Free Physics NEET Video Lectures for Motion In A Straight Line

00:00:00 e today we will start with kinematics kinematics is a branch of mechanics where we study motion of a point or a particle and we describe the motion without describing what causes the motion so when we study kinematics we do not go in the details of what is causing the motion but we just analyze the motion but before we start with kinematics let us look at some basic mathematical Concepts you will realize that while studying physics we do need the help of mathematics and the mathematical Concepts which we will talk about you

00:01:39 will study details about them in maths but here we will introduce some of the concepts as and when we need them so we are talking of motion of a point and we will restrict our discussion to planer motion that means a particle is moving in a plane so here to describe the position of a point we use what we call as a coordinate system and what we do is we take two mutually perpendicular directions let's say one of them is X the other is y so we call them the two cartisian xes and they are X and Y and these

00:02:47 axes are mutually perpendicular that means they are at an angle of 90° to each other now the intersection of these is called called the origin represented by o any point P which is at a location here is described by what we call as the coordinates of these points X and Y and what we mean by X is that if we look at the X distance the amount of distance this point is from the origin along the x-axis that is given by X and the distance the point is on the along the Y AIS is given by y so X comma y are

00:03:41 called the coordinates of the point P so let's write this the position P of a point is given by the coordinates X comma y now clearly as the point P moves that means it's moving along some path then the values of X and Y will change and this is the analysis of this is what we will do when we talk of kinematics now there's another term which we Define this is the point p in if we draw a line segment from o to P then this o2p which is a directed line segment from origin to the position of

00:04:55 P this op is what we call as the position Vector now this is a directed line segment because what we realize is the quantity op has two quantities one is the length op which is also called the magnitude of the vector and the second thing is the direction of Op with the same length if I travel along a circle I can get different points P which have the same length but uh to give the precise direction of Point P we then give the uh we mark from o to p and this direction is what gives us the directionality of

00:05:50 the vector therefore you can think of a vector as a quantity which has two characteristics one is the magnitude which in case of a position Vector is the length and the second is the direction and together both of these they Define a vector and we can uniquely then Define a position Vector with these qu ities let us also see some concepts of what we call as calculus and uh to you could call them branches of calculus first we will see elements of differential calculus and once again what we are describing here is very limited

00:07:01 mathematics this is not to be taken as a class in maths the mathematical details of these and more involved mathematical problems Concepts you will get to see in your maths course so to understand the first we have what is what we call as a concept of a derivative in differential calculus is all based on derivatives now suppose we have a line where which is varying with X so this we call as Y is equal to function of X so we Define a curve Y is equal to function of X that means for different values of X

00:07:54 we have different values of Y and when we join them together we get this curve which we call as Y is a function of X now let us look at two points p and Q on this curve so we have a same curve there are two points p and Q on this curve now let us say the coordinates of Point P are X and Y the coordinates of Point Q Point Q is located at a distance Delta X away from p in the X Direction and it is located at a distance Delta y away in the y direction from point P so therefore the coordinates of point Point Q so

00:09:01 coordinates of Point P are X comma Y and coordinates of Point Q the x coordinate will be x + Delta X and the y coordinate is y + Delta y so these are the coordinates now if we look at a straight line joining PQ then if we look at the make a right triangle like this that means we continue P up till where Q the extent of Q along x-axis and then go vertically up to point q and if this angle is Theta then we can clearly see that tangent of theta will be equal to Delta y upon Delta X so now what we say

00:10:03 is if the point Q approaches Point P so we let Q approach P which means we are now taking it closer and closer to point P but it does not exactly go to point P so this we say that Q approaches P which we will mean we say that Delta X approaches zero and we will have Delta Y which is the distance along y this will also approach Z 0 but Delta y by Delta X the uh division of these two this will not approach zero in a general case in a specific case it could go to zero but generally Delta y by Delta X this is a small quantity this

