 Now, let us begin to today's lecture on particle kinematics. We will discuss what particle kinematics is, we will solve a bunch of problems and see how these concepts form the founding stone for any advanced concepts that will be developed further. So, this one thing I want to emphasize is that, that the new addition vector mechanics for engineers were Beer and Johnston, the 10th edition is extremely good for dynamics. So, my lectures for dynamics will be exclusively based on Beer and Johnston 10. Indian edition is available if you wish to purchase. Slide contents are also essentially from Beer and Johnston. If you are interested, you can contact these guys and they can guide you further about how to use their resources and so on. There are also some interesting problems as usual from Merriam and Craig. So, I am going to use a few of those problems in the tutorials and there are these excellent online resources especially for dynamics. You will see that on this website, demonstrations.volfram.com, you will see some really nice demonstrations about various problems in mechanical engineering machines and so on. So, if you want to understand just to visualize how particular mechanisms work and so on. You can have a link, you can check out this link and look at the demonstrations that are shown here. Also another amazing resource is lecture notes on dynamics by Professor Alan Bauer at Brown University. Just in my opinion, they are totally worth it if you are interested in going into depths of dynamics. It is a good idea to go through these notes by Professor Alan Bauer at Brown University. There are some very nice, easy to understand lectures on dynamics on YouTube. So, if you want to learn some topic in an easy way, then please follow this link and also there are many nice animations for projectile motion, for in general impact of two bodies and so on. On some example problems, these are example problems from dynamics by Hebbler and this is a Prentice Hall website on which some resources are openly available. So, you can go and have a look at these resources. Additionally, there are some other links and as I when I keep finding them, I will post them so that you can have and look at these problems to get some deeper insights into overall dynamics problems. So, the topics that we will cover in this class will be so, what is position, velocity, acceleration, extremely basic things. Then we will discuss about reclinear motion with some examples. Later on, we will discuss what is curvilinear motion is and ultimately at the end of the class, we will discuss what is motion relative to a moving frame, how to find out relative velocities, relative accelerations, both for example, when there are constraints in the system and when there are no constraints in the system. What is kinematics now? Coming to the basic question that kinematics essentially means, so think about say a golf ball, think about a satellite, a jet. So what happens is that in any of these objects or moving trains, moving cars, flying cricket ball, jumping acrobat, 50 hundred things you can think of where you have a particular object in mind and you want to understand that what is the particular trajectory in space and time that is taken by the object. So kinematics essentially is the study of what are the trajectories, what are the velocities, what are the accelerations that are taken by a particle or by a body. Now kinematics essentially mean that something that you can see. For example, you can see a tennis ball and you can see a cricket ball and both may follow a same trajectory. So kinematics essentially means is to understand how does the trajectory of that object change as a function of time, what is the velocity, what is the acceleration and so on. So that is kinematics which studies the geometry of motion, it relates displacement, velocity, acceleration and so on without any mention to what is the cause. So tennis ball is hit by a racket, cricket ball is hit by a bat, some x amount of force is applied to the tennis ball, some y amount of force is applied to the cricket ball but we do not worry about what is that force that is applied to the ball or any other for example, a bike is moving or a truck is moving. We do not worry about the forces or the causes of the motion or the effects that are caused by that motion in kinematics. That is studied in kinetics, where what is the reason, what is the effect that is produced by that will be studied in kinetics. For example, we can look at a tennis ball, we can look at a cricket ball. Kinematics for them is the same but kinetics is drastically different. For one example, if we are forced to face a tennis ball or a cricket ball then the answer is clear in which ball we want to stand in front of. We do not want to stand in front of a cricket ball as much as possible because it is heavier, much higher momentum and so on and what are the effects that are caused by or what are the forces that arise in the motion of objects that will be studied in kinetics in the next lecture. So in particle kinetics, we will think about what is called as rectilinear motion? What is a rectilinear motion? For example, a train moving along a straight track that is that means that the particle is following a straight line or for example, if you take a coin and just release it, make it drop downwards, take a ball drop it downwards then the trajectory that is followed by that particle that ball, coin anything is just in a straight line and that particular motion is called as a rectilinear motion and another type of motion which is more common than rectilinear motion is the curvilinear motion. What happens in the curvilinear motion is that position, velocity and acceleration, they do not follow a straight line, they move along a particular curve in space. Now for example, if you are moving on a level ground say for example, if you are walking on a ground which is flat then that motion is a curvilinear motion possibly if you take turns move around for example, run on a track then that motion is curvilinear but in a 2D plane on the other hand if you have a road which is bumpy which is going up and down and you are riding a car on that bumpy road which is going up down moving sideways taking curves and so on especially for example, travelling in ghats then that particular motion is a curvilinear motion in 3 dimensions. But in this particular course especially keeping in mind that we have limited time and the syllabus that is typically covered in dynamics of these courses typically do not involve 3 dimensional motion we are going to deal with curvilinear motion only in a plane. Curvilinear motion in a line and curvilinear motion in a plane is what we are going to discuss as far as both the kinematics are concerned and the kinetics are concerned. Now first thing is we are going to talk about position, velocity and acceleration of a particle. So let me briefly discuss that what is the idea about particle, we keep on talking about motion of a particle kinetics of a particle kinematics of a particle really in real world there is nothing like a particle. Dealing as 3 dimensions and we do not have any real particle in real world. So what is the deal here? So is it any useful that we discuss about particles and a discussion is as follows and a usefulness of dealing something as a particle is as follows. So for example take this range which is used to tighten nuts and bolts. So what we can do to this range we can provide it some angular velocity or to begin with what we can say is that the angular velocity of this range is 0. Let me make it 0 here. Now what we do is that we put some horizontal velocity on this range here we adjust it and now see how does this range move in a 2 dimensional space. You will see that the range moves like this note it will follow a parabolic trajectory this is the center of mass of the range what that is will come to that very soon okay and it will just go down without any rotation. Now in such kind of motion for example when there is absolutely no rotation or when the rotation is irrelevant for solving our equations then we say that this entire object I can just concentrate it at its center of mass and then say that this is a particular motion because I do not worry about each and every point of this range but say that I just keep track of this one point and we are done. So that is the reason that for example if there is no angular moment angular velocity or angular momentum we will discuss that in the next one or two days then that particular motion can be effectively modeled as the motion of one particle because motion of all the points is same. So we say okay why do we worry about anything we just take the center of mass and then model what is the motion of that. On the other hand we can provide some angular velocity okay then in that case what happens is that that while in addition to translating you will see that each and every point do not have the same velocity okay some points have more velocity some points have less velocity and as a result there is some kind of angular motion angular velocity angular acceleration and we say that this particular motion cannot be directly idealized as a particle motion and for that we have to go to what is called as rigid body kinetics and we will discuss that topic on 4th of December okay. So with this preamble okay that what is meant by a particle motion why do we care about kinetics and kinematics of particle when in real world there is nothing really which can be called as a particle the answer is that that if the angular