Free Physics NEET Video Lectures for Motion In A Plane

00:00:44 Today we'll start a discussion on motion in a plane. Now to study motion or kinematics as we have seen the quantities which are relevant to us they are the position displacement velocity and acceleration. And till the last unit what we saw was we saw motion along a straight line. And when we described motion along a straight line then we took care of the direction by a positive or negative sign. If the if something was moving along one direction and if we call that as positive. If it was coming back or

00:01:49 reversing the direction if we said this as positive we call the coming back direction as negative. But now we will extend the discussion to two or threedimensional motion. So we will extend this to two or three-dimensional motion. And when we do this then the direction has to be specified in a general way. In one-dimensional motion, the direction was specified by the plus or minus sign. But now it has to be a general way. And this we do using quantities called vectors. So therefore the first part of study of

00:02:45 motion in a plane will actually be studying about vectors. Uh so before we describe what we will see here is how do we add vectors? We will see how to subtract vectors. We will see how to multiply a vector with a scalar. Now I've not introduced the term scalar. Scalar I will show is something which is got just a magnitude or rather than scalar we could say if we have a real number how do we multiply vector with a real number. Now the question which may also come to you is how do we multiply

00:03:35 vectors? Can we multiply them? And that we will refrain from answering that question in this unit. Later on we will talk about two types of multiplication of vectors and that will follow after some time. And after we do the vectors then we will look at motion of a body in a plane and when this motion of a body in a plane takes place with constant acceleration. Now because this is motion in a plane it will have multiple directions and if acceleration is constant then this is what leads us to a special type of

00:04:21 motion which we call as projectile motion and which we will explain. And finally in the last part of this unit we will study circular motion uh point which is moving and as it moves it traces a circular path. So these are things which we will study in this unit. So we start the discussion of this unit with scalers and vectors. A scalar is a quantity which has a magnitude. So how much of that quantity is there? That is what represents a scalar. And there is no sense of direction in a scalar. So this is basically a scalar

00:05:31 quantity is specified by a single number in a particular sets of units. So it's always any quantity unless if it is dimensionless will be specified with a unit. In a scalar quantity, this will be just a single quantity. For example, when we talk of mass of an object, so we say the mass is 1 kilogram or 2 kilogram or 1,000 g, 500 g etc. So there is no sense of direction when we talk of mass. Other quantities which are scalers are temperature of an object. When we measure the temperature then we say let's say when

00:06:29 we measure someone's body temperature we say it's 37° centigrade or 98.6° F. The normal temperature of human body we talk of again there is no sense of direction. Even when we talk of motion in a point then if we look at the quantity which we called as distance or the path length this was the total distance uh a point moves in going from A to B and when we measure the distance there is no sense of direction involved. So this quantity is also a scalar and scalers follow the general or the common

00:07:17 rules of algebra like scalers can be added addition subtraction multiplication addition and division. Now addition and subtraction we have seen we can take two scalers and we can add them or subtract them. When we add or subtract scalers then the units have to be identical. So for example if you have two masses you can add two masses. But if I have mass and a temperature of a body. Now these two things are scalar. Adding mass to a temperature will not give me anything meaningful. So therefore that cannot be done. When we

00:08:12 have addition and subtraction then the units have to be identical. But when we multiply and divide scalars. So now if suppose I'm multiplying a scalar A with a scalar B. Now scalar A may have its units of A. scalar B will have units of B. Now the product of this when I write this as A multiplied by B the units of the product will be the product of the two dimensions of A and B. So we can then work it out as we have seen when we multiply a and b then the units of the product will be the units the product of

00:08:55 the dimensions of these two units and similarly we can carry out a division of a scalar a by division b. Now these when we do the division these quantities a and b may have different units and the uh quotient which we'll get this will be the division of these two units and for example let us look at the density density is equal to mass divided by the volume. So now if the units of mass are in kilograms and if the units of volumes are in me cube then the units of density would be in kilogram per me

00:09:49 cube. So this is what we have and sometimes let's think let's take an example suppose we have a rectangle of dimensions a and b then here when we look at this if we look at the perimeter of this rectangle perimeter is equal to 2 * a + b so we are adding up the two ve two scalers here the two lengths a and b so perimeter is equal to 2 * a + b if we look at the area Then the area is equal to a * b. And here if a and b are in meters then perimeter is also in meters. But if we look at the area a * b then

00:10:33 the unit of area will be m * me which is m²ared. Now let us talk about vectors. A vector is a quantity which has firstly we'll talk of it has a magnitude and it also has a direction. So vector will have both magnitude and direction. And then there's a second property associated with vector which is very important and this is that it obeys a specific law of addition which I'll describe and this law of addition we can either call it as the triangle law we can express it in that term or we can call it as the