00:10:58 will not approach zero and here what you will see is the straight line PQ this will approach tangent to the curve at Point P so the line PQ approaches the tangent of the curve now if we denote slope of the line and slope of the line is again something which you will see in quadrate geometry and so this in mathematics course if we denote this slope of the line by m then what we have is the slope of the line M is equal to Delta y by Delta X in the limit Delta X goes to zero so now what we do is we Define

00:12:00 the derivative of a function y with respect to X where Y is equal to F ofx this is defined as so we use the symbol Dy by DX and this is written as we write it as derivative of f ofx with respect to X this will be equal to function of x + Delta x minus its value at X so this is the value of y at x + Delta X which is Delta y minus f of x which is y divided by Delta X and this we take and this we call as the process of limit as Delta X tends to zero so this we write as in the limit Delta X going to zero

00:13:01 and this is termed as the derivative of the function at the position X and physically this is equal to the slope or this is equal to the slope of the tangent at that point so this is how we Define derivative and once again more of this concept of derivative you will see when you do your uh when you do the mathematics course now if we have two there are some simple formulas for derivatives which we may need so let's just I will just give you some of these formulas so suppose we have two functions U of X and V of X there are

00:13:46 two functions uh just as we had f of x there are two of these then what we have is derivative of U + V is equal to the sum of the derivatives so any quantity which is defined as sum you can take the derivative of individual functions add them up for product we have a different rule if we have the product of u and v then the derivative of this is given as U * DV by DX + V * du by DX so this does not follow the same rule as we had for addition but the derivative rules uh for product is in this form

00:14:30 and then we have a derivative rule for a division for quotients D of U by V / DX or derivative of U by V is equal to 1 / V ² du by DX minus U * DV by DX and once again these derivations you will do in detail in your math course now if we have a power if we have X the^ of n then the derivative of x the^ of n with respect to X is given by n * x ^ of nus1 so and if we have a function U If U is equal to U of X then what we have is uh then derivative of U to the power of n with respect to X is given by n * U ^ of

00:15:33 nus1 * du by DX and actually this formula which I have done this is a generalized form of a formula which is which we have which is called the chain Rule and in chain rule what we have is if we have a function f which is a function of U and U is equal to U of X so now now here if we want to find F is known to us as a function of U let's say f is equal to uh u² and U is equal to 2x so U is a function of x f is known as a function of U and if we want to Now find the derivative of f with respect to

00:16:20 X now here notice we do f is known as a function of U and U is known as a function of X so here what we do is this is given by first we differentiate f with respect to U and then we multiply it by the derivative of U with respect to X and this is what is termed as the chain rule this is often it will come out very handy and you should keep this in mind that that's how we use this now we also have a couple of trigonometric functions derivative of sin x with respect to X is equal to cosine of x X and derivative

00:17:01 of COS of X with respect to X this is equal to - sin x there are formulas for tangent and cotangent SEC Etc uh they can be all worked out from here and using some of the rules we have done we have the product rule so for finding the derivative of tangent we can take s divided by cosine and work it out so these things can uh so these are some of the differential uh calculus which may be needing right now there will be more of maths we'll be needing but those we will see as and when we need the other

00:17:39 thing which we need slightly is some elements of what we call as integral calculus and here if we have this problem if we have a function f ofx and what we want to find is the area between F ofx and the x-axis when X is equal to a or between xal to a and X is equal to B that means at this point this is the xaxis x equal to a at the other point x equal to B there's a function FX we want to find the area of this shaded portion which lies between the x-axis and FX and on the left side we have the limit X is equal to a on the

00:18:39 right side it is x equal to B clearly if f of x is a straight line then the area will be the area of a rectangle which we can easily work out but how do we work this out if this is a general curve F ofx and to find this area between F ofx and ab and the x-axis from A to B what we do is we divide this area so we have this curve F ofx this is the x-axis this is x equal to a this is x equal to B we divide this distance from A to B in many small intervals let us say a position intermediate position is given by XI and