motions or the angular rotations angular velocity angular acceleration of the body okay of the three dimensional body are not relevant for the analysis of its motion okay or if there is no rotation okay then in that case we say that we will just idealize this entire thing as a particle and just track okay the motion of its center of mass for example and be done. Now let us look at what is a rectangular motion so in a rectangular motion particle moves along a straight line and we see that we can keep one origin okay we can keep some point O as an origin and track what are the coordinates what is the coordinate of this point P with respect to this origin okay. Now the point can be on the plus side okay or on the minus side we can define positive or negative coordinates the motion of a particle is known if the position coordinate of a particle is known for every time for example this particle moves and if we know that what is the time history of motion of this particle then we can track the complete motion of the particle. Now when we know what is the time history of the particle then what we can do is we can plot the curve that how does the position of this particle vary as a function of time this particular curve okay is called as the particle trajectory that as a function of time okay on the x axis this is how the particle moves. Now what we define is position is just one quantity okay in principle what we can do is we can just define the particle position as a function of time and say okay I know how the particle moves as a function of time but we will see that this is not enough okay just define the position of a particle okay it is perfectly fine in principle but if we want to get something useful out of it then what we really want to do is to find out what is the velocity of that particle and the way the velocity is defined okay using the calculus language that if in some tiny interval delta t the particle moves by distance delta x then the instantaneous velocity is you take the limit when this time goes to 0 which is delta x by delta t and which we call that which in the calculus language will become dx by dt and that is called as a instantaneous velocity of the particle and now note that velocity is a vector. So in a rectilinear motion if this quantity is positive then it means that the direction is to the right of the velocity if this quantity is negative the direction of the velocity is to the left and absolute magnitude with without reference to the sign is called as the particle speed which is a scalar quantity. So what is a scalar quantity that it is just a number and we do not have to define a direction with respect to it. Now what is velocity we can now so velocity is just dx by dt so if this is x okay that this is a position as a function of time then what is velocity velocity is nothing but derivative of this if you take the derivative of this with respect to time what do we get 2Lt minus 3T square and if we remind our self of our calculus then we will see that dx by dt is nothing but the slope of this line okay if the slope of this line is high velocity is high if the slope is positive the velocity is positive and according to the sign convention that we are using here it is to the right whereas if the velocity is 0 the slope is negative which means the velocity is in the opposite direction okay. So this is what we mean as velocity so this is the expression for velocity we have plotted and this is nothing but the slope of the displacement graph. Now additionally okay what we see is that one very simple example that we take a ball and just leave it from our hands okay the velocity when we leave the ball is 0 but we see that the velocity does not remain 0 okay the ball will start moving down. Now why does it move down even if the initial velocity is 0 we know from our high school physics that there is a force from the gravity that acts on the ball but what does that force do that force changes the velocity of ball with respect to the time and by change in velocity of ball with respect to the time the position is changed. So what is happening is that in this case the governing effect the thing that is causing the motion is the acceleration due to gravity because of the acceleration okay how do we quantify it the velocity changes okay and as a result we define acceleration of a point as limit delta t which is a small time interval that goes to 0 okay and what is the change in velocity in that tiny interval is what we define as the acceleration and in the language of calculus this can be written as dv by dt and that is called as acceleration and what does the acceleration mean that we move from point p to point p prime okay the position changes at point p suppose the velocity of the particle is v as we had discussed earlier at point p prime the velocity of the particle is v prime which has changed then this v prime minus v divided by delta t okay is the acceleration. Now if v prime which is the velocity at a later time is higher than the velocity at the earlier time we say that acceleration is positive or in this direction according to our sign convention look at it this is another interesting point is that which is not immediately clear for position but look here that if now the particle goes from here to here and the velocity okay the velocity was higher here okay and it decreases after time t then what happens is that then we can see even in this case okay even the position of the particle is in this direction the velocity was higher initially and lower later okay even in this case the acceleration is positive why because the velocity is actually because the change in velocity okay with respect to time is still positive okay and as a result the acceleration becomes positive but on the other hand if as a function of time while moving in this direction the velocity decreases then the acceleration is negative and the direction of the acceleration is in this direction or if the particle is moving in this direction and the velocity increases as a function of time then we also say that the acceleration is negative. So essentially what acceleration means is that that we have to take into account both the direction of velocity and the magnitude of velocity and this four cases show okay that the velocity direction is positive here it increases in this direction so acceleration has to be positive in this case the velocity is negative but it is decreasing as a function of time so what does it mean that the acceleration has still to be positive one simple example is that that if you take a ball if you throw it upwards then the if the y coordinate is positive upwards then what then the velocity is decreased as a function of time while moving in the positive while moving in the upward direction the acceleration is still down if you drop the ball down the velocity is increasing downwards still acceleration is down okay and so on okay. So these are all four possible cases that can happen for acceleration so note one thing that the acceleration direction depends both on the direction or the sign of velocity and how does the magnitude of that velocity change as a function of time and depending on that we will get some acceleration that is either positive or negative. Now if you plot this acceleration as a function of time in some simple case you will see okay that our velocity which was a simple case that we had discussed earlier velocity was position was x equal to 60 square minus t cube whereas the velocity was 12t minus 3t square in this case you will see that the acceleration comes that you have to take a derivative again with respect to velocity and that will give rise to what is acceleration. Now in this case you will see that the acceleration is initially positive okay it acts in the direction of positive x and later on as the time progresses the acceleration becomes negative and negative. So now one very simple thing okay very small quiz like question so what is true about kinematics of a particle so there are various things okay so let us brush up our concepts so first thing is that the velocity of a particle is always positive is it true okay of course it is not okay velocity can be positive or negative the velocity of a particle is equal to the slope of the position time graph is the only true statement here and it is a very important statement okay very simple statement in elementary calculus but when we come to mechanics the slope okay when you take the slope of position displacement curve then the slope has a particular geometric meaning a mechanical meaning and that is nothing but the velocity the third option is that if the position of the particle is 0 then the velocity must be 0 of course we know that for example if we take a ball okay take our origin at our hand and then drop the ball okay in that case the velocity will be 0 okay if we just drop it but we can also push the ball down and then the velocity need not be 0 so very simple point okay but very important point that the velocity need not be 0 if the position is 0 okay that is one point of confusion a lot of times okay but position velocity position can be 0 but since velocity is a derivative it need not be and ultimately if the velocity of a particle is 0 its acceleration must be 0 is another miss misnomer that for example if you take a ball no velocity is given to it initially we just drop it initially the velocity is 0 but what is the acceleration the acceleration is coming from the acceleration due to gravity G which is 9.8 meter per second square and that clearly is not 0 why because