00:11:32 parallelogram. program law of addition. So a quantity which has a magnitude and a direction that is the first requisite and secondly when we add this quantity two of these quantities then they have to follow a specific rule the answers given by both of these will be the same and we will explain this. So a quantity has to do both of these for it to be qualifying as a vector. So a vector is spec because it has two attributes. It has a magnitude and it has a direction. So therefore a vector is specified by its

00:12:20 magnitude and its direction. And there are many ways of doing this which we will show. Now to represent a vector in a textbook normally you will use a bold letter notation. So in printed text you'll find vector being denoted by a bold letter notation. But when we write it then we use a letter with an arrow on top. This is what? So for example, if I write V with a arrow on top, this represents the vector V and the magnitude of this vector V. This is represented by just the letter V without the arrow

00:13:32 or sometimes we show it in between two parallel bars with a vector V. So the vector is indicated by the sorry start the magnitude of the vector V. This we will write by either V or V without the arrow. Now one of the vectors which we have seen earlier also and let's write this. This is the position vector. Suppose a point P is moving along a path. So at this instant point P is here. At the next instant it is at point at P P prime. So that means at time T the position the the particle is at P

00:14:34 at time t prime. This is at P prime. So what we do is to to find the position of this let's choose a coordinate axis. We choose two mutually perpendicular direction. We call them as X and Y. The cartisian axis. The intersection of these is the origin represented by O. Now what we do is we draw the line. So we choose X and Y which are mutually perpendicular directions and O is the intersection of X and Y called as the origin. So we'll draw this again. We have here X Y. This is O. This is P. Now the

00:15:44 line from O to P if I draw this. So this along with this direction, this is called R the position vector of point P. So this we can write vector R is equal to O P and because we are going from O to P along a particular direction. So therefore we represent this as a vector sign. This is the position vector of P at time t. Now if this is P prime I can draw a line from O to P prime and this I can call as the position vector R prime. This is OP prime. This is position vector of P at time T prime. And actually if I join P to P

00:16:54 prime, this is P, this is P prime. If I join them by a straight line, P P prime, this is what we will call as the displacement vector for P. So the distance this straight line from P to P prime which has a particular direction going from P to P prime is the displacement vector. And as we can see the even if the particle is traveling on any path from P to P prime it's traveling from this path or this path the displacement vector from P to P prime will always be the same independent of the path. And it's clear

00:17:37 from here that the magnitude of the displacement vector will be less than or equal to the length of the path. So now let's define if we have two vectors A and B then equality of the vectors means if we say vector A is equal to vector B then this will mean that magnitude ude of a is equal to magnitude of b. This is one thing but because it's a vector quantity and it is has two aspects magnitude and direction. So this is one and direction of a must also be equal to direction of b. So to ensure this if we

00:18:46 have a vector A which we denote by OP and there's another vector B which we denote by QR. So now if a is equal to b then what you do is to check this to check if a is equal to b we shift b we shift the vector either of the two vectors either a or b parallel to itself. we shift the vector till the two tails touch each other. So we'll shift B, we shift it towards A till O and Q coincide and then we look at if P and R coincide. If points P and R coincide, then we say the two vectors are equal.

00:19:48 If they don't coincide then the vectors will not be equal. So till the two tails touch. If the two heads coincide then we say vector A is equal to vector B. Now it's possible sometimes that magnitudes of A and B may be equal. And this we will write as A is equal to B because magnitude of A is A. Magnitude of B is B without the vector sign. But vector A may not be equal to vector B. And this will happen suppose we have vector A like this and a vector B which has the same length but a different

00:20:42 direction. So now when you shift the tail of B to A you will get the two tails will be at the same point but the two heads will not coincide. So therefore vector A will not be equal to vector B. So this is how we define equality of two vectors. Next we look at multiplication of a vector by a real number by a real number and let's say not only a real number we'll make it as a positive first we'll take a case of positive real number let's say the number is lambda So lambda is a scalar because it's a

00:21:37 just a number. So that means it's just a magnitude positive real number. And we look at lambda* a a scalar multiplied by a vector. So now what we get is this will be a vector. The so let this product which we have this product is a vector whose direction is the same as the direction of A. But magnitude is lambda times the magnitude of A. So for example, let's look at this with an example. Suppose I have a vector A and I want to say I want to write 2 A. So 2 a will be a vector which is twice the length of

