Lecture 3 | Instantaneous Speed & Instantaneous Velocity | Kinematics | Class 11 Physics | NEET, JEE

[00:00:00]Hello everyone, welcome back. This is third lecture of chyomatics. In the previous lecture, I have taught you about velocity and speed. Then I told you what is instantaneous velocity, instantaneous speed, average velocity, average speed. And we have done question a lot of question based on average speed and average velocity. Instantaneous I told you what is instantaneous. I gave you a feel of that. But questions we have not done on that. In this particular lecture, we will do question based on what? Instantaneous velocity.

[00:00:26]Okay. So I want you all to revise basic maths at least differentiation before seeing this lecture. Okay. So [clears throat] let's have a quick recap of speed and velocity that we have discussed. So I told you what is a speed? Speed is distance per unit time. Is it total distance upon total time? And I told you we have two types of speed. One is average speed and one is instantaneous speed.

[00:00:58]Instantaneous speed. In the same way, what is velocity? Velocity is displacement per unit time. And we have two type of velocity. One is average velocity and one is instantaneous [clears throat] velocity. Is it? I have taught you this. I told whenever we have a time gap, whenever we have what? a time gap. Then we talk about average. So average speed will be what? Total distance.

[00:01:28]Total distance by total time.

[00:01:33]This is your average speed. What is sorry? Yeah. And what is instantaneous speed? Speed at a particular instant.

[00:01:41]Speed at a particular instant. at a particular instant which I can write as what but instant means what very small time a time tending to zero is it a time which is very very small 0 0 1 second in that small time whatever distance particle will cover That distance upon time is known as what? Instant speed.

[00:02:17]Speed at a particular instant.

[00:02:20]Is it? Let's say a particle is going from A to B following this path. So if someone ask you, let's say he starts with 2 m/s.

[00:02:32]Then the road was completely empty. He speed up 2 4 6 a turn came slow down 1.

[00:02:39]Again he speed up 2 4 10. Again traffic came. He came to rest again. He speed up, slow down, speed up, slow down.

[00:02:47]Finally, he reaches to be. This is the practical thing. Now, when you go from one place to another throughout the journey, your speed keeps on changing.

[00:02:56]If you look at the speedometer, it will show different value at different time.

[00:03:00]Yes or no? If I ask you from A to B, what is the average speed? Let's say the average was 5 m/s.

[00:03:10]What is the meaning of this average speed? It means if you also go from A to B with 5 m/s whatever time this person has taken same time you will take the 5 m/s is the average of all the velocities he had while going from A to B. It's not the reality reality is something else. Sometime he had 2 m/s, 6 m/ second, 1 m per second, 2 m/s, 4 m/ second, 10 m/s. At every moment he had a different speed. The average of all is five. So what this is average when you have a time gap 2 second gap, 4 second gap, 10 second gap, we talk about average. Next is instantaneous speed.

[00:03:53]Speed at a particular instant. If I ask what was the speed of this person exactly at 2:30 p.m. So exactly at 2:30 p.m. let's say he is here. What is the speed? Let's say 4.5 m/s.

[00:04:09]exact speed at that moment that is known as instantaneous velocity. What is the velocity at that or what is the speed at that particular moment that is known as instantaneous speed.

[00:04:21]So moment means what? At a particular moment means what? In a very small time interval in a very small time interval the distance covered will be very very small. So we can write this as what we can write instantaneous. I'm rubbing this part and on this we have discussed a lot in the previous lecture if you all remember please go and see that lecture before seeing this. So instantaneous speed will be very small distance which I will write as dx by dt. This is known as instantaneous speed in which this dx is what? Very very small distance.

[00:04:57]And this dt is what? Very very small time. Very very small distance. Very very small time. dx by dt. A small distance upon a small time. So this was what? This was speed. Average instantaneous. Now come to velocity.

[00:05:16]What is velocity? Velocity is displacement per unit time. This was distance actual path covered. This is displacement.

[00:05:25]Is it? Let's say let's say a particle has position vector R1 and that particle went from A to B here and now the position vector of the particle is R2.

[00:05:43]Clear? So what will be displacement? You will say so displacement is shortest path from A to B which will be straight line. This is displacement of the particle.

[00:05:52]And I can write this displacement as what? Position of B R2 minus position of A R1. Yes or no? You start from O. R1 + S will be same as R2. Is it R1 + S will be same as R2. So S is so R1 + S is equal to R2. So S is equal to R2 - R1.

[00:06:14]This is the displacement and let's say this the time taken to go from A to B is t.

[00:06:21]So again here we have two parameters one average velocity one instantaneous velocity. Average velocity means in the time interval you have a time gap you have a time interval then only we can talk about average is it? So you will say sir average velocity will be average velocity will be displacement upon time displacement will be r2 - r1 / t. This will be displacement of the particle. This is average velocity which also you can write as delta r by delta t.

[00:06:59]change in position upon time or displacement upon time or you can say time gap. What is instantaneous velocity? Again same velocity at a particular instant in a very small time interval.

[00:07:15]When the time interval is very very small, it will become what?

[00:07:18]Instantaneous. So what will be instantaneous velocity? The instantaneous velocity will be like what is average velocity?

[00:07:27]R2 - R1 change in position by time. So here time will be very very small. You you will call call that time as DT. In very very small time the change in position will also be what? Very very small. You call it DR. DR is a very small change in position. So instantaneous velocity is what? DR by DT. Where R is what? Very small change in position by DT. Or you can also write as very small displacement ds by dt.

[00:07:59]This is your instantaneous velocity.

[00:08:03]All these things we have discussed in the previous lecture. H on this we have not done question we will do in this particular lecture. Clear? If you want you can pause and write. So let's have a quick recap. So we have speed velocity.