even though the velocity now is 0 if we just drop the particle the velocity will slowly increase in a small time interval and that derivative is equal to G so very very basic simple concepts but they are prune to a lot of errors that is why so much emphasis now very simple example okay the same example let us look in terms of graphs okay so x is equal to 60 square minus t cube this is a position graph as a function of time we take the derivative of x with respect to time and we get the velocity we see that initially the velocity is positive positive velocity by our sign convention is velocity to the right and then velocity slowly decreases it starts having direction with respect to the left its magnitude also increases okay so simple question okay what are x v and a at is equal to 2 second just one line answer we just substitute the values and we get what are the velocities what are the acceleration in what is the position okay same what are the x v and a at is equal to 4 seconds okay we emphasize to the students okay that we can have a graph like this and just read out what are the velocities positions and everything from these graphs now the next thing is that that typically okay because we are doing a mechanics course there is a source of a force and what we know from our high school physics is that Newton's for a second law tells us that force is equal to mass times acceleration so one class of problems is that if the force on the particle is given to us then we know if the mass is given then what is the acceleration of the particle then many a times what we know is that acceleration is the most basic thing so note one thing that even though we started with position from that went on to velocity from velocity we went to the acceleration typically okay in any real life problems it doesn't go that way so conceptually the progression is fine but really in in actual world it goes the other way around that it is the acceleration the source of that acceleration is given to us and from the acceleration we need to do appropriate integrations and so on okay and need to find out what is the velocity and then what is the position as a function of time now note one thing that there are three quantities that we are looking at one is x which is the position one is v which is velocity and third the most fundamental quantity okay is time which we are the which is the most basic quantity and so the acceleration can be given as either a function of time okay or it can be given a function of x or the position or it can be given as a function of velocity okay so one simple example that the acceleration is a function of position is the force from a spring so professor Banerjee today taught you about or discussed about the concept of potential energy so if you pull on a spring the resisting force that a spring gives is k times x where k is the stiffness of the spring and so we know that the force in the spring is a function of the position another example okay which we will discuss when we discuss vibration of bodies okay which are subjected to damping then in that case there is a drag force and drag force for example whenever for example a ball flies to the air or when a move vehicle moves or when a ship moves okay whenever there is a relative motion between an object and a surrounding atmosphere we get a drag and a drag typically is a function of velocity now the problems can be categorized into these various types one is a simple kinematic relationship okay what do we have so if acceleration is a function of time then straight away we know that dv by dt is equal to aft simple integration we can find out velocity as a function of time by this simple integration the next class of problems when the acceleration is a function of x in this case we just play a very small trick okay not a big deal a very simple trick that dt is equal to dx by dv and a is equal to dv by dt so we write these two expressions and use this two together and what we will see is that that vdv is equal to ax dx do integration on both sides and then we can get some extra relation that how does the velocity vary as a function of space for example and once we know how does velocity vary as a function of space we can also function of position then we can also find out how does the velocity vary as a function of time and how does position vary as a function of time and ultimately when acceleration is a function of velocity then what we say is a dv by dt is equal to a of v we just bring acceleration to the other side and we can do a simple integration like this from initial to final conditions and say that how does the velocity vary as a function of time by doing this direct integration we can get our answer and once we get that we can also write it like this dv by dt can be written as dv by dx into dx by dt now since dx by dt is velocity we write it like this and instead of getting a relation between velocity and time we can also directly now get a relation between position and velocity a very simple problem a ball is tossed from okay so very simple problem I know that it's quite elementary but why I am discussing this so that everybody is on the same page the ball is tossed out with a velocity of 10 meter per second from a window 20 meter above the ground determine the velocity and elevation above ground at time t highest elevation reach by the ball and the corresponding time and a time when ball will hit the ground and the corresponding velocity so we write a very simple equation okay what we know here we know that g or the acceleration is a constant function so we can easily write dv by dt is equal to a we can integrate velocity okay we can integrate this expression to get velocity as a function of time velocity is dx by dt we can do another integration and can find out position as a function of time then what we are asked to find out okay the velocity and elevation about the ground at time t that just comes from this highest elevation reach by the ball so we solve for t when the velocity reaches 0 because what we know is when we throw a ball it goes up up with decreasing velocity finally it stops and then starts coming down with acceleration g and lastly you want to find out time when the ball will hit the ground all the dimensions are given to us we find out that when you throw it from here it will go up stop and then come down because the expression for x as a function of time is known to us we can immediately find out that how much time does it take for it to reach the 20 meters from here to here okay the simple integration dv by dt is equal to a we integrate it we will see that this is initial velocity v0 to vt so v at time t minus v0 will be minus 9.820 why minus 9.8 we have taken upwards as positive as a result since the acceleration due to gravity acts downwards that is why the negative sign and we will see that this is the velocity as a function of time if v is positive it means it is upwards v is negative it means it is downwards okay that is a convention which we had discussed just a few moments ago then the next thing is you want to find out what is the y coordinate or what is a position as a function of time we say dy by dt is equal to v straight away we know that this quantity is given by okay 10 minus 9.820 this is the 10 is the initial velocity upwards this is velocity we integrate it again and what we get we have to make sure that the initial constant okay that the integration for y or the vertical position should be done from its coordinate at time t time 0 and you integrate it up to time t so we will get yt minus y0 and what is y 0 straight forward okay this is 20 meters is the if you take this as the origin then this is 20 meter is y0 do this simple integration this simple formula comes out now the second thing which we want to find out that what is the velocity what is the time when velocity equals 0 straight forward just put this equal to 0 we get that corresponding time and ultimately we want to find out what is the time t for which the altitude equals 0 so this should be equal to 0 okay we substitute that what is the altitude yt should be equal to 0 because this was our origin it where we had fixed fixed our origin of the coordinate axis this was where we threw the particle from so it has to go from here okay start from 20 and come all the way fall back to y is equal to 0 so we just substitute y is equal to 0 we get a quadratic equation in time and from that you have to solve this for t is equal to 0 and from that quadratic equation we will get the corresponding time and we get that time and we then find out what is the corresponding y coordinate okay so this is yt is the expression we obtained earlier so yt is equal to 0 so this y of t is equal to 0 is a quadratic equation are the two solutions when is t is equal to minus 1 to 4 3 now this minus 1 to 4 3 is a time reversal problem which is not meaningless physically speaking but as far as this problem is concerned okay our time 0 is when the ball motion starts and in that context okay for our purposes this time reversal has no physical meaning whereas this time t is equal to 3.2 seconds will be the time when the ball will come and hit the bottom ground one simple example of uniform rectilinear motion what is uniform motion that acceleration is constant so for example one motion is for if you have a jumper who jumps during this paragliding for example okay or when somebody jumps from the air and then deploys a parachute then due to the combination of gravity and air drag acting on it the particle or the body reaches what is called as a