00:19:32 each of these interval has a length Delta XI so now that means I'm looking at XI and the next interval Delta XI so this area of this small portion I write this area as Delta AI so Delta AI will be equal to this area will be equal to this height here which is f of whatever is here that is f of x i multiplied by Delta X so this is the width of the strip and this is the height so product of these two gives me the area of the strip so the total area this will be equal to summation of all these Delta AIS so that means I make

00:20:42 all these areas here and add them up that will give me the total area so what we have is if I write this the total area this will be this is a symbol for summation capital Sigma Delta AI so this will be equal to sum of I going from 1 to n f of XI Delta X now if we want all the summation of area to match the exact area that means we should make these rectangles smaller and smaller so therefore what we to do this what we say is if n tends to Infinity that means n becomes very large then the summation sign which

00:21:37 we have we call this as the integral and we use a symbol which is like an elongated s this is the symbol of integral so this is when Sigma when n tends to Infinity we call that as the integral and what we say is the area is the integral of of f of x and Delta X we write it as DX and this goes when X is equal to a to x equal to B and right now here to explain to you I have written X is equal to a and X is equal to B but normally we write it as integral from A to B FX DX and this is what we call as the

00:22:27 integral so integ integral of a curve is the area under the curve between the lower limit and the upper limit this is the the first limit here this is the lower limit of integration and this is called the upper limit of integration so and such an integral this what we have done when we have area under the limits this is called a definite integral and this definite integral is the area under the curve now what can what we can also show is that integration is the inverse of differentiation which means that if we

00:23:23 have a function G of X such that the derivative of GX is equal to F ofx then G of X is equal to the integral of FX with respect to X so in that sense integration is uh inverse of differentiation because if DG by DX is is equal to F then integral of f is equal to G and when we write some this we can write if we say integral from A to B FX DX this will turn out to be this is equal to G of X and in terms of when we put definite integration we put limits here then this will be equal to GB minus G so and this is how we put the limits in

00:24:32 integration if we just integrate it and leave it in form of a function then the integral function is evaluated up to an arbitrary constant and if we look at some of the again some of the formulas for integral functions this you can see in your in any book and they take the form integral of x the^ of n with respect to X X this is equal to x ^ of n + 1 / n + 1 plus a constant any of these indefinite integrals where the limits are not put we always put it up to an arbitrary constant and in this formula n

00:25:16 is not equal to minus1 because if n is equal to minus1 this will become zero and this doesn't work for n is not equal to minus1 then we have another formula integral over 1 /x DX that means this is the case when x n is equal to minus1 this is given by a function called the logarithmic function natural log of x you may not know too much about this function right now but you will see this later on in some of your maths courses and when we need it here we'll explain this so and once again because these are

00:25:55 indefinite integrals we put an arbitrary constant uh two more things when we look at we need these integral of sin of X DX will be equal to minus cosine of x + C and integral of cosine of x DX this will be equal to sin of X+ C and these functions these two formulas you can directly get from the uh from the inverse rule which I talked about that derivative is is inverse of integration now the these are some of the mathematical preliminaries we have seen let us now come back to mechanics so a small break which we took but we

00:26:37 require all this so that is why we have done now let's so we come back to mechanics and physics and we talk about kinematics let us start with definition of a particle in our initial discussion on mechanics we call it as particle mechanics particle is an entity of very small size which means we will idealize this to be a point but of a finite Mass so such an object what we call we treat this as a particle now it may be uh if we look at it from a physical Point finding something which is a very small

00:27:45 uh size may not be possible and what we will see is in when we study uh mechanics we will uh treat in often we would treat things like ball which may be a cricket ball or a football they can also be treated as a particle so when can we treat these things as a particle if the distance moved by these qu by these entities by the individual bodies is much large as compared to its size and uh we assume that everything in there is moving like a same point so then we can treat these as particles so here what we have is

