# Class 11 Physics: Projectile Motion Part II, Range, Motion in a Plane; IIT JEE #ExemplarPhysics Hello Everyone! How are you all? Today, we are going to continue our discussion on projectile motion. In the last video, we have discussed projectile motion as, projectile motion as two-dimensional motion. We’ve discussed the projectile motion which has two dimensional motion and in order to represent this two-dimensional plane we have used horizontal axis and vertical axis we call horizontal axis as x axis and vertical axis as Y axis after that we have derived an expression further trajectory of motion you had arrived at the equations for the trajectory of motion equation for trajectory missed the path the path the path of the projectile then we derived an equation further time of flight time of flight we represented it with the capital T and capital T is equal to 2 u sine theta by G which we got in the last discussion it is a very simple question what are the factors on which the time period ends if you have seen the video properly for the derivation of these components this from the derivation of these equations I have considered only one axis at one time either vertical axis or horizontal axis but one axis at one time you might have water and out that is it correct over well is to control only one divergent at one point one time are the two dimensions or the to access the component of two axis components means I need to say displacement velocity acceleration along a certain axis so this when I say components of an axis also a comprehensive y axis I mean that displacement along Y axis velocity along Y axis acceleration along y axis and when I say one of the components on me the displacement along horizontal axis that is x axis velocity along x axis and acceleration along x axis so all the components of the vertical and horizontal axis are independent to each other so let me put some light on this but they are really independent of not further and making here two case studies ok and to make you understand friend of mine okay so let’s I know the case studies first is study number one our intention is to see whether the vertical component and the horizontal components are really dependent or independent of each other so that is our intention to do this case study so let’s see our case study one in case anyone what I am doing is I am asking one of my friend again asking one of my friend to go to the top of my guess what every again go to the top of the – or the breathing and it’s standard so this is my friend our friend so he is going to do the activities bus he’s going to do actor dispenser I haven’t to name him anything if you have any suggestion please sir white it in the comment section because he is going to be very useful for us all the activities experiments will be done by this friend so see that our friend I have asked him to go to the top of the my destroy everything okay so he went there I gave him two boys I gave him a two iron boys I have two idols faster I told him to drop the ball drop one ball first I told him to drop one one so he improper this is black ball okay proper this is black ball so when the black ball has dropped off what see ya the motion the black polish proper it will fall downwards right it will fall downwards so this bitch action is this this is vertical axis so which axis is this vertical axis so what is the initial velocity when he drove up to the ball the initial velocity I’m representing it by UI is equal to Cu and since what is the acceleration along this axis have such a Y which is equal to acceleration equal gravity and the distant razor because of gravitational pull so gradually the velocity is going to increase as the body moves down that is a new data this fighter from which is dropping of his a height H is of height H and I have told Anna also to measure that time taken for the body to reach the ground and he said that the is capital T the height from which is thrown at this potential and the time taken to reach the ground s capital T I have put them to measure the time also now C initial velocity 0 acceleration is G so what is the velocity at any time v v whether city at any time how did I write it uy – let me do one thing let me write little bit sign so what is the velocity of rewind at any time whether city at any time V by is equal to uy plus 80 correct at any time T so here in this case it is going to be V by u is 0 you have dropped it when you drop into the initial velocity is 0 0 plus a is G and on my axis G and time’ is any time T so this is the expression for velocity and any time say this point measure the time into that type of G will give the velocity at this point next what is the distance distance powered by it so distance covered by the point hello my axis let’s call that displacement along the y axis y is equal to ux t plus half ay T Square you extremis half a t-square and here in this condition Y is going to be u x is 0 0 into T 0 half a why if g is per so this is the displacement of the body and in a time T at any time T okay next what if either a question V Y square is equal to uy square plus 2y so here also in this condition V Y square equal to you guys 0:02 a if – today is G is G so – guh so this is the third equation so I’m writing the loss the the equations of motion further our body which is a proper down so I have measure time also tie that is T is equal to time is capital D to cover the distance H yeah so what is that relation of time with respect to H how to find out I can find it from this equation right so if I write T Square equal to 2 y by G so T is equal to 0 to Y by G for the total distance enter so T will become total time of flight capital T is equal to 2 mu H by G so this is the expression this is the expression they have in God so please remember these things which I will return in the boxes okay let’s just call it as so one mole of fell down cover distance H or displacement H time taken P so when one is over now you have another ball in history I have asked him I have asked him threw the ball to throw the ball with the initial velocity horizontally equal to in width so when he is throwing there is no vertical component for this there is only horizontal movement and I have said time to throw it with velocity equals in horizontal direction now if you see if you see what is the motion of the body the body is this the body is this right and it is going to slowly it is going to fall you now again I have told him to find the time having given to them to find the time you might have to think