terminal velocity and so the particle or the body keeps moving at a constant velocity so this is a uniform rectilinear motion and a second simple example is a uniformly accelerated rectilinear motion now uniformly accelerated rectilinear motion simply is moving with some uniform acceleration which happens when you drop a particle or for example when you are going down a slope and when the drag is negligible now these are the famous examples that we learn in our high school mechanics that when the acceleration is constant then we can just do simple integration and say that velocity at any time t will be v0 plus a times t where a is the acceleration and the corresponding position dx by dt is equal to this so x is given by x0 plus v0 t plus half a t square just simple integration so these are the formula which we know okay like the back of our hand that position as a function of time is given like this and ultimately if you want to find out that how does the velocity depend as a function of time then we can say that d dv by dt can be rewritten as v dv by dx equal to a and we can find out that v square is equal to v0 square plus 2a x minus x0 where x0 is the initial position okay now these expressions they apply only for uniformly accelerated rectilinear motion now let us look at motion of several particles this is a relay race for example then the baton is passed from one runner to the other and so on so there are many many objects okay in this case okay the runners are can be taken in physics to be objects or in even simpler idealization to be particles and they are all running so we can define various quantities that what are the relative velocities of this with respect to each other and so on okay so when so we may be interested in motion of several different particles with motion may be independent as in this case or it may be linked together for example if you have various particles okay ABC which are linked by a mechanism of pulley like this then we will see that the velocities or the speeds of all these particles are intricately linked to each other now the first concept that I want to discuss is what is a concept of relative motion now we have two particles a and b okay they are moving along this coordinate axis so first we define what is a relative position of one particle with respect to the other the simplest case is that the relative motion okay the relative position of particle b with respect to a okay now what we say is that absolutely speaking o is the origin so in our lab frame which we say is the frame at rest o is our origin and with respect to this frame the position of this point b okay is given by x of b and the position of point a is given as x of a and what we say is that that the relative position of b with respect to a simply will be xb-xa or in other words we can also say that the position of b with respect to a is nothing but position of a plus the relative position of b with respect to a the same concept can be also applied for velocities that the velocity of point b is nothing but velocity of point a plus velocity of point a with respect to a so this if you think about it mathematically speaking this is triviality we just define velocity of b with respect to a as vb-va and then just add them together you get 1 is equal to 1 or 2 is equal to 2 so even though mathematically speaking this is a cliche in a proper engineering sense okay this is very important because what happens is that many a times okay you have for example I will give one simple demo here if we if we have for example two cars okay so from the point of view of the road okay taking road as our coordinate axis from the side of the road we see that is blue car is moving at a speed of 70 kilometers per hour in this direction red car is moving 30 kilometers per second in the right direction and we will see that from the point of view of the road okay both are moving together but from the point of view of the blue car okay we will see that it is stationary and the velocities will add up and if a observer is in this car okay then they will see that velocity is just some of these two velocities from an absolute point of view so it will see a much higher velocity then a observer on the side of road we will see whereas if you are in car a you are driving the car then you will see that this is coming at an opposite velocity okay but with the same speed towards it and the speeds will be added up as compared to a person who is sitting on the side of the road that he will see. Now the simple idea is that that frame of reference is an extremely important quantity that even though we may say that okay this is my lab frame but there is nothing wrong with having some other frame okay which is also moving at a different at a different velocity with respect to the first frame and we can always define that what are the relative velocities from one frame of one particle or one body with respect to one frame so when we change frames then we can also define velocity from one frame we can also define velocity from other frame of reference so these are the concepts of relative velocities same will go for relative acceleration okay that acceleration of B with respect to O can be written as acceleration of A in the absolute frame or in this O frame plus acceleration of B with respect to A. Now these kind of concepts become extremely important when we look at moving frames of references okay so this is a simple problem okay that ball thrown vertically from a 12 meter level in elevator shaft with initial velocity at same time open platform elevator passes at 5 meter per second moving upwards at 2 meter per second now we want to find out when and where ball hits the elevator and the relative velocity of the ball and elevator at contact. Now note one thing is that previously we had just thrown the ball and we are looking that when it hits the ground now that ground was a fixed frame of reference for us. Now in this case the frame of reference is a moving elevator so what we want to find out is that that the elevator is moving okay upwards the ball is trying to move it will move downwards okay first it moves upwards and it moves downward and when we want to find out when it hits the elevator we want we should definitely now take the motion with respect to a moving frame that is a moving frame of reference of the elevator okay so how do we do this problem first we find out okay the initial position and velocity and constant acceleration of ball into equation of motion for a uniformly accelerated motion. So from a fixed frame of reference with respect to a fixed point O we first find out what is the velocity of this ball as a function of time and what is the position of ball as a function of time this is what we had done earlier. Now we also find out that what is the velocity of this elevator with respect to this fixed frame which is 2 meter per second and not only that we also find out that what is the displacement of this elevator why E is the elevator as a function of time. So now from this fixed frame of reference we know what is the velocity of the ball in this fixed frame of reference we know what is the position of the ball and the same we know for the elevator velocity and the elevator position and now what do we do we need to find out what is the position of the ball with respect to the elevator which is nothing but yB minus yE and what do we want is when this relative position becomes 0 that is when the ball hits the elevator and when we put that equal to 0 we solve the quadratic equation get 2 times one is a minus time which we had just seen that is a reversal time which has no meaning for this problem and the next time okay is a positive time which has a physical meaning for this problem we take that problem and we see that at this particular time is when the relative position of the ball with respect to the elevator becomes 0 and that is the time when the ball hits the elevator. Now take this time okay what do we want is that that at that particular time what is the absolute position of the elevator just put that time inside we will see what is yE or the absolute position of the elevator with respect to the fixed axis okay then we also want to find out what is the relative velocity of the ball with respect to the elevator at the point of impact now what was the point of impact point of impact was when t is equal to 3.65 seconds V of B with respect to E is V of the ball minus V of the elevator we put t is equal to this we will see that at the time of impact velocity of the ball with respect to elevator is minus 19.81 second which means that okay from the point of view of the elevator you will see that the ball is actually coming down with a velocity of minus 19.81 or coming down with a speed of 19.81 meters per second okay so that is the idea about frame of reference that this is the velocity okay what will be seen from elevator E of ball B that V E plus V of B with respect to E is the absolute velocity of the ball so this is a person standing on the elevator will see that this is the velocity of the ball now the next question is motion of several particles okay or dependent motion now look at this particular problem what we have is that we have masses A B 2 masses A and B okay they are connected by set of pulleys C and E and there is a rope which is inextensible that is going from this around pulleys CD around pulleys EF and coming back at point G now what is the observation here let us do one simple thing what we will do is that we will fix a fixed fix a reference coordinate frame at the top which