[00:08:16]Speed is distance upon time. Velocity is displacement upon time. We have two type of speed. Average instantaneous. Average means in a time interval whatever distance particle has covered divide by total time. This is average speed and speed at a particular instant that is known as instantaneous speed. Same velocity average velocity instantaneous velocity. Average velocity when there is a time gap.

[00:08:43]Time interval. How much displacement particle has covered in what time?

[00:08:49]Displacement upon time. And what is instantaneous velocity? Velocity at a particular instant. Velocity at a particular instant. Clear? Now one thing you can write this if you want. We'll move to next page. Okay. [clears throat] So average we have done a lot of questions. I will write definition of instantaneous velocity and I want you all to write in fact can I say instant before writing definition definition instantaneous velocity magnitude will be exactly equal to instantaneous speed. This we have discussed in the previous lecture.

[00:09:28]Yes or no? Listen why instantaneous means at a instant. Instant means very small time. In very small time displacement of the particle will be very very small. If displacement is very very small you can assume displacement to be a straight line. In 000 0 1 second the particle will have very small displacement. If displacement is very very small it means the particle will go straight. Is it very small means what?

[00:09:55]Straight. If particle will go straight its distance and displacement will be equal. That's why speed and velocity will be equal because speed is distance upon time, velocity is displacement upon time. Only difference is in instantaneous velocity you will have direction also. That's why I put a mod their magnitude will be same. In fact if I say just instantaneous velocity so GPS GPS navigator tells you what instantaneous velocity. It gives the velocity at an instant. The moment your car is going with 60, it will say the speed of the car is 60 kilometer per hour in west due west. It will specify direction with your speed with the magnitude and the speedometer.

[00:10:40]Instantaneous speed is given by what?

[00:10:44]Speedometer.

[00:10:46]Speedometer of the car. If you look at the speedometer, it will not tell what direction you are going. But if you look at the speedometer, it will tell with what magnitude you are moving. What is your speed? How much distance you are covering in a given time? So speedometer of a car gives what?

[00:11:03]Speed. Instantaneous speed. No direction. But GPS gives you what?

[00:11:08]Instantaneous velocity.

[00:11:11]Magnitude plus direction. Clear? So keep this in mind. In the entire chapter whenever I will ask what is the instantaneous velocity or what is the instantaneous speed their magnitude will be same. Clear? Understood? You can pause and write. Pause and write. Okay.

[00:11:29]Now, what is this instantaneous velocity is? Let's write definition.

[00:11:35][clears throat] Instantaneous [clears throat] velocity.

[00:11:45]It is the It is the instantaneous It is the instantaneous change in position with respect to time. with respect to time that is [clears throat] velocity at a particular instant you can write this definition and if [clears throat] it's instantaneous velocity direction will be there if it's a speed direction will not be there is it clear okay and I showed you that this instantaneous velocity is nothing but the rate of change in position how the position is changing d by dt is it how the position of a particle is changing this is our what instantaneous velocity or instantaneous speed right now I'm talking only about the magnitude this is instantaneous velocity okay you can pause and write let's discuss this more with the help of graph you will understand better okay let's say I have a position time graph I have a position time graph for a particle.

[00:13:09]And the particle is moving like this.

[00:13:13]This is how the particle is moving.

[00:13:15]Okay. If you look at this particular diagram, what is instantaneous velocity?

[00:13:20]You will say sir velocity at a particular instant. And that instant time is very very small. So very very small means if I take one time here that instantaneous time is very very small. The gap is very very small. So if I take one time here the other time will be just on the right of this. If I call this time as t_1 and this time as t2 this t2 will be t1 + dt.

[00:13:53]This t2 will be what? t1 + dt because from t_1 to t2 the time gap is very very small tending to zero. So this gap is very very small. If I take t_1 the other time is what? TS2 which is what ts2 equal to t1 plus dt. A very small change in time very small. If the change okay I I have drawn a bit far this is tending to zero dt is somewhat this dt is somewhat 0.00000000000000000000000000 1 second something like this. So if time is very small the position will also change by a very small amount. So if at t1 time particle is here I am zooming and drawing it. This is zoomed zoomed version. So here the position is r1 and at t2 time the position of the particle is r2.

[00:14:50]And this time is so small so small that this displacement whatever you are seeing I am zooming it this displacement whatever you are seeing what is displacement at t2 time position is r2 at t1 time position is r1 so displacement will be r2 minus r1 and this r2 - r1 this r2 - r1 is nearly very very small tending to zero which I will call as dr yes or no in a very small time displacement will be very small. A small time is dt. A small displacement is DR. Clear? Clear. So if I join this like this, if I join this like this and then after this, what will be this? This pink line, this is change in time dt. Yes or no? And what is this pink line? This one. This one. You will say sir this is change in y which is r2us r1 which is dr r. So this is very small change in displacement. This is very small change in time. Yes or no? And I can say although it's a curve. Although it's a curve but this time gap is so small on this curve. On this curve these two points are so close that this sky blue line seems to be a straight line like earth. Earth is a sphere. But when you stand on a ground what you feel it's a straight line because you are seeing a very small segment of that. So the small segment will be like what? Like a straight line. Yes or no? So this part I'm drawing here like this. This is dr. This is dt and this is a straight line.

[00:16:37]Yes or no? If I say this angle is theta, what will be tan theta? Tan theta will be d r by dt. Yes or no? Oh, one second.

[00:16:48]I think d by dt is what? Look here. What was d by dt? Instantaneous velocity. dr by dt was what? A small change in displacement upon a small change in time was known as velocity at an instant. And here if you look in this triangle tan theta is giving me what? dr r / dt. It means tan theta is basically velocity.

[00:17:13]and tan theta. If you look at if if this is theta, we all know tan theta is nothing but what? Slope.