that less distance this is this is more distance so that time taken allow this motion will be more compared to this can I do you think that now the time is greater than three but if you either do this activity you will find that and even if there is a horizontal velocity and the body is powering your horizontal distance even then that I am taken to which the grounder still is P still it is e there is no change in time I’m of flight whether the horizontal velocity is present or not here let me explain the F of U let me explain with the help of these equations Cena first up when the body is back to own it in this direction see there is an iron ball and the air is going is not going to affect ax I am what is having enough weight so that there is not going to affect my devotion that’s why I taken these fireballs and I see that so if the what is the acceleration acting on the body at any point of time it is equal to G that – suppose here G downwards suppose here G downward suppose here G downward and that G is always constant the G sorbus constant in a constant direction uniform acceleration uniform acceleration and what is the so the velocity what is the initial velocity what’s the initial velocity there are y-axis my axis V is equal to 0 VL also few hours of you what the acceleration vertical axis G so initial velocity same acceleration also same so are there not see that they’re not going to give the same the equations for this also can’t I say that even the horizontal component present or not vertical components are going to give me the same result see verticality will cover how much height vertically high-touch so I can conclude here that the vertical components are independent of vertical components are independent of horizontal components horizontal components yeah now one more thing one more important thing when I told my friend to do the same activity I also told him to find the distance of a horizontal distance covered I am calling it as things I have told him to find the horizontal distance covered and this horizontal distance is equal to UX into D so the equal to UX into P see that it is request thrown by velocity UX horizontally so velocity into time that means what no acceleration along the horizontal axis acceleration at all or you know tell axis point number one and number two this displacement component this displacement is not at all dependent upon vertical component that means water not only vertical components are independent but also horizontal components are also are independent over what together component that if you do in even if you do analysis you need one component alone any one component was cooperating is hello hi there are no vertical axis or horizontal axis the result is not going to change I hope it’s clear so here by seeing all this what you can say you can say that horizontal components horizontal components and vertical components ah independent offer each other independent of each other or you can consider this two-dimensional motion see that this is moving the plane x and y this two-dimensional motion you can consider it as two one-dimensional motions so do the analysis of any one dimension which is required are given in the question you know you know that is enough okay I hope you have understood let me give you another case study ok let me give you another case study this time I will ask my friend to bring it god okay this time I have asked my friend to bring it hard so you brought the star wheels of God again sports car in our sports car is sitting inside is sitting inside okay now again I think I gave him her when I don’t want it again I’ll give them one I don’t know so I told him throw the I am particularly awkward with the velocity you might through the I fell vertically upward with velocity uy so he had fun with the ull city you might it went up it went up to a certain height inch it went up to a certain height and children’s consider this has height H right so it went up to a certain height and the time of flight for going to the topmost point and they came running back at captivity so I am streamer when you throw it with the velocity u my vertically upward please note down the height as well as NASA their time period and he has no put them down now what did I say you see that I asked him to okay so I am asking them one thing so I want them to driver to point a Avastin to tribal until point a so this is point a reference disappointed Forester and there is a point to be here there’s a point to be here there’s a point B here so this is point B so all are the same vehicles I cannot draw all identical should be smarter for that alright so this is the point B do you know one of the distance between a and B however okay let me let me let me explain you the conditions and what’s the distance between a so what I did what I said is I asked my friend to drag the waiter to try the vehicle with the velocity UX with velocity UX right so when you read your pointed when you reach with velocity U at some point they have started by angle and when the time is equal to capital T when the time is equal to capital T my friend has which of the position be if you don’t that means it will travel with UX a constant velocity UX then you will reach B and type capital D ok so the distance which are covered by a our friend our that is represented by X because it is along the x axis or horizontal axis X is equal to nothing but u X into T because our friend is a crime constant speed constant velocity is driving with constant velocity so displacement is equal to UX in 2d plane now it said that now I said to my friend take this iron body take this iron ball when you reach a when you reach a then through the iron wall what did any of what I would remember I’m saying him to throw it vertically upward it is equal to because the velocity equal to uy and here velocity you were now asked him to throw it upward all right so he did that he did that he was driving he was coming with the velocity UX when he reached it suddenly he has thrown it upward the velocity U I see our friend is very precise whatever we say it does suggest some name for hammer will use it okay so as soon as he reaches 8 it throws it over to Kelly of world Hannah and what he will have put with you why and when he when he when the ball came to his hand back ok with the wall went up and it came back so when the wall came right back and in your head his hand the homeward he found himself at position B our friend through the balcony of each particular upward with velocity u v and the ball