is a which is a rigid portion okay we will fix our coordinate frame here and with respect to this coordinate frame we will write down that X A is the position of point A with respect to this frame and X B is the position of mass B with respect to this top frame and what is the total length of this rope you will see that this is X A then neglecting this constant values okay these are some constant values which we do not need to bother about but the total length will be X A plus X B plus this length is the same as this so the total length of the spring of the string is X A plus 2 X B which is constant okay so now because this length is constant this position of A and this position of B are not independent as a result this really now is a one degree of freedom problem and now if you have 3 blocks for example 1 2 3 seemingly these are 3 degree of freedom problems that this has one position X A this has another position X B and this is another position X C so seemingly it is a 3 degree of freedom problem but in reality it is not okay it becomes only a 2 degree of freedom problem why because the length of the string is constant so what do we have X A plus X A 2 X A plus X B okay this is X B so this will also be X B so X B plus X B plus X C is equal to constant which is the length of the string so 2 X A plus 2 X B plus C is constant so even though there are 3 variables there is one equation which relates them and that equation makes this seemingly 3 degree of freedom problem into a real 2 degree of freedom problem now what happens is that we know that these are the positions and this is the relation between the independent positions so what we can do is you can just take derivative of this with respect to time straight away and we immediately see that D X A by D T is equal to velocity of A so for velocity of B and velocity of C and sum of all these velocities 2 V A plus 2 V plus 2 C V C equal to 0 take further derivative we will see that 2 times acceleration of A plus 2 acceleration of B plus acceleration of C is equal to 0 and from that this seemingly 3 degree of freedom system actually becomes a 2 degree of freedom system okay now let us look at this simple problem it is a very nice problem very simple problem so what do we have here pulley D okay look at this mechanism okay just stare at it what do we have pulley D is attached to a collar so this pulley can slide up and down on this collar and this collar is pulled down at 75 millimeters per second so this is the given condition in this problem that you are taking this pulley and it moving down okay by a speed of 75 millimeters per second that the pulley that is a given condition in this problem at T is equal to 0 collar A starts moving down from K okay so look at T is equal to 0 okay that this is the configuration at T is equal to 0 and at T is equal to 0 what is said is that this collar A also starts moving down from K with constant acceleration and 0 initial velocity so 2 things so the given condition in the problem is that in this configuration starting at T is equal to 0 this has a downward velocity of 75 millimeters per second and K has a constant acceleration what that acceleration is not known to us but it has a constant acceleration and 0 initial velocity but what we also know is that the velocity of collar A is 300 millimeter per second as it passes L okay what is this point L this point L is at a distance of 300 millimeters below point K we are asked to find out the determine the change in elevation velocity and acceleration of block B this block when block A is at L okay so the problem is straight forward this is a many degree of system okay so 1 2 3 this is seemingly 3 degree of freedom problem okay but now what is given to us is that this is provided this acceleration is provided we want to find out given that these conditions are given at T is equal to 0 what happens when this block this collar at K when it moves down 300 millimeters then what is the position velocity and acceleration of block A so what we do first we define the origin at this upper horizontal level with a positive direction coming downwards now collar A has uniformly accelerated uniform motion okay so we are told here that is the condition of the problem that this one okay has a uniform acceleration of A now what do we do we know the uniform acceleration but we also know that its initial velocity is 0 and it reaches this point L okay which is at 200 millimeter okay sorry are at 300 millimeter pardon me so which is at 200 millimeter so how much so we can easily solve for this quantity also okay so we can solve for acceleration and the time that is required to reach L okay because we had seen the previous expression that V is equal to V naught plus AT and V square is equal to V naught square plus 2A times X minus X naught and X is equal to X naught plus V naught T plus half AT square okay so we have we know all these equations and we can use those expressions and then ultimately pulley D has a uniform rectilinear motion so we can find out how do the position of pulley D change as a function of time and block B commotion okay we will see that because the length of the strings is constant block B motion is independent is dependent on motions of collar of this collar and at and this collar so collar A and collar D okay the block commotion is these motions are linked with the motion of pulley B now what do we do first we want to find out what is the acceleration of this particular collar at A how do we do that we know that the initial velocity is 0 okay what else are we told we are also told that the velocity when the collar is at A okay when the collar comes here okay when the collar A comes here the velocity is 300 millimeters per second so what we do is that this velocity is 300 millimeter per second so this square is equal to 2A which is the acceleration which is unknown times 200 millimeter what is 200 millimeter 200 millimeter is nothing but the distance it falls through okay when it reaches this point L from point K so we find out what is the acceleration which will come out to be 225 millimeter per second square now we need to find out what is the velocity of A how do we do that we know that VA is equal to VA0 plus acceleration times multiplied by time now VA0 is 0 VA at A at this point is 300 acceleration we found out from the previous step just put in all the values we will see that what is the time okay at which okay this top collar at A okay reaches the bottom point at L the top collar A which was initially at point K reaches point L we get that time next what do we need to know we need to relate how is the position of pulley D link as a function of time we know that position of pulley D is already given to us what is HD0 okay what is velocity of D is given to us velocity of D in the problem is given as 75 millimeters per second use these two values and what do we get that this difference okay from here to here is just 100 millimeters so given this position okay this is going to move down by 100 millimeters now note one thing we do not really need to know what is HD0 okay it does not matter what we only need to know is that at given this is the initial position how much from the initial position it has moved down by okay so that is what is this difference and now ultimately what we need to find out we have found out what is the velocity and what is the position okay for both this collar at A and and this pulley at D and ultimately then we want to find out by using this expression what is the velocity and what is the position of pulley B and the corresponding acceleration at pulley B how do we do that we write down what is the total length here so this is x A this is the position of collar A x A this becomes x D x D so x A plus 2 x D this is x D this is the same as this so x A plus 2 x D plus x B is the total length of the pulley so what do we know is that that x A minus x A 0 plus 2 x D minus x D 0 plus 2 x B minus x B 0 is equal to 0 why because this remains constant and at any instant x A should be equal to this some of this coordinates at time t is equal to 0 why because the total length of the cable remains constant so do all these subtractions from previous expressions okay we have found out that with respect to this position this x A is nothing but 200 we also found out that given the initial position this x B minus x D 0 is nothing but 100 millimeters and what we need to find out now what is x B minus x B 0 substitute all the values and you will see that the relative displacement of point B with respect to its initial position is nothing but minus 400 millimeters what does this minus mean that our coordinate system was fixed at this horizontal plane and downwards was taken to be positive so negative means this point B is moving up next you want to find out what is the velocity how do we get that just differentiate this with respect to time since these are all constants we will get x dot A which is the velocity at A plus 2 x dot D which is the velocity at D plus x dot B should be equal to 0 but what is x dot A we have found out previously that at the bottom point the velocity of A is given to us okay when the velocity of A at that bottom point was 300 millimeters per second velocity at B is uniform and given by 75 millimeters per second just substitute everything in and you will see that the velocity of B is minus 450 millimeter per second or 450 millimeter per second going upwards to find out now the accelerations what do we do again differentiate this expression with respect to time you will get acceleration of A plus 