[00:17:21]This is what? Slope. Yes or no? Yes or no? If you draw tangent at this point, how how will be the tangent? The tangent will be like this tangent. And this theta and this theta are same. What is slope? Slope is tangent.

[00:17:40]If this if at this point you want to know the slope draw tangent find this theta then what you say you say that sir slope is tan into this theta so this theta and this theta are same is it is it these two are what same it means the tan theta which I got this is nothing but what slope slope at this point tangent at this And which is also our drt. If you look at this triangle, what is tan theta perpendicular by base? Which is d by dt?

[00:18:17]It means slope of displacement time.

[00:18:21]Slope of position time graph. Slope of position time graph is dr by dt which we call as what? Instantaneous velocity.

[00:18:31]Very very important. All of you write this. Pause and write. Pause and write.

[00:18:35]Pause and write. Okay, you can zoom it.

[00:18:38]It means it means I got instantaneous velocity as d by dt and this is also what tan theta.

[00:18:51][clears throat] This tan theta is known as slope or tangent or tangent. And we also call it we also call it what? Differentiation of R with respect to t. Yes or no? Like what you write? You write dy by dx. What is dy by dx? Differentiation of y with respect to x. In the same way what is r d by dt?

[00:19:19]Differentiation of r with respect to t.

[00:19:21]It means instantaneous velocity is all of you listen and repeat four times.

[00:19:25]Instantaneous velocity is rate of change of position with respect to time which I can call as differentiation of r with respect to t which again I call as tan theta again I can call as slope again I can call as tangent all these are same.

[00:19:43]It means to find out instantaneous velocity at a point I will draw what slope. If a displacement time graph is given, If a displacement time graph is given, I will draw what a slope, a tangent, I will draw a tangent at that point. And that tangent will give me what? Velocity. Because velocity is dr by dt, which is tan theta.

[00:20:08]Is it clear to everyone? You can pause and write. Just pause and write. Pause and write. I will zoom also. This also you can done. Fine. Next. [snorts] So what we learned just now which is very very important. If position and time graph is given to you at this point if you want to know velocity. So just now we got to know that instantaneous velocity is dr by dt and this dr by dt is what? Slope.

[00:20:43]Velocity is what? slope of RT graph at that point. The point where you want to know the velocity, you make a slope, a tangent. You draw tangent at this point.

[00:20:56]The tang, the tangent which you will draw at this point. this tangent find this theta and you can write directly that velocity at that instant will be tan theta because on the previous page I showed you that tan theta is what d by dt and what is dr by dt velocity but you tell me you tell me practically is it possible this is correct 100% correct but practically is it possible if I give you any equation Any function I give you, let's say I give you position is changing as t + 2t - 6.

[00:21:38]Will you go to plot graph? And I ask you find velocity at 4 second.

[00:21:47]Will you go and plot graph? If I say t power 4, let's say t power 4, you cannot go and plot graph. Huh? Time taking you will plot graph then at that particular point you will draw tangent then you will find theta then tan theta you will get velocity 100% you will get but that will be time taking we cannot use that approach but if they ask let's say in a certain region they ask in J to find the velocity we will plot a graph at that particular time I will draw a tangent and I will find the slope of that I will find tan theta will it be the velocity you will say Yes, it will be the instantaneous instantaneous velocity at that moment but that will be calculative agree. So what we will be doing instead of that we will very simple thing we will use instantaneous velocity is dr by dt which is also known as what differentiation of r with respect to t. You just differentiate this r with respect to t.

[00:22:51]You know differentiation you can easily differentiate differentiate you will get the instantaneous velocity as you get the instantaneous velocity then at whatever time they are asking put that time clear understood but you are right you are right to find out velocity we will make a graph and the point where they're asking velocity at that point I will draw a tangent I will find the slope that will be the velocity this is a method accurate method this is also Perfect. But the thing is time taking so you will use this method. What is instantaneous velocity? d by dt. Differentiation of r with respect to t. Differentiate. Get the instantaneous velocity. Then at whatever time they are asking substitute. Don't worry I will show you.

[00:23:37]Don't worry. Can I move to next page?

[00:23:39]Next part. Fine. All of you write.

[00:23:44]You can just pause and write and make a box and write one very important thing.

[00:23:48]Pause and write this. Fine. Make a box and write which is very very important that slope of slope of position time slope of position or you you will also see sometime displacement time slope of displacement time [clears throat] graph will give you will give you velocity very very important okay and then we got instantaneous velocity as d by dt I have repeated multiple times because I want that if you listen three four times it will go into your brain okay because these things will come multiple places pause and write good now we'll take one more question like let's say displacement of a particle or position of a particle is changing at as with time as r= t².

[00:24:52]The position with time is changing as r = t². Okay. So can you tell me what will be the value of r at different different time? When time is zero, position will be zero. When time is 1 second, position will be 1. When time is two, position will be four. Let's say the particle is moving on the x-axis. Particle moves on the x-axis. And this is how the coordinates are changing. Then if I say that time is three, position will be 9 m. Time is four, position will be 16 m.

[00:25:23]Can you plot this? Yes. So you can see r is equal to t². It will be a parabolic graph. Is it? It will be a parabolic graph like this.

[00:25:33]Clear? You can mark the points now. 1 0 0 comma 0 then 1 comma 1 in x1 in y1.

[00:25:42]Then in x how much?

[00:25:45]2 but y4 x2 but y4 x3 but y9 so something like this is it so you can just draw with free hand like this you can see from here one thing that sir in the first second how much the particle has moved 0 to 1 1 m in the next 1 second how much distance the particle has covered 1 2 4 3 m in the next 1 second 2 to 3 how much distance particle has covered 4 to 9 means 5 m. It means in every second, in every 1 second the distance cover is changing.

[00:26:23]It means the particle is moving fast.