00:28:32 that if if we need overall estimates of path moved by the body without bothering about details of different parts of the body then we can treat this body as a particle and the distance moved by the body has to be much larger than the size of the body now we also Define another term when we study Physics is a rigid body and rigid body is a body in which distance between any two particles is always the same for example if we look at a rubber band and if I stretch a rubber band I Mark two points on a rubber band when I

00:30:23 stretch it then I find the distance between two points these two points will change change so a rubber band cannot be treated as a rigid body whereas if I look at a cricket ball in motion and I Mark two points on the cricket ball now as the ball moves the points may move but if I look at the distance between any two points of the ball that will always be constant so that is what we call as a rigid body now whenever we talk of mechanics here we are talking of motion of a point and to observe how a point

00:31:03 moves or how it is moving we require a concept of what is called a reference frame sometimes also called as a frame of reference it's very important to understand this concept in a reference frame is a location on which distance between two points is always constant so we for example if I'm standing on the ground then I will say ground is a reference frame and if I look at any two points on the ground they distance between them will not change so this is how I Define a reference frame and in a reference frame

00:32:04 what I do is I have a device with which I measure the length and I have a clock with which I measure time so there are two things which I will need along with the frame that in a reference frame we have a device to measure length and a div and a device to measure time which typically would be a clock so now when I fix myself on a reference frame and I want to observe so now I can observe the motion of a point p in this reference frame so let's say this is the ground I there's a person here I sit

00:33:15 down here I have and I there's a point P this point is moving with time so at time T is equal to Z I measure the distance of Point p with time and I keep on measuring how P changes with time I have an instrument to measure the length and I have an instrument to measure the time so using this I can see the observe the motion of Point p and this is this motion has been observed with respect to reference frame one now the motion of the same point can be observed with respect to a second reference frame

00:33:56 so reference frame there can be multiple reference frame so let us say we have a reference frame one which is the ground and on the ground suppose a car is moving so we observe the motion of the car with respect to ground so that is what I will say I'm observing with reference with respect to ref reference frame one or with respect to ground I see motion of the car now let my reference frame two be the car that means there's another Observer who someone sitting on the seat of the car so now to the second Observer who is

00:35:02 sitting in the seat of the car whatever is there in the car is not moving at all whereas a person who is on the ground if so let's think of this so what we have is suppose a car is moving we have a car a car is moving on the road reference frame one is on the ground and reference frame two is attached to the car and let's say there's a point the a point on the rear seat of the car now if I observe this from reference frame one now one thing what I'll have to do is if I if I want to observe the

00:35:59 motion of a point P what I will do is I will attach a coordinate axis to the reference frame one to the reference frame where I want to take measurements and suppose I observe I attach coordinate axis to reference frame on the ground and I observe the motion of this point P which is on the rear seat a fixed Point p what I will find is as the car moves if the car is moving along a straight path I will observe this point P to be moving with the car whereas a second reference frame let us attach the reference frame

00:36:41 two on on the car itself and let the obser AES be mounted on the front seat of the car now from there from this reference frame when I observe the same point p then I will have to go in the negative side because I'm sitting on the front side but if I look at the position Vector p in reference frame two then as the car moves I'm moving along with the car and in this Frame what is going to happen is that the point P will appear fixed whereas the point P appears moving from reference frame

00:37:19 one and we have looked at the point P if suppose I look at a point Q which is fixed on ground then the point Point Q from reference frame one will not be appearing to be moving at all whereas if I look at this point Q from the car then as the car moves the point Q will appear to move so Point Q which is fixed with respect to reference frame 1 is moving with respect to reference Frame 2 and point P which is moving with respect to reference frame 1 is fixed with respect to reference Frame 2 so therefore Define finding a reference

00:37:58 frame all these measurements which we take of the position vectors are always with respect to a reference frame now let's pause a bit and think about this is there any reference frame which is absolutely stationary that means now we are talking of the reference frame can we find a reference frame which is not moving at all so in the example which I took earlier it seems to us that if I have fix the coordinate axis to ground then in that reference frame nothing is moving which may be good for some