went up and up and up and they gave it written the market to his hand but when he saw this position even B if you see it from the ground up if V people see it from ground what is the motion of the ball the motion of the ball does you seem like this the motion of the ball at him like this right same time same time same velocity moon upward and it reads the back of the time with the time T with the time of right after T but this time he will he started his position he is not constant a given position he started at a he reached with peace how is this possible how is this possible he has owned the father of word and he when he reaches down when the ball reaches down but by that time the ball as well as the whole bodies and P how is this possible so let us understand this how this is possible see that he is moving with constant speed you X right so the vehicle is moving with constant speed u X inside the body our friend is there he is also moving with u X and the ball inside volley which is whole data is also travelling but the UX is also new X so when he has 1 October together though he is coming over today upward the bone itself possesses of the velocity so when he reaches they what we see that is it is this not the projectile motion it’s just not the projectile motion that means if you throw vertically upward and if you throw vertically upward the time taken to reach again with the speed u my is capital T here the time is also capital even thrown with velocity u ye and if you feel me see this at the highest point if you could measure it this would be the same as height I hope now you can understand that this food is or another corporation shall see this auto drivers all right all are the vertical it’s moving on deeply this plane plane has a board work with body moving like this right like this address this but for doing analysis if you take anyone access is enough our result is not going to change because these two accesses are independent to each other they are not affecting each other they are not affecting each other clear so doing an answer along one axis and this correct one axis is correct right moreover it makes our analysis simple because at one point what we only consider one axis now let’s come to the question today’s question okay today’s question is the range of projectile depends upon a projection be acceleration due to gravity see mass of projectile be magnitude of measure should be of project now I will suggest that if you haven’t seen the past video that is the last video first part of this if I haven’t seen that it has been questioned now to see the video and after thing the video coming into this time to make it through this time and solve the question clear so what is that project type right axis vertical axis V X horizontal axis thrown with us sorry thrown with the velocity u so it has two components or a lot of poverty u X u cos theta vertical component u sine theta now how does our body move a body moves in a project time body moves in a project time okay more emotion a project plan so this is let me write this as of me X so B X is always constant response rate equal to u cos now what is it what is not he is not asking the range the range makes what is not a distance and covered by the body however the distance covered by the body are represented by capital R so this distance if you see how to find the distance is this distance or displacement again saying how to find that since our body is moving with constant speed along the horizontal direction just because you are not in a direction from now what diggle direction is not important vertical the components are not important why not one not one dimensional you know that is the horizontal dimension if the velocity is constant then the displacement how to write the displacement u cos theta sorry horizontal velocity VX into time taken what is the time taken see from starting point to the point when it again which is the grub what is that time that time has nothing but time of flight yeah the velocity into time of flight now see what is V X V X equal to u cos theta what is T from the last video to u sine theta by mu G to C let me write u u u square 2 sine theta cos theta excuse me by so use where to sine theta cos theta sine 2 theta okay they said American identity 2 sine theta cos theta is equal to sine 2 theta divided by G so range R is dependent upon what are the factors number one vaccine this answer depression the first first phase first option angular position I am a prediction theta yes first one is acceleration due to gravity G gee yes second one correct mass of the director any mass component here no the third one is one-fourth of the magnitude of velocity see that you use for my yes so the answer is a B and the answer is that see how they are related to each other and the output I get that this maximum distance range of max maximum each so to get maximum range I have two options can exchange II know so I have two options either I can increase the you have to throw away the separate is a particular angle theta so that is the regard will become maximum so more you more than I do are you the more is the value of ha more is the value of you more or is a rally of you Square and some more is a value of fringe so if u is fixed enough you is fixture of you is fixed and so what do you have for example I can give an example here suppose you have a tan and water okay there are you have a cavity ever ready so cannon is this is a cannon which is respect upon these okay two wheels so when you fire a pall from this when you find a ball from this the velocity of the ball is fixed but what you can change you can change the angle you can change the angle in the same way if you is fixed the velocity with which it is gone is fixed then what you can do you can use then you can use the angle that is change the angle so that what either this value will be maximum at the ball either the value is going to be max or that said you is fixed right you maximum so sign to to theta will be maximum better water what are the maximal value of sine theta is 4 so this is maximum only when our 2 theta is equal to 90 degree remember sine 90 is 1 so 2 theta is equal to 90 degrees so theta will be equal to 90 by 2 45 degree so what is our max u square by G u square by G if theta is equal to 45 essence of maximum is the maximum value when u is fixed I hope that today’s discussion you understood nicely and if you have not still if you have any doubt please free to please feel free to ask okay take care of yourself thank you