[00:26:25]You only think in the first 1 second it move 1 m. In next 1 second 3 m in next 1 second 5 m. It means speed is increasing velocity is increasing that's why particle is covering more and more distance in a given time. Yes or no? You can cross check from the graph also. I told you now displacement or position time graph slope gives you what?

[00:26:50]Velocity. And what is slope? Slope is tangent. Draw tangent. Draw tangent here.

[00:26:57]Draw tangent here.

[00:27:01]Okay. Here. Draw tangent here.

[00:27:07]Draw tangent here.

[00:27:10]Draw tangent here. What you are seeing?

[00:27:13]Tangent means what? Tan theta. You can see theta is increasing. Theta is increasing. So displacement time slope is velocity. And as slope is increasing, velocity is also what? Increasing. And same thing I told you from there as well in every 1 second the distance cover was increasing. You can say from here also sir as I'm going up the slope is increasing. slope displacement time slope or position time slope is velocity. It means velocity is increasing because the slope is increasing.

[00:27:46]Is it clear? Is it clear? Now if I ask you what will be velocity at 6 second you will say sir do one thing. Listen very carefully from here you will get the feel what what is happening. You will say okay sir you find the time 6 second. Let's say this is time 6 second.

[00:28:06]At 6 second you draw a tangent. I will say okay at 6 second I will draw a tangent. So I will draw a tangent at 6 second.

[00:28:17]At this 6 second I will draw a tangent.

[00:28:19]Then I will find this angle theta because displacement time slope is velocity. So I have drawn a tangent. If I draw a tangent what will be the value?

[00:28:30]We know tangent is tan theta or slope is what? Tan theta.

[00:28:35]And that tan theta is velocity tangent slope tan theta all are same and that's the velocity for displacement time slope is the velocity. So just find this theta you will get the velocity at 6 second draw tangent get theta tan theta will be velocity but again it will be calculative plot the graph mark the point find the angle find tan theta better you use we also know velocity is d r by dt differentiation of r with respect to t. When you differentiate t² what you will get? 2t. It means the instantaneous velocity. Instantaneous velocity is what? 2t.

[00:29:13]At what time they're asking the velocity? They're asking the velocity at 6 second. 2 into 6 velocity will be 12 m/s. Very easy to get.

[00:29:24]Differentiation will make our work very very easy. Pause and write. I hope you understood. Is it clear to everyone? Is it clear to everyone? Good. Now we'll do some questions. Let's do some questions.

[00:29:36]You will 100% get the feel of this topic. Question number one, [clears throat] displacement or position versus time graph is given. [clears throat] Okay. And the graph is somewhat like this.

[00:29:59]Okay. Let's mark some point A, B, C, D, [clears throat] E. Okay, I want you all to find out at which point velocity is zero. At which first question is all of you pause and try. At which point velocity is zero. You will say sir velocity is displacement time slope is it? And slope is tangent. So you will draw a tangent here. Is it? You will draw a tangent here. Uh let me draw it like this. Tangent here. Yes or no? You will draw a tangent here. You will draw a tangent here. You will draw a tangent at this point. Yes or no? And you will draw tangent at E. Draw tangent at all the points. Next velocity is zero. So velocity is nothing but slope of displacement time. And you can see at B and D slope is zero. Slope is what? Tan theta. Theta is zero. Theta is zero.

[00:31:09]Slope is zero. Yes or no? Slope is what?

[00:31:12]Zero. So velocity is zero at B and D. At B and D because what? Slope is zero. And displacement time slope is velocity. It means velocity is zero. At that two point the particle comes to rest for a moment. Clear? Then they can ask you at which point velocity is positive and at which point velocity is negative.

[00:31:35]All of you pause and try. You will say sir velocity positive means velocity is slope of displacement time. Slope should be positive. So you will say sir here slope is positive. What is slope? Tan theta and theta we take from where? From positive x direction we take theta. In what sense? Anticlockwise sense. Keep in mind theta we take from positive x in anticlockwise sense. This theta is less than 90.

[00:32:02]So any value of theta less than 90 will give you tan what? Positive tan 30 tan 40 60 70 all are positive. So it means at a point velocity is positive.

[00:32:14]At d point again angle is less than 90 velocity is positive.

[00:32:19]Then where is the velocity negative? At C the angle is more than 90 10 135 140 negative. So at uh this C now yeah at C velocity will be negative. A and D velocity will be positive.

[00:32:34]At C velocity will be negative because slope is negative. So they can ask you this even they can ask you compare the speed. Okay. I I have asked you velocity. If I ask you what about speed?

[00:32:46]What about speed? The speed you will say this V is velocity. You will say such speed can never be negative. Speed is distance upon time.

[00:32:56]Speed is always positive. So at A B speed is zero. E speed is zero. A C D speed is always positive. Speed is always positive.

[00:33:09]They can ask you to compare the speed.

[00:33:11]Compare the velocity. How will you compare? You will say like if I say compare compare one more question I can make compare velocity at uh A and D a and D. Okay you will say sir velocity is slope what is velocity slope of displacement time. So when slope will be more which means velocity will be more and slope is tan theta in simple words slope means slide more slide more slope less slide less slope so which is more sliding you can see this is more sliding more dangerous slide this is a bit less dangerous clear so more angle means more slide so where the slide is more at A as compared to okay at D I'm comparing A and D. So at A slope is more, slide is more so velocity will be more. So velocity at A will be more than velocity at D. Is it clear to everyone? So they can ask you at what point velocity or speed is zero, you will find tangent. Where the slope is zero. They can ask you where velocity is positive. Find slope. If angle is less than 90, you will say that velocity is positive. More than 90, you will say that velocity is negative. Speed will be always positive. They can ask you to compare the velocity or speed. You will check the slope. The point where slope will be more speed or velocity will be more. I hope it is clear to everyone.