2 times acceleration of D plus acceleration of B is equal to 0 from the first step we found out that the acceleration of A is downwards to 25 millimeter per second square since point D moves at uniform velocity its acceleration is 0 and acceleration of point B okay then can be easily obtained from this expression which comes up to be minus 225 millimeter per second square or 225 millimeter per second square upwards okay so this is one problem which is conceptually simple but you have to just connect a lot of pieces together and then all these connected problems can be solved very easily once we recognize that how are the motions are motions of one part linked with the motions of other parts okay there is just one point which I want to mention here is that that for graphical solution of rectangular motion problems you will see that if x is given as a function of t then the slope is the velocity if B is given as a function of t then the area under this curve is nothing but the distance traveled and so on okay so just keep that in mind so two very similar questions one is from center 1331 that what is the difference between kinetics and kinematics and why do we have kinematics and kinetics studies separately okay so first thing is we all know that there is this famous Newton's law for particles which says that for a particle okay force is given by mass of the particle times the acceleration of the particle okay so now note two things that force is a quantity which you cannot just by looking at it or by using a video camera you cannot find out what is the force that one quantity is exerting on other quantity means for example if you look at a at a truck okay moving at some particular speed and you also look correspondingly a bicycle moving at the same speed so the kinematics okay or the geometry or the motion for both of them is the same okay but we clearly know that if at all we are pushed that we have to come in front of them we do not want to come in front of the truck why because we will see in the next topic that truck has very high mass and as a result it has very high momentum and when a truck hits you okay the amount of force that a truck generates okay will be much larger as compared to for example when a bike unfortunately I hope this situation does not happen or if the bike hurts hit somebody then the force that is generated is significantly lower so what we need to know note down is that that we need to find out acceleration and then multiply that with the mass and that will tell you that what is the force causing that acceleration so force is the governing cause okay which causes acceleration or if you stop some acceleration okay that a particle is undergoing some velocity and if you try to stop it then you have to exert a force to accelerate or to decelerate that particle so force is the cause and effect and the result is the acceleration or the velocity or the position change but what we see is that that in order to figure out okay what is the force that is acting on the particle or what force we need to put on the particle to move it with that particular velocity on acceleration first we need to know okay that how does the acceleration change depending on the path or depending on the speed change what is the acceleration and unless and until we understand the acceleration unless we and unless and until we understand that what is the velocity because before understanding velocity we cannot understand acceleration and unless then we understand that what is the definition of position because because unless and until we understand what is the quantity called position we cannot understand velocity so from position we understand velocity from velocity we understand acceleration and once we know that in different coordinate systems we are going to discuss now curvilinear coordinate system that how to understand what are the acceleration that are faced by the particles only then will be in a position to understand what are the forces that are acting on the particle or what are the forces that are required to keep the particle in that particular motion. So kinematics that is why is the most basic study because unless we study kinematics we do not understand what is the acceleration of that particle and until we do not understand what is the acceleration of particle there is no way we can understand what is the force what is the angular velocity sorry what is the force what is the momentum what is the angular momentum okay and what is the energy okay so these quantities we cannot understand. So we study kinetics first okay sorry we study kinematics first and from kinematics okay we understand all these quantities and then we link them using Newton's laws to figure out what are the forces that are acting on the particle or on a mass or what are the moments what is the angular momentum all these quantities then we go ahead and figure out from this basic quantities that is why we need to study kinetics and kinematics separately because many different particles may have same kinematics but because their masses or because their moment of inertia may be different the kinetics did not be the same okay. So I hope this answers the question okay so center 1321 and center 1148 they have asked to elaborate instantaneous velocity instantaneous velocity is nothing that instantaneous is redundant velocity at the time of interest okay or at that particular time which you want to find it out that is the instantaneous velocity and nothing else okay instantaneous velocity is just an elaborate term it is simple velocity and velocity at a given time that you are interested in is the instantaneous velocity at that time 1108 ask the question is it possible in some cases that the numerical velocity becomes negative. Negative velocity means that the particle move in the opposite direction of the constant motion perfect so you have yourself answered the question see negative positive everything okay these are for our convenience okay because we want to quantify things okay we are very happy putting a coordinate system okay and then measuring everything with respect to that coordinate system okay we are we really can think very properly in that way for example we name particular station for example we name particular station as Mumbai CSD we name station as New Delhi say that this is our origin and then measure distances with respect to that why that just becomes easy for us and now because that coordinate system is arbitrarily fixed okay it may happen that the particle sometimes move in one direction or in the other direction and then we say somewhere the velocity is positive and somewhere the velocity is negative so the positive and negative is purely for our convenience but if you mentally think about it is just a geometrical picture that the velocity as a vector has a arrow pointing in one direction or velocity has a vector negative means it has a arrow pointed in the exactly opposite direction that is what it means for positive and negative velocity okay as simple as that positive negative just so that our mathematics okay and our calculation become easy okay there is one question which is asked by 1085 okay said could you please explain the concept quiz again so let me go quickly to that what is R there is that what is true about kinematics of a particle okay this is general statement why this concept question okay this is a very simple question but this concept question so that once we think about all these four points and we convince ourselves that this statement is either true or false then our concepts automatically are very clear even though they are very basic concepts but we have to realize that all the difficult concepts that will later on come in mechanics okay they are essentially based on these simple concepts first is velocity of a particle is always positive that was the first concept which we just know for sure that velocity can act in the direction as we had just discussed in our direction of the presumed coordinate frame we take that to be positive but we know that it can also go in the opposite direction and the velocity can be negative we also solved a lot of examples okay from which it was clear that velocity can be negative second the velocity of a particle is equal to the slope of the position time graph so for a regular motion this definitely is true that the velocity of a particle is equal to the slope of the position time graph okay clearly if you come to this position time graph what you will see that here the slope is 0 so velocity is 0 okay what is velocity here dx by dt dx by dt velocity at t is equal to 0 is indeed 0 look at the top point slope is 0 so dx by dt is equal to 0 this happens at time t is equal to 4 note that at t equal to 4 you plot this again this is 0 okay so clearly this also happens now what does this statement means is that the velocity of the particle is equal to the slope slope is positive velocity is in a direction of our coordinate system slope is negative velocity is in a direction opposite to the coordinate system if the position of a particle is 0 then the velocity must be 0 now this is an obvious question that a position may be 0 but velocity need not be 0 why because for example if I take a ball and throw it from my hand so my origin is at my hand so at t is equal to 0 the origin is 0 because I have defined that as the origin but clearly if I throw the ball the velocity is not 0 okay that it has some velocity at that instant okay so clearly the position of the particle is 0 then velocity need not be 0 okay clearly simple other