00:38:50 practical observations but actually if you see the Earth itself is rotating about its own axis so therefore so if we have clearly if we see a reference frame fixed to car this is moving with respect to ground now I take a reference frame which is fixed to Earth or fixed to the ground then in an absolute sense this reference frame is rotating with the Earth so that means even this Frame is moving so then what I say is maybe I F fix my reference frame not to the Earth but to the center of the sun if I do that then in that frame

00:39:49 Earth will move but this reference frame will not move but Sun itself is moving around some other body so there therefore whether it is possible to find a reference frame which is absolutely stationary we do not know the answer but the question is why do we talk of stationary reference frames and the reason for that is that later on will talk about Newton's Laws of Motion where we'll relate Force to acceleration and that relation is valid provided acceleration of a particle is is measured with respect to a frame which

00:40:27 is stationary which we also call the inertial frame of reference if acceleration is measured with respect to some accelerating frame Newton's law is not valid and so but then if we have said that if we do not know absolutely stationary frame exists that means Newton's law may never be valid but then the issue comes can we treat the motion to be negligible for example in most of the calculations which we do for motion of bodies on Earth if we neglect the rotation of the earth that works out to be a fairly good

00:41:01 approximation so therefore then we treat the ground as an inertial frame or as a fixed frame but if motion of Earth has to be accounted for then this Frame cannot be treated as inertial so these sort of considerations one have to keep in mind so now having observed a reference frame I what I'm going to do is before I start the discussion on kinematics of a simp discussion on kinematics of a point I would like to give a basic definition of two quantities which will come in which are Velocity and

00:41:44 acceleration we talked of motion of a point now velocity of a point will give us an idea we come to the exact details of what velocity means later but it gives us an idea of how fast a point is moving a moving point means its distance is changing with time how fast is this distance changing that idea is what is given to us by velocity and I'll talk of in a moment about the exact meaning of what we call as the velocity and similarly how fast the velocity is changing with time this notion is what we understand

00:42:25 by acceleration so velocity acceleration gives give us rate of change of distance and rate of change of velocity respectively so now let's see we are again taking planer Motion in this case suppose this is a path on which point p is moving so at time T the point or the particle is at p and this is the origin so what we do is we draw op is the position Vector at time T we denote it as Vector r as a function of time at time t plus Delta so this is at time t at time t + delt T the particle is at P

00:43:39 Prime so the position Vector is given by op Prime this Vector is the position Vector at time t+ delta T so this is r at t + delt T this is r at T now the vector p p Prime this is called as the displacement vector and if I look at r Vector at t + delta T minus the r Vector at T / delta T and the limit that we keep on making delta T very small in the limit delta T goes to zero this is what we call as the velocity Vector at time T that is the basic definition of velocity Vector so the velocity of Point

00:44:56 P at time Mt this is the velocity Vector this is equal to the derivative of the position Vector I'm talk I'm giving you very general definition here I've not we've not even talked about the how do we add or subtract vectors that will come in later but before we move on ahead at least let's go through the general definitions so we have the position Vector p position Vector P Prime uh so we look at the displacement Vector between these two now one thing you'll realize and this is again some thought

00:45:37 which I would like to leave with you that if there is another set of x's on the same reference frame where with which you are measuring this and that axis could be oriented anywhere in that set of axis again if you look at the vector p p Prime that will be the same the position vectors in the new so for example if there is let me use a red pen here so Suppose there is another coordinate system let's say x Prime y Prime mounted on the same reference frame the origin is different the directions are different now if I

00:46:16 look at the position Vector of Point p position Vector Point P Prime they are different from R and R at T and R at t plus delt T but if if I look at the vector p p Prime this is the same even in the new coordinates and the velocity is given as p p prime vector divided by delta T limit delta T going to zero so therefore the velocity Vector will be the same it is independent of the coordinate axis provided they are both fixed on the same reference frame if they're fixed on the same reference frame even if the origin