00:22:47 a in the same direction. This will be equal to 2 a. Now lambda as we said so lambda a is a another vector in the same direction. If lambda is greater than one then the magnitude is of the product is bigger. And if lambda is less than one then the magnitude is smaller. And if I look at so the new vector let's write this r as the product is equal to lambda a if I take the magnitude of r which is equal to r. This will be equal to magnitude of lambda a and this we can write it as lambda times the magnitude

00:23:37 of a or is equal to lambda * a. Now here we have talked of positive lambda. If lambda is negative then we get a vector whose direction is opposite to a. It is in the same line but it is in opposite direction. So for example, if we have a vector A like this, then a vector of the same length in the opposite direction, we write it as minus A. So and if I want to have if there's a vector which is twice the length in the opposite direction, that will be equal to minus 2 A. Now, so we are talking of lambda * A.

00:24:38 Lambda can have its own dimension and the dimension of lambda A will be product of dimension of lambda and dimension of a and actually the way we have described this if suppose we have a scalar if there's a scalar beta which divides a vector a which means you want to write a divided by beta then this is just a special case of multiplication of a scalar where 1 / beta is lambda. So therefore vectors can be multiplied or divided by scalers with this rule. Next comes the key rule which we described in the uh property of a vector

00:25:58 and that is addition of two vectors because we said a quantity or two a quantity is a vector if it follows the addition rule of the vectors. Now as we said this addition we can describe by two methods with which we get the same answer. The first of these we call as the triangle law of addition. So we have a vector A and there is a vector B and we want to find vector R which is equal to A + B. So here we choose it's A + B. We choose vector A. We draw the vector. So we draw vector A. Draw the vector.

00:27:06 Now at the head of A that means at the tip of A which is at the end we draw the vector B with its tail at the head of A. So we we have vector we draw vector B with its tail at the head of vector A. So we have vector A here. This is A. This is B. So now the third side of the triangle seen in reverse order. See we are going from A to B. Now the third side of the triangle is this side. And what I mean by reverse order is if I continue from A to B then if I follow the arrows I should be taking this line I take the

00:28:20 reverse order and this re this third side this gives me R which is equal to A + B. So this is how we get vector addition from the triangle law. So this is called the triangle law that we draw the two vectors in same order. We complete the triangle and the third side of the triangle in the reverse order gives us the sum of vectors a and b. Now let's see we have seen what is a plus b. Let us reverse the order of addition. So with these two vectors let's see vector b + a. So if I have to do this

00:29:07 then as I have described first I will draw vector B and then on the tail of vector B I will draw vector A. So this was B this is A and then I look at the third side in the reverse order. This gives me the vector R1 which is equal to B + A. Now what we realize is that vector R1 is nothing but the same as vector R that is B + A is equal to A + B and this is what we call as the commutative law of vector addition. So and we so this is the commutative law of vector addition and we see this happening with scalers

00:30:04 also that the sum of a + b is equal to the sum of b plus a. We have a third property if we add three vectors together. So now let us add up three vectors A, B and C. And this way as you will see we can generalize it to adding of more vectors. And sometime this as you will see is called the polygon law of addition. So here let's see so we have a vector A to that we add vector B and uh so now what we are looking at is A + B and to that we add C. So A + B if we look at this this will be given by this dotted line. And now the third

00:31:03 vector C which has to be added to A plus B has to be placed at the tail of B. And now when we look at this, if I add A + B to C, I complete the triangle. And when I look at it in the opposite sense, so this is A + B. So this vector here, this gives me a + B plus C. So we form a polygon of all the vectors which are there. And the last side which will complete this will give us the sum of all the vectors. Now let us see how this is related to if suppose we interchange the addition that means we look at vector

00:31:45 A plus B plus C. That means first we add vectors B and C. So if I add vectors B and C I will get this dotted line and I have to take it in the opposite sense. So a b c. So I look at this line. This is vector b plus vector c. And when I add this to a, I again get a + b + c. So what we get is vector a + b added that to c will be to equal to vector a plus the sum of b and c. And this property is called associative property. So we can add vectors in any order. Now we also have if we look at a plus minus A and we add the vector to

00:32:54 negative of itself. Let's try to do this. So this is vector A. Now at the head of this I place a vector which is minus A. That means I come back to the same point. So the sum of these two will be now a vector which we call as the zero vector. So the result of adding a to minus of itself is a zero vector. So what we get is a and we can also from here reduce a plus the zero vector gives us a lambda* any scalar times the zero vector will give us the zero vector itself. and a zero scalar multiplied by

00:33:40 a vector a will give us the zero vector. So we have defined a zero vector in this way. And just as we have addition of vectors, if we try to look at subtraction of vectors, then this is a that means we are talking of a minus b. Then this is just a special case of addition which we can write it as a + minus of b. So that means we take vector a and we add it to minus of v which means reverse of v that will give us a + minus b and uh now as we have discussed this was we have seen addition using the