[00:34:47]You can pause and write. You can pause and write. Good. Next.

[00:34:53]Let's do some numericals.

[00:34:55]Okay. Question is particle move along X.

[00:35:01]Write fast. Particle move [clears throat] along X according to according to position function according to position function [clears throat] XT A. This means position changing as a function of time.

[00:35:30]Okay. is equal to 3t ² - 6 t + 5.

[00:35:40]Okay. Find first part instantaneous velocity as a function of time. as a function of time.

[00:35:54]Second, velocity at 2.

[00:36:02]Third velocity into second fourth time for which particle comes to rest for a moment. Rest for a moment.

[00:36:26]All of you pause and try. Pause and try.

[00:36:29]Done. Good. So they have given what particle move along x according to the position function they have given the position of the particle as a function of time the particle is moving in x and how the position is changing how the xcoordinate is changing they have given and this x coordinate is changing as 3t² - 6 + 5 question is instantaneous velocity as a function of time we all know instantaneous velocity is rate of change of position here they we have told with x. So instantaneous velocity is d r by dt. Rate of change of position. Here x is given. So dx by dt.

[00:37:08]So instantaneous velocity is dx by dt.

[00:37:11]Rate of change of position is equal to just differentiate. So instantaneous velocity will be 60 and two will come down 60 minus differentiation of 60 will be six and this is constant. If you differentiate what you will get zero.

[00:37:26]This is the answer of the first part.

[00:37:27]This is the instantaneous velocity of the particle.

[00:37:31]Means if you want to find velocity at any instant just put the value of time.

[00:37:35]It will give you instantaneous velocity.

[00:37:37]I hope it is clear to everyone. Good.

[00:37:40]Second is velocity at. I want you all to make a box and write one thing in at the corner. At means instantaneous.

[00:37:50]Whenever you see the term at, you have to understand instantaneous. Add 2 second, add 4 secondond, exactly at that moment. And if you see in, in or from 2 to 4, from 2C to 6 second, in 2 second, in means 0 to 2 in means a time gap, a time gap, which will be what? Average.

[00:38:17]So if you look at question number two and three, velocity at 2 second is nothing but instantaneous velocity and velocity in 2C is nothing but average because in means 0 to two from 0 second to 2 second. How many intervals are there? Infinite. From 0 to two, we have infinite number of time intervals. 0 0.11 0.112 0.113 like this. Infinite number of time intervals. So we will talk about average clear. So second second is instantaneous velocity. We know we have differentiated instantaneous velocity came 60 - 6.

[00:38:56]They're asking at 2C put t = 2 6 into 2 - 6 12 - 6 6 m/s. Exactly at 2 second this particle is going with 6 m/s. I hope it is clear. Good. Third is very very important into second in means average. What is average velocity?

[00:39:18]Average velocity is change in displacement upon change in time or change in position upon change in time.

[00:39:25]Here they have represented with X. So change in position upon change in time.

[00:39:30]This is average.

[00:39:32]So what is the meaning of change in position? Final position minus initial position divide by final time minus initial time. Whenever you will see average you will use this.

[00:39:44]What is the final position of the particle? So you can see 0 to 2. So initial time is 2. Final time is 2 second. So and this position function is 3t² - 6 t + 5. So final time is 2. If you put here two you will get final displacement. So final displacement will be you put two you put two what you will get 2 2's are 4 4 3's are 12 minus 6 2's are 12 + 5 so displacement final will be five minus initial displacement initial time is zero you put time 0 if you put time zero you will get the initial position of the particle 0 0 5 okay initial position is also five divide by what is final time.

[00:40:35]Final time is 2, initial time is zero.

[00:40:38]Your answer will be 0 m/ second. Average velocity is zero.

[00:40:44]Now you will think sir how it came zero.

[00:40:47]Average means from 0 to two whatever velocity we have throughout average of all why it cannot come zero for some time velocity might be positive. For some time velocity might be negative positive plus negative they might have become zero. That's why we got zero. So at 2 second it was going with six.

[00:41:07]Okay. But in 2C the total average is zero. Average is not the reality. It's the summation of all upon total number.

[00:41:15]Is it clear? Is it clear? Fourth part.

[00:41:19]Time for which the particle will come to rest for a moment. Okay. Particle will come to rest. Keep in mind if a particle is at rest, I will ask you something. Is the displacement zero? Is the velocity zero? Is the acceleration zero? Your answer will be velocity zero. If I throw this pen up, it will come down. At the maximum point the pen came to rest, the displacement was there. Gravity was acting. Velocity was zero. Rest means velocity zero. You all keep in mind small things but important. Rest means velocity zero. What is velocity function? 60 - 6 make it zero. You got t is equal to 1 second. Exactly at 1 second the particle came to rest for a moment. I think now you can understand why zero came. I asked in 2 seconds. So from 0 to 1 at one particle came to rest you know in one dimension motion if a particle comes to rest it means the particle is changing direction or the particle will come to rest forever.

[00:42:23]But in this case you can see the particle is displacing always. There will be some displacement. Displacement is changing with time. It means particle is moving. If it's coming to rest, it is it is coming to rest for a moment. So at 1 second velocity becomes zero. It means at 1 second you you only tell particle is going then coming back. So at this moment particle has to come to rest like this pen. If I throw this pen up, it will go up come to rest. The point where it will come to rest, at that point it will change direction. Yes or no? At the maximum point it came to rest, at the maximum point it will change direction.

[00:42:58]Yes or no? Yes or no? At the maximum point it's coming to rest for a moment then changing direction. So in 1D motion to change direction particle has to come to rest. So at 1 second the velocity is zero. It means at that 1 second the particle is changing direction.

[00:43:14]That's why in 2 second velocity average came zero because before 1 second if the velocity was positive after 1 second the velocity was negative positive negative overall zero.