example if you take a billiards ball for example if you are playing pool snooker billiards okay the ball is at rest okay so put a coordinate frame at the position of the ball so coordinate the position is 0 now you hit it with a stick okay you give it an impulse we will do these problems tomorrow what happens because of the impulse you get an instantaneous velocity but the position is still 0 but you have some velocity so position 0 does not mean velocity is 0 the velocity of a particle is 0 then the acceleration must be 0 is again wrong okay why because simple example you take a ball and just drop it from our hand if you just drop it then the velocity is 0 but we see that acceleration is not 0 why because acceleration is G 9.81 meters per second square because the moment you drop it in a tiny interval the velocity actually increases so dv by dt is not equal to 0 so if you look at this plot of velocity you will see that dv by dt is not equal to 0 okay that velocity is 0 but there is a slope so dv by dt is not 0 which is 12 positive slope okay only when dv by dt is equal to 0 at this slope does the acceleration become equal to 0 okay so there is a question raised by 1073 which is about explain little bit on jerk and its relation with displacement velocity and acceleration now jerk is a very simple concept so we saw that derivative of displacement with respect to time that how does position change with time in plain english is velocity how does vector velocity change in time is acceleration now going one step further how does acceleration change with time is jerk simple example why is it called as jerk okay if for example you are sitting in your car and you suddenly like see a simple stretch of road ahead and decide that you want to accelerate it you step on the accelerator what you will see that you will feel that you are going backward and the car is zooming forward but now suddenly what happens is that suddenly some stupid person cut you across okay and it comes in front of you what you do you slam on your brakes when you slam on the brakes what happens you are like going behind you suddenly jerk forward why because there is a change in the acceleration okay in our roads in India which are typically pretty bad we can also never move with a constant velocity or acceleration you will see that the acceleration increase decrease if you are going in a bus you will see that the bus stops starts stops starts so not only there is a velocity change also the acceleration change so you will see that you keep moving forward backward forward backward so that jerking motion you translate it forward translate in the mathematical language and say that dA by dt is jerk okay as simple as that so jerk is just that okay so there is a question again raised by this center if the reference frame also moves from positive direction to negative direction what about relative velocity calculation so if you stick consistently with a with a with a convention that what is our convention telling us okay so if I move to the whiteboard you will see that there is a particle A there is a particle B let us for the time being just think about rectilinear motion it has some velocity VB what is this velocity this velocity is with respect to a fixed point okay fixed point O and this A has a fixed velocity with respect to the fixed point let me call it as V of A now what is V of B can be thought of as velocity of A plus velocity of B with respect to A what is velocity of B with respect to A it is essentially that if I am particle A then what is the velocity that I will see for particle B now once you write it like this and fix the convention if velocity of A particle A is moving in the opposite direction then only thing you need to do is to make sure that that minus sign is taken care of in this otherwise this expression always remains valid so a positive negative in the positive sense minus sense once we go to two dimensional motion then is it in one direction or other direction if we take the directions appropriately we do not need to think or break our head on that take the directions appropriately the relative velocities okay will be just vector sums and automatically will follow we will do a couple of problems also in the tutorial that we just need to look at the vectors properly proper direction proper or a proper sign convention we take that proper sign convention then just use this expression and we are done we do not need to think too much about it and once we get the answer then we may put our self in that position and have an intuition if whatever answer we get is right or not there is a question that is asked by center 1314 please explain dependent relative motion and there is a question by center 1260 in case of three masses how does it have two degree of freedom look at this okay this problem once we have a look at this everything will become clear now think about it we have three masses a b and c let us fit a coordinate system on the top that is our origin now think about it okay that in principle if you naively if they are not connected with respect with this chord okay all this a b c are not connected from with the chord then what happens a is one degree of freedom b is another degree of freedom c has third degree of freedom but now what has happened is that they are connected by chord like this now when they are connected by chord like this what will happen is that if a moves down b and c also will have some motions which will not be independent because of the presence of this chords so what we do is that we just write down a simple mathematical relation that links these three positions now what is a common connecting ground between them it is this wire or this chord now this chord we take it to be inextensible in principle it did not be if the chord is not inextensible then this does not become a simple problem this become more difficult but as far as e make is concerned let us say that a chord is not extensible or very little extension compared to all the displacement now what is the length of the chord in terms of the position coordinates of a b and c you will clearly see that at this point a okay the length of the chord is nothing but x a and you may add some dimension this length and this length and so on but see note one thing that these are only constants they are not variables to the chord length the only variables that are contributing is this x a so just note that the length of this portion is x a length of this portion okay again is x a now you may ask me that what this portion is also there but think about it okay write down some expression you will see that it will be x a minus some constant so that constant we do not worry because we want to ultimately differentiate or take relative separations so this distance does not matter so the length of this portion is 2 x a plus some constants the length of this portion is nothing but with respect to this frame what is the position of this which is x b same this goes up this is also x b so length of this portion is 2 x b plus some constants which are this some radius of the pulley and this distance and ultimately the length of this portion is simply x c plus some distance plus or minus some constants but the only variables that play in the length are x a plus x a plus x b plus x b plus x c plus some other random constants so essentially what does what do we have that 2 x a plus 2 x b plus 2 6 x c is equal to constant and now once we know that we can just straight away differentiate that with respect to time and we will see that dx a by dt which is velocity that the velocities of a b and c will also be related like this even though it seemingly is a 3 degree of freedom system this expression makes it now a 2 degree of freedom why because there is an additional relation which relates this 3 coordinates or we will have 2 v a plus 2 v b equal to v plus v c equal to 0 differentiate again with respect to time you will see that 2 acceleration a plus 2 a b plus 2 ac equal to 0 that is why even though it is seemingly a 3 degree of freedom system because of this one relation that is connecting all this coordinates x a x b x c we see that that this is not a strictly speaking 3 degree of freedom system but actually because of this relation a 2 degree of freedom system now curvilinear motion where do we see curvilinear motion for example when a cricketer hits a ball this is from beer and johnson so of course there is a baseball here the baseball for example or the cricket ball or a tennis ball or a projectile for example when you throw a stone all of them they are not moving in a straight line they are not moving in a straight line now in a straight line what is the thing in a straight line the direction can be only in 1 plus or in the minus whereas when you go in a curve the direction for example keeps changing okay the velocity direction keeps changing the position if you measure the position with one particular origin you will see that the position vector keeps changing and because the velocity the particle is moving along a particular trajectory the velocity vector also does not have just plus or minus it has a vector direction similarly for example if a car moves on a curved road you will also see that the direction velocity everything keeps changing both the speed okay and the direction can keep changing all the time and as a result to find out the kinematics of that particle means given the position what is the velocity and what is the acceleration it requires a little bit of more specialized thinking than the simple calculus that we