00:34:28 triangle law. Now there's a second way and which is called the parallelogram law of addition of vectors. And in the parallelogram law, what we do is we arrange vectors A and B with tails coinciding at the same point. Earlier if you remember we first placed A and at the head of A we placed the vector B with its tail coinciding at the head. Now here what we do is I have a vector A. I want to add this to vector B. So I'll place vector A and vector B with the tails at the same point. Now these

00:35:35 are two sides or rather two adjacent sides. So these are two adjacent sides if we look them as a adjacent sides of a parallelogram. So we complete the parallelogram that means we we draw a vector A from B and from this step we draw uh another vector which is equivalent to B. So this is a parallelogram. Then the diagonal of the parallelogram starting from the tail. This diagonal which starts from the common point of these two vectors. This is equal to vector r which is equal to the sum of a + b. Sometimes the reason why

00:36:24 we are using sin r because often the sum of these is called the resultant. So that's why but you can use any symbol. So here the diagonal of the parallelogram gives us the sum of vectors a and b and this is what is termed as the par. This is how we use ve addition using parallelogram law of vectors. And you can clearly see here if you look at this second side it's a parallelogram. So this side is nothing but vector b. And so the diagonal which we have got here is nothing but the third side of the triangle if a and b

00:37:04 are placed in order. And so therefore we see that the parallelogram law is nothing but will give the same result as the triangle law of addition. And if we sometimes as we said if we want to look at a minus b so if we have a vector a and this is vector b. We want to find a minus b. Then from B on this side if I draw a vector which is opposite this is minus B and I complete this triangle. So this vector will be equal to vector A minus vector B and this was A and B. So that is how we can do the subtraction. Okay. And

00:37:54 now what you'll realize here is a minus b will not be equal to b minus a. If I have to do b minus a then this is b. I move in the opposite direction here. I will get this will be minus a. I add these two. So that means I complete this triangle. I do this. So in fact it will fall on the same line. this vector will be equal to B minus A. In fact, it will turn out B minus A will be minus of A minus B. So they will fall in opposite directions. Now we also have if we have a scalar lambda multiplied by sum of A +

00:38:38 B, this will be equal to lambda * A plus lambda * B. And this can be verified. Next we look at what we call as resolution of vectors. Suppose we have two vectors A and B which are non-zero vectors in different directions in a plane. And we have a third vector a in the same plane. Right? Now we are talking of uh all the vectors lying in the same plane. So now what we can say is this vector. So we have a vector A, a vector B. These are in different directions. They need not be perpendicular to each

00:40:03 other. They are just two different directions. And we have a third vector A. So we have three things A, B and A. Now what we can say is that the vector A can be expressed as a sum of two vectors. Now what is the specialtity of these two vectors? The first of these is a multiplied by a real number. Let us say this is lambda. And the second number is vector B multiplied by a real number which is different in general say mu. So what we are saying is we can express a as lambda * small a plus mu * small b.

00:41:11 So this we can do and the way to do this to understand this is let vector A be equal to OP. So this is the vector A which we denote by OP. Then through O we draw a line parall to A. So if the original vector A was like this. So through O we draw a line which is parall to A. And if the original vector B was like this and through P we draw a line parall to B. So now B was in this direction. We draw a line parallel to B. So now this one will be lambda * A. This one the second vector this will be mu * B because this parall to B. So

00:42:23 therefore whatever is the length that will come in this factor mu. The length uh magnification factor will come in the factor lambda. So lambda a plus mu b can be written as vector a. And what we say is that so once we do this we say vector A has been resolved along two components two component vectors lambda A and mu B along A and B. Now in general A and B can have any orientations but they cannot be parall to each other. So this is what we call as resolution of vectors. Now resolution of vector becomes more

00:43:24 common if vectors A and B are perpendicular to each other. And here what we have let's so before we talk of resolution along perpendicular direction we first talk of something called a unit vector. A unit vector is a vector of magnitude 1. It has a directional sense but its magnitude is always unity. And so we denote a unit vector in general. Whenever we'll talk about this we will use a symbol hat. And whenever we use hat that means we are talking of a vector whose magnitude is one and we

00:44:25 call it as a unit vector. So now let us see the cartisian system. Our cartisian system. And here we will now extend this to three dimensions. So let's denote a vector a along its three dimensions. So we have for example the x-axis, the y-axis and the z axis. There's a general point p whose coordinates are x y z. And if we take the vector from O to P, let's say this as vector A is equal to OP. Now if we drop a perpendicular from P on the XZ plane and let this hit the XZ plane at P prime. Now if