[00:43:23]Clear? Understood? Are you understanding what I'm saying? Pause and write. Good.

[00:43:29]Next.

[00:43:31]Now this question I want you all to solve by yourself.

[00:43:35]Okay. Again the particle move along x-axis according to position function.

[00:43:40]The position function x is given as t cq - 60 t².

[00:43:46]Okay. I want you all to find velocity at 1 second.

[00:43:53]Second in 3 second.

[00:43:58]Fourth sorry third from t = 1 to t = 2. All of you pause and try. All of you pause and try.

[00:44:11]Okay. So how will you solve? So displacement as a function of time is given. If displacement as a function of time is given, you will find out first instantaneous velocity. What will be instantaneous velocity? dx by dt. So position is given. I differentiate it. I got 3t² - 12t. This is the instantaneous velocity equation. Is it? This is the instantaneous velocity equation. Fine.

[00:44:39]Next from here if you look at the first question velocity at at means instantaneous so instantaneous velocity at 1 second will be put t = 1 3 into 1² - 12 into 1 you will get - 9 m/s minus means the particle is going left x clear second in 3 second in means what 0 to 3 it's average how to find average velocity so Average velocity will be what you will write change [clears throat] in displacement upon change in time. Average means what? A time interval, a time gap. So delta x by delta t. So final displacement delta x is final displacement minus initial displacement upon final time minus initial time. From where to where they're asking 0 to 3 second. So from 0 to 3. So final time will be three. If you put three here you will get final displacement. Yes or no?

[00:45:38]Partial going from 0 to 3 seconds. So final time is 3. Put here 3 3 cube 27 minus 3 3's are 9 9 6 are 54. So final displacement will be what? Uh 27 minus uh 9 54. So minus 27 minus initial displacement put zero. So this will give you what? 0 upon final time 3 initial time 0. And what you got? you will be getting -9 m/s as the average velocity.

[00:46:11]Is it clear? Third question is from 1 to 2 second again 1 to two a time gap is there a time interval is there. So again what you will find again you will say sir they're asking average. So what is average velocity?

[00:46:27]Change in displacement upon change in time. Final displacement minus or final position minus initial position upon final time minus initial time. So change in x upon delta t. Change in position upon change in time. What is final position? So here final time is 2 seconds. So put t= 2. So 8 minus 2 to the 6 24. So 8 - 24 - 16. So final displacement will be minus final position will be -6 minus initial position will be initial time is 1 second here put one. So 1 - 6 - 5 divide by final time 2 initial time 1. This will be giving you what? So -6 + 5 - 11 m/s.

[00:47:15]This will be what? This will be your average velocity from 1 to 2 second. I hope it is clear to everyone understood.

[00:47:25]Can I move to next part? You can pause and write. So whenever you see at it means instantaneous. Whenever you see from n it means average. Average velocity will be delta x by delta t.

[00:47:38]Instantaneous will be dx by dt. In this case differentiate in delta x by delta t case x final minus x initial upon t final minus t initial. That's what you have to do. Clear? You can pause and write. Understood? Okay. Let's do the next problem. So I think you can do question based on this. I want to change the type of question. For example, I want to ask you let's say a question is there which is x = t² - 2t and x is what? Position. X is what?

[00:48:13]Position.

[00:48:16]Find distance [snorts] and displacement of the particle in five second. I want you all to try this distance and displacement in 5 second.

[00:48:31]Now tell me one thing before solving this. Let's say I'm saying my coordinate my x coordinate is 1 and after 10 second my x coordinate is 4.

[00:48:41]You will say sir your displacement will be three. You went from x= 1 to x= 4.

[00:48:46]Displacement will be three. Can you tell my distance?

[00:48:51]No, you cannot tell. I can start at 1, I can go to 10, I can come back to four. I can start at one, I can go to 100, I can come back to four. I can start at one, I can go at four directly. In all the case, displacement is three. But distance will be distance different because distance is the actual path covered. In all the three the actual part covered is different. So whenever you get a question which asks distance and displacement and position function is given. They have given you how the position in X is changing with time.

[00:49:25]They have given you how the position in X is changing with time and they're asking you distance and displacement. So displacement you can directly find out because position is given. You can find initial position, you can find final position and you can just find the displacement that is fine. What about distance? So whenever you get this type of question, what you have to check is the particle changing direction or not?

[00:49:49]What you have to check first is the particle is the particle changing direction.

[00:50:00]Changing direction [clears throat] second the point. How to check that? How to check that? You will check velocity zero. I told you now in one dimension motion in one dimension motion to change direction velocity has to become zero.

[00:50:14]Throw the pen up it will go up then while coming down at the maximum point velocity became zero it change direction. If I'm going to I'm coming towards you and I want to go back I have to come to rest then only I can go back.

[00:50:28]So in one dimension motion if velocity will become zero for a moment at that instant the particle is changing direction. You check that clear. Clear. So is the particle changing direction? You will check velocity zero. Then what you will do?

[00:50:45]Let's do do step by step. So displacement is given. So what will be velocity? You will say sir instantaneous velocity is dx by dt. If you differentiate 2t minus 2 and you make this velocity zero. So you will find out that 2tus 2 is equal to 0. 2t = 2t = 1 second. At 1 second the particle is coming to rest for a moment. It means at 1 second the particle is changing direction. If particle change direction, distance and displacement are not equal.

[00:51:16]Distance and displacement are only equal if particle goes in a straight line without changing direction. If I ask find distance and displacement in 0.5 second, your answer will be same because it is changing direction at 1 second. So till 0.5 second it has covered equal distance and displacement because it has moved in a straight line but at 1 second what happened? So particle was going let's say left at 1 second particle came to rest and then particle went back. So displacement will be less distance will be more because distance is the actual path. Yes or no particle went particle was going right then particle went left. So displacement will be only this shortest path from initial to final but distance will be overall. Are you understanding? So first you will check is the velocity becoming zero or not in the time interval. Your time interval was 5 second and velocity was becoming zero at 1 second. Yes. So distance and displacement will be different.

