 So after circuit motion, in this chapter we have got projectile motion. So projectile motion is the motion of an object when what happens, when gravity is acting. So if you drop an object, gravity is acting, is it projectile motion? No, when the initial velocity which is not equal to 0 and 180. Ok, please write down when del motion, alright? So with uncomfortable absolute vertical, should not be equal to 0 or 180. If I am throwing it down, it just goes down in a straight line. So if it is a straight line, it comes under motion in 1D, motion in a straight line. Ok, but it goes like that on motion in 1D. Ok, but definitely entire motion can be contained in a single plane. So that is why we are studying it in motion including on motion in a plane. Understood? Ok, now when this kind of thing happens, we are assuming here the only force acting on an object is gravity. That is our assumption, that is gravity force. No. And these are gravity, more acceleration, but here is not a force itself. This is space-time continuum and space-time fabric is creating a hypothetical force. It feels like a force, ok? More is the mass, more is the deformation in space-time fabric. But we are not getting into all that. We are studying gravity as a force whose magnitude is what? Mg. Bigger the mass, alright? So suppose you have a mass of m kgs, its velocity is like this. Let's say this is the velocity, Mg force, like this. Its acceleration will be… So m into g is m into a, so from here, a is equal to g. So it does not matter what is the mass of an object, acceleration will… The acceleration is g. Direction of the mass, irrespective of what is the velocity, direction of gravity, alright? And when we are studying motion in 2D, under motion in 2D, we are assuming that the object is near the Earth's surface. When small g can change its value, if you go very far from the Earth, g will be in the motion near the Earth's surface. So g is uniform across flat, ok? Same level, it is possible project time. For example, this one, this is the simplest possible project time. So things I need to know. Suppose I am… So if there is… So when I am studying project time, time will take to hit the target, how far it goes, what is the maximum height will reach depending on initial velocity until I am less specified. When something is moving again, distance is ignored. Air distance is ignored, ok? It is not a factor. The entire projectile motion is nothing but… It is a relation between initial velocity, angle and gravity g. Anything else you can think of on which it can depend on? Nothing. If something happens, you change that. Let's say you go to mass and throw a projectile motion. g will be the force of g. So suppose I tell you g is now g by 6. So in the formula itself, you just write g. So whatever the length between initial and final points. Any doubts? Simple, right? Whatever the projectile is, suppose I am… So I need this example. I am throwing a stone like this. What do you think? What else could be offered? Any question is given to you. What is the maximum possible height it can go up to? You can say that is h max. Fine, let it out. Right now let's keep it simple. We are going to take example of the simplest formula