00:45:40 we join OP prime then it's very clear OP prime plus P prime P is equal to OP. Now if we look at the coordinates of the point P prime then they will be equal to X 0. This is a point in the Z plane. So Ycoordinate of point P prime will be zero. It's x 0. And what we do is if from point p prime if we draw take a perpendicular to the x-axis and a perpendicular from p prime to the z axis. So now this distance here this will be zed and this distance here will be x and this x is what we will say is the x will be the x

00:46:44 component of the vector op. zed will be the z component of the vector op. And similarly P prime P this is in magnitude equal to Y. So this will be the Y component of the vector OP. Also what we see is suppose if this vector OP makes an angle alpha with the X-axis. So vector OP makes angle alpha with the x-axis. I'll show it in the figure here. So OP makes an angle alpha with the x-axis. It makes an angle beta with the y-axis. And it makes an angle I'll use a third colored pen. It makes an angle

00:47:47 gamma with the Z axis. Then what we have is the X component. This will be equal to magnitude of O O P cosine alpha. Y component will be magnitude of OP cosine beta. and the Z component will be equal to magnitude of OP cosine gamma. So these are what we call as the XY. So this is X component of the vector OP. This is Y component of OP. And this is the Z component of OP. Now what we do is we write unit vectors along X, Y and Z axis. Now these axis have a fixed direction. So for a unit vector along x-axis we

00:49:05 will use the symbol i. This is very common. For unit vector along y-axis we use the symbol j. For unit vector along the z axis we use the symbol k. So that means if these are are x y and z axis a vector of length one along x-axis is i. A vector of length one anywhere along the J axis, this will be J. And a vector of length one along the Z axis will be K. And if I want to find the magnitude of I, magnitude of J and magnitude of K, think about it. Obviously all of them have to be one. And suppose if I have a vector

00:49:56 A in some general direction. So the unit vector along A will be a vector of magnitude I shouldn't use the word length I should say of magnitude one along the direction of A. This is what we'll call as unit vector of a. Either we use a symbol n for this or sometimes we use e sub a with a hat. e represents a unit vector in many notations and a is to give the uh give the sense of a and with a hat tells us that it's a unit vector. So unit vector along any general direction can be written like this. And if that is so

00:50:48 then what you'll realize is if vector a if unit vector along this is n then vector a can be written as magnitude of a * n or magnitude of a * n. This is how we'll have the unit vector. And let's now take the resolution of vector A along X and Y axis. So let's look at a planer case. Now we have a vector A. This is the X axis. This is the Y axis. Let the angle between x-axis and a be theta. So now if this is the point p this is o p is nothing but the vector a. So now if I drop a perpendicular from here let

00:51:52 this point be px. I drop a perpendicular on the y-axis. Let this be point py. Then it is very clear that A which is equal to OP vector this is nothing but O * PX plus O O O O op OPX plus OP Y sorry not start vector A will be equal to OP is equal to OPX plus OP Y where PX and PY are the points on the X and Y axis and OPX one can Write it as a sub x * i that is the magnitude of opx which is the x component of a times unit vector along i d along x direction which is i and this I can write it as a y * j. So here what we have is I can

00:52:49 write the vector a as a x i plus a y j and ax and a y are called the x and y components of a. So ax and a y are the x and y components of the vector a. Now we can also if we want to find the so we have vector A we can resolve it as along X and Y as a X and a Y. So we see that there are two ways to write a vector A in a plane. The first way is we use the magnitude of a and we specify the angle theta which a makes with x-axis. We specify these two things and that gives us the vector a. And the second way is that we compute the x and

00:54:19 y components ax and a y. And then we write the vector a as equal to ax i plus a y j. And from the figure which makes this, it's quite obvious. If this is vector a, it makes an angle theta with the x-axis. This is ax. This is a y. Then what we have is ax² + a y². In fact, ax will be equal to a cos theta. a y will be equal to a sin theta. So ax² + a y² is equal to a² cos² theta + a² sin square theta which will be equal to a². And so therefore we can write the magnitude of a in terms of ax and a y as

00:55:18 ax² + a y square. And tan theta is equal to a y upon ax. If you look at this figure this height is a y this is ax. So tan theta is equal to a y upon ax. And in all these you realize ax and a y can be positive or negative. So that is how we work out the this thing. Now in the next class we will continue with a bit more on vector. We look at the analytical method of vector addition in terms of unit vector and then we will move on to the description of motion in a plane using vectors. Thanks. [Music]