[00:52:12]Okay. So second rule will be you take that time take time when velocity becomes zero and solve accordingly. What is the meaning of this and solve accordingly? I will tell what does that mean? So in this particular question we understood that particle starts at x equal to0. So when sorry particle starts at t equal to0. So when time is zero what is the position?

[00:52:41]0 0. Okay good position is also zero.

[00:52:45]Then at 1 second at 1 second particle come to rest. So at 1 second where is the particle? Put t equal to 1. 1 - 2 - 1. So when time is 1 second position is minus1 it means particle has moved left particle start to move towards left and at 1 second when x was minus1 the velocity of the particle was zero at this moment particle came to rest and after this the particle will change direction you will find out how now you see at 5 second what what will happen at 5 second what is the position you will say sir at 5cond the position will You put five 5 are 25 25 - 10 15 m particle came to 15. It means from here the particle went to x equal to 15.

[00:53:39]So what is displacement of the particle?

[00:53:41]What is displacement? You will say sir initial point is here final point is here. This is the displacement 15 m. So displacement will be 15 m. And that 15 m you can get from the equation also. It's a position equation. Now the equation tells you position. So change in position is displacement. What is displacement? Displacement is change in position.

[00:54:06]Delta x final position minus initial position. So time final is 5 put five 5 - 5 to the 10 15 minus initial time is 0 0. So displacement you will get 15 m from the equation also. But distance you will not get. For distance you have to do this. So what will be the distance you will say? So distance will be actual path. So from here to here 1 m distance cannot be negative. Again from here to here 1 m 112 and then 15 m. So 1 m 1 m 15 m your distance will be 17 m. Very very important question. This may come simple question but you can do silly mistake. So keep in mind whenever position is given and they ask distance.

[00:54:56]You have to understand position will not tell my distance. I have to be smart. I have to check is the velocity becoming zero or not. Because when the velocity becomes zero particle change its direction and then I will check where is the particle at initially and when the velocity becomes zero where is the particle and finally where is the particle and I will find the distance.

[00:55:18]You can pause and write.

[00:55:20]Clear? Understood? I hope you all understood. Okay, one more question like this.

[00:55:29]Again, position as a function of time is given as t into t minus 2. I want you to find out distance and displacement in 2. All of you pause and try. Pause and try fast. Find distance and displacement in 2C.

[00:55:50]Okay, done. So what you will do? You will say again sir x is equal to open this bracket t ² - 2t. First I will find out instantaneous velocity. I have to check particle is coming to rest or not.

[00:56:03]Instantaneous velocity will be if you differentiate this 2t and this will be two. This is the instantaneous velocity and part and I will check when particle comes to rest. So I will make velocity zero. Okay. I will get 1 second. Again at 1 second particle is coming to rest and I have to find displacement and distance in 2 seconds. It means they will not be equal. If they ask find distance and displacement in 0.5 second in 0.5 second distance will be equal to displacement they will be equal in magnitude but here not because at 1 second particle is reversing its direction. So they will not be equal they will be different. How to find out same you check when x is zero where is the particle? When x is zero when sorry when time is zero check x. So when time is zero x is also zero particle is at origin.

[00:56:54]Then after that then you check at 1 second velocity is becoming zero at 1 second so you put one 1 - 2 okay so at 1 second particle is at 1 - 2 - 1 and then they're asking in 2 seconds. So find in 2. In 2c particle will be at 2 4 and again - 2 4 0. Okay. Again at 2 second is it correct?

[00:57:21]Yeah. And the first part one correct. So again at 2 second what will be the position? 2's of 4 minus 2's 4. Okay. 0.

[00:57:34]Again particle came back to here only x= 0. [clears throat] So at t=0 particle was at x=0.

[00:57:43]At t= 1 particle went to x= minus1. It means particle first went left and again at 2 second I got the particle at the same position x equal to0. So what will be displacement from the point where particle started at the same point particle came displacement will be zero. What about distance? Distance is the actual path covered that will be 1 1 2 m. Distance will be 2 m. And displacement you can get from here also. Now it's position given. What is displacement? change in position. So at 2 second put two. So 2 and 2 - 2 0 minus initially when time is zero position [clears throat] is zero.

[00:58:26]So position has not changed in 2 second.

[00:58:28]So displacement is anyway zero. But this will give you feel clear understood.

[00:58:34]Let's do one more problem and pack this.

[00:58:38][clears throat] Okay. Right now I am doing 2D two dimension sorry one dimension. Right now we are doing one dimension but I want to give you one question of two dimension which can be done here not difficult. We'll study 2D I know we'll do a lot in that but if I say that position of a particle in X is changing as 5 sin omega t [clears throat] and the same particle position is y in y is changing as 5 1 - cos omega t.

[00:59:10]This is how the x coordinate is changing with time. This is how the y-coordinate is changing with time. You have to find find distance covered by the particle.

[00:59:22]Distance covered in 5 second. All of you pause and try. It's a really good question.

[00:59:29]Really good question. Okay. So, how will I do? See, first I will see sine and cos. You will get a sir what the sign and cos you are giving. You know see but what you can do here is you look at the problem distance I know only one thing that distance is speed into time okay time is given to me time I know 5 second fine if I get speed I will get distance okay fantastic so here what you can do is x is equal to 5 sin omega t and position as a function of time is given Can you find velocity in x? So velocity in x will be instantaneous velocity. Which velocity?

[01:00:11]Instantaneous velocity in x will be dx by dt. How the x coordinate is changing.

[01:00:17]Rate of change of x which will be five.

