[01:18:56]>> Okay. Then let me see.

[01:19:00]Gift Nva.

[01:19:03]Are we now okay with interception?

[01:19:08]>> Yes, sir.

[01:19:10]>> Okay. So, I encourage you to complete the remaining items so that tomorrow we do another concept called closest approach. Otherwise, we are good to stop.