had done previously so we need to do little bit of multidimensional calculus but that is not a big deal let us have a origin okay we all like to have a coordinate system okay we always like to say that this is the center and measure the trajectory of a particle with respect to this coordinate system so let us say we have a particle which is at p from p position it moves to p prime now clearly when it moves to p to p to p prime this is r is the position vector of this particle means for example if there is a person who is sitting here at o and he has a he has some way he has some radar for example or he this is striking a flight and he has a radar to keep track of that flight okay then what he is measuring strictly speaking is that that person is measuring what is the position of this particle p as a function of time okay now initially it is here it goes here now what happens the position when it goes to p prime is r prime and so to go from r to r prime okay what do we need to do we need to take a displacement of delta r now these are all vectors and this is nothing but simple triangle law that r bar plus delta r bar is equal to r bar prime okay so we know that this is a vectorial displacement now this is not a simple scalar it is not like moving on a straight line where the display is this positive or negative here this displacement can have a particular direction depending on what is the trajectory that the particle is undergoing then what do we define we define now okay now let me define our velocity how is that velocity defined that if this distance if this displacement happens in a small time delta t then the velocity is defined as delta r bar by delta t when the limit tends to 0 now what is this delta r bar in the limit when distance to 0 this is nothing but just dr bar by dt okay this is just dr by dt and you will see that this dr by dt is nothing but tangent okay this will be tangent to this curve just think about it that is dr by dt will be tangent to the curve okay which the particle is undergoing when this time is finite okay then this is this acts like a called but when this time become infinitesimal this can be thought of to be a tangent to this particular curve or v is equal to dr by dt or tangent to this curve now instantaneous speed now this speed is a scalar quantity now what is the scalar what is this scalar quantity is think about it that this particle moved from p to p bar now to move from p to p bar just note that the displacement is joining this point p from here to here okay just join this but what is this distance s this distance s is nothing but the length of this line the length of this line is s and ds by dt is nothing but the instantaneous speed of this so if you say that the velocity of the particle is nothing but the instantaneous speed v acting along the direction which is tangent to the curve okay that is a simple way of thinking about it that the instantaneous velocity again instantaneous why do you say instantaneous instantaneous is just an additional word instantaneous means that at this instant when I am interested in looking at this I want to know what is the velocity I know the trajectory okay I know the trajectory and what do I do in a very tiny interval I just join r to r bar I say this is delta r bar delta r bar by delta t in the limit when delta t goes to 0 is my vectorial velocity if I just look at the magnitude ds so ds by dt is the speed and then you can also think of velocity to be equal to that the particle has this speed and the direction of motion is in this direction which is the tangent to this curve now is a more interesting concept and more important concept so I will not go into extreme details of that because it requires some visualization but the simplest thing you can do is that just stare at the slides if you have not already done that what you will see is that unlike simple rectilinear motion when you are undergoing a curvilinear motion what happens that when a particle is moving along this curve it moves from point p to point p prime what we saw just a few moments ago that instantaneous velocity or the velocity at point p at given time t is nothing but the tangent to this curve okay that the direction is tangent to the curve and magnitude is nothing but ds by dt so magnitude and direction will give you this velocity vector now when the particle moves further okay from p to point p point p prime then what may happen is that if the motion is not rectilinear if there is a curve for example if you move the your car along a curve you see that this not only the magnitude of the speed may change means the magnitude of the speed is v that v may become v prime okay not only the magnitude may change but if you are going along a curve then the direction may also change that initially the direction is like this but at point p prime the tangent okay changes the direction so you can have a change both in the direction as well as in the magnitude and when we talk about acceleration now acceleration is not simply change in magnitude with respect to time but it is the change okay this is v bar this is v prime bar so delta v bar which is this this vector minus this vector divided by delta t so how much that the actual vector change by okay in and small time delta t that is how we define the acceleration and in the language of calculus this acceleration will be simply dv bar by dt we call that as the instantaneous acceleration again instantaneous is just an extra word that if we want to find out that here at any given time what is the acceleration we need to find out that how does this velocity in a very tiny interval both with respect to magnitude and direction which is essentially signified by the actual vector so how does the vector velocity change okay in a tiny interval okay divided by the tiny interval that is essentially what is called as the acceleration and typically okay you will see that if you do appropriate geometry you will see that if the velocity is like this okay the direction changes then the acceleration will not be just tangential okay if the speed changes then there will be a component of the acceleration which is tangential but if the speed does not change only the direction change for example if you are moving for example if the scar is for example moved in a circular direction okay in a uniform circular motion then the speed always remains same okay the tangential speed remains the same if you hold a stone on a string okay and rotate it then in that case if you rotate it at constant speed okay the angular velocity is constant you will see that the velocity of the speed of the stone is always constant but the direction changes continuously and as a result you get what is the acceleration okay which has a direction which depends on the vectorial change in the velocity and not just in the scalar change in the velocity okay so this is what you can think of the geometrical visualization for velocity that R1, R2, R3 all these three tangents will tell you what happens to the velocity and this is called as a velocity curve with respect to some arbitrary point C each point here okay this determines what is velocity this is called as hodograph the name is not important this slide is taken from Merrim and Crick 7 and what you see is that that the rate of change of vector velocity this is V1, V2, V3 is actually the acceleration and before we stop the session just now what I want to do quickly is just want to show a simple animation okay if I have a circular motion and this orange is the acceleration blue is the velocity and somebody had asked that this pink thing is the jerk, jerk is the rate at which the acceleration changes now if we just run this simulation what do we see okay we start from t is equal to 0 this is the particle and this is the trail this is how the particle has moved in history and you will see that when the particle moves this is a uniform circular motion so what happens to the velocity okay the velocity is blue it keeps changing direction all the time look the size of the arrow is constant the size of the arrow is constant but the direction changes constantly so the speed is constant but the direction changes and as a result we get what is called as a centripetal force you see that when we come back afterwards so this is the centripetal force which is the acceleration and because the acceleration is changing constantly you see that this is the jerk because acceleration keeps on changing constantly so jerk is nothing but the rate at which the acceleration changes and we see that for example if you are moving in a car which is taking a turn in a guard for example then we keep seeing that our direction in which for example our body moves keep changing as a function of time the acceleration we feel also keeps changing as a function of time so this simple simulation to tell that just speed remaining constant but the direction of the speed changing or the vector velocity changing can give rise to acceleration now instead of having a circular motion if you have a accelerated circular motion what is accelerated circular motion that the speed also changes so note now okay let the simulation run okay so we start this simulation okay you will see that as time progresses look at the velocity the velocity vector keeps increasing look at the acceleration the acceleration has both tangential component and a normal component component that goes inside and as the velocity increases the acceleration keeps changing okay so these kind of problems we will see when we come back okay so we will have a lunch break