00:46:55 or orientation of the coordinate AIS are different the ve velocity V of the point will be the same as a vector because that just depends on p p Prime which is the same whether it is measured in XY or X Prime y Prime so this is the definition of velocity of a point p and I'll just give the formal definition of acceleration these definitions are General and they will work for onedimensional two dimensional all of motion so acceleration of Point P this is given as velocity of Point P at t+ delta T

00:47:42 minus the velocity of the same point P so this is you could call it as P Prime because that's what we call the it's the same point actually so p and p p Prime are just different locations of the same material point so you and divided by delt t in the limit delta T goes to zero if we take the difference between the two velocity vectors then we get acceleration of Point p and which in terms of our starting definitions we will call it as instantaneous acceleration so with these Concepts we

00:48:15 would like to move on to the concept of the simplest type of kinematics which is motion in a straight line so now what we look at is a particle moves along a straight line so we have some line like this we have a particle which we say it's p it moves at different location now it need not move forward only maybe after some time it can recede its path so this is this is the particle P which is moving along a straight line now what we can do is we can align the x-axis along the straight line so for

00:49:32 example the particle could be moving like this if it is moving it may look incline but what I can do is I can always align my x-axis in this direction along that incline as long as it is moving in straight line so then what we will have is the particle when it moves along a straight line it we can generalize it by saying it travels along x-axis or minus xais so we'll tilt and this is called rectilinear motion and this is as opposed to curved or planer motion which will come if the particle is let's say going along a

00:50:23 curved path now in rectal linear motion the position is the location in space where the particle is at time T So as the particle moves then we also so for example suppose if this is the path so at time T is equal to Z the particle is here at let's say there's a distance of 200 M this is point P this is point Q which is at another 100 m from P so at time T is equal to zero the particle is at o at time T is equal to 1 seconds particle is at P at T is equal to 2 seconds the particle is at Q and at T is equal to 3

00:51:41 seconds particle is at P that means the particle moves from o to p p to q and then comes back to P so if we call if we write what we call as the path length let's write the path length path length is the total distance which the particle moves or the length of the path so now at T is equal to 1 so path length is the let's find the PATH length of this particle so what we find is if we try to find the PATH length at T is equal to 1 the path length is 200 M at t equal to 2 the path length moved is 300

00:53:00 M and now comes the thing at T is equal to 3 the particle has moved 200 100 and then it comes back from Q2 P so that means it has moved another 100 m so the path length is equal to 400 m so path length is the scalar distance moved by the particle and this has no direction this quantity is always positive because when a particle is moving it covers a distance and as much distance as it covers it is always positive now path length is opposed to what we call as quantity called displacement which is

00:54:01 the net change in X position so suppose if the particle is at coordinate X1 at time T1 and is at X2 at time T2 then displacement is defined as Delta X by delta T the symbol Delta which I've used here this means the change in which is X final - x initial so this will be equal to X2 - X1 ided by T2 minus T1 so this gives us the displacement and in uh in St if X2 is greater than X1 that means the particle is moving along the positive xais and if X2 is less than X1 the particle is moving along the

00:55:08 negative x-axis so if the particle is moving forward so displacement can be either positive or negative so displacement can be positive or negative it is positive if particle moves along the x-axis and it is negative if particle is moving against the x-axis or it is moving as we would say along the minus xais it is moving in the minus X Direction then we say this is displacement so now displacement is a vector quantity but in one dimensional motion we do we do not need to give its direction in the sense that it is always

00:56:18 along X so therefore in 1D motion direction of displacement is given by by the side if displacement is positive then it is along plus xais and if it is negative then it is along Min - xais so this is how one write the displacement as a vector and it is clear from here that displacement is not the same as path length there are two different quantities so in in the next class we will continue from here we will talk about how uh how we look at the graph of x versus T what what is the meaning of those graphs and uh we'll then talk

00:57:18 about how fast a particle is moving that means we'll introduce the concept of velocity and how fast the velocity is changing that is acceleration for one dimensional Motion in the next class thank you [Music]