[01:00:21]Differentiation of one more thing you can take uh omega value here to be two.

[01:00:27]You can take omega to be two units.

[01:00:29]Okay. Radion per second. You can take omega no problem. So five differentiation of sin omega t will be cos omega t. Then [clears throat] again you differentiate omega t you will get omega this omega will come out. I hope this much differentiation you know this is the velocity next direction. Can I write in the vector form? Yes [clears throat] I can write 5 omega cos omega t icap. In the same way you find velocity in y. So rate of change of y.

[01:01:01]at what rate the y-coordinate is changing. So this is basically y is basically if you open the bracket 5 - 5 cos omega t. So if you differentiate differentiation of five will be 0 - 5 differentiation of cos omega t will be minus sin omega t minus will come you know so let me write like this 1 minus will come and omega t differentiation will be again 1 omega will be multiplied so multiply by minus omega multiply by minus omega - cancel and velocity in y came how much 5 sin omega t In vector form can you write? Yes. Y means jcap.

[01:01:44]So I was having how the position is changing in x. I got velocity in x. I was having how the position is changing in y. I got velocity in y. Can you write the net velocity in vector form? Yes sir. We can write velocity in x 5 omega cos omega t icap plus velocity in y 5 sin omega t jcap.

[01:02:06]Yes or no? Okay. I will ask you some more question without before solving. Is the velocity in X changing with time?

[01:02:12]Yes or no? Tell yes or no. Yes. Velocity in X is changing with time because it depends upon time. Velocity in Y is changing with time. Yes, because it depends upon time. Net velocity is changing with time. Yes, because it depends upon time. They depend upon time. So they will change with time.

[01:02:29]Correct. Perfect. But what I want? I want speed. I want speed. In the starting of the lecture I have told you that instantaneous speed instantaneous speed is nothing but magnitude of instantaneous velocity.

[01:02:50]Yes or no? At any instant the magnitude of velocity will be same as speed. I told you because in that small moment small instant the particle is going what? Straight. So instantaneous speed is equal to instantaneous velocity.

[01:03:04]Okay. So we have velocity as a function of time. If I ask you find its magnitude just just find out its magnitude. So what will be the magnitude of this velocity? You will say sir magnitude will be one velocity is in icap one in jcap. This is 5 omega cos omega t. This is 5 omega sin omega t. So vector addition we will apply under root of a² + b² and you will find out that 5 omega sin omega t² you can pause and write that part. Okay plus 5 omega cos omega t².

[01:03:49]So magnitude of velocity will be under root of can I take 5 omega 5 omega square common from here? Yes. So square is there then root is there. So only 5 omega will come out inside I will be having sin² omega t + cos² omega t. Ho this will become what? 1 sin square theta plus cos square theta is one. It means the speed which I am getting or the magnitude of velocity which I'm getting is 5 omega I have told 2 5 2 are 10 m/s what I got 10 m/s and you know you know the magnitude of velocity the magnitude of instantaneous velocity is the speed you take any instant magnitude you take at time 1 second 2 second 3 second 4 second 5 whatever time you put these two are equating to one means they are adding up to one sin square omega t cos square omega t you are correct velocity in x is changing with time velocity in y is changing with time but the thing is this quantity sin square omega t plus cos square omega t whatever be the value of time it will be always one it means the velocity magnitude is not changing with time it means the speed is constant I will give you more feel. Let's say you are running in a circular park. Circular park. In the park you are running, you are running with a constant speed 10 m/s. You are running with 10 m/ second.

[01:05:24]In a circular park, in a circular park you are running always with 10 m/s. At any moment your speed is 10 m/s, 10 m/s, 10 m/s, 10 m/ second, 10 m/s.

[01:05:37]But what is changing? You know your direction is changing. Here your direction is like this. Here like this, here like this, here like this. Every moment your direction is changing. So you know velocity direction is changing.

[01:05:51]Velocity is a vector quantity. You will say sir velocity depends upon time.

[01:05:55]Velocity depends upon time. Correct?

[01:05:56]Velocity is changing but in direction not in magnitude. Because when you go to find the magnitude of velocity the sin square omega t and cos square omega t is becoming one.

[01:06:08]And you will learn in the upcoming part.

[01:06:10]It's what it's equation of a circle.

[01:06:13]This particle is moving in a circle.

[01:06:16]When a particle moves in a circle, this is basically uniform circular motion.

[01:06:20]Without starting the chapter, I gave you a question of uniform circular motion.

[01:06:24]In uniform circular motion, speed does not change. But as direction is changing, velocity change. That's what happening here. Velocity depends upon time. Velocity is changing with time.

[01:06:34]But with direction, not with magnitude.

[01:06:36]If magnitude of the velocity is constant, we call that what? Speed. It means the particle is continuously going with what speed 10 m/s. What I have asked? Distance cover in 5 seconds. So distance is what? Speed into time. Speed is constant. Speed is not changing. What is the speed? 10. What is time? Five. So distance cover will be 50 m.

[01:07:01]Distance cover will be 50 m. Multiple I can give this question in advance. I can ask you is velocity changing in X? Yes.

[01:07:09]Is velocity changing in Y? Yes. Is the net velocity changing? Yes.

[01:07:14]Is the magnitude of velocity changing?

[01:07:16]No. Is the speed changing? No. Is it a uniform circular motion? Yes.

[01:07:24]Multiple questions I can frame from here. So all of you try this again. This question I will take when I will teach some uh circular motion and I will take this question again. No need to worry.

[01:07:32]In 2D also I can take some questions like this. I hope you understood. Next class I will be coming with acceleration and we'll do a lot of problem based on integration, differentiation from acceleration to velocity, velocity to displacement, displacement to velocity, velocity to acceleration. All these things we will do. Okay. So just pause and write. All the best. We'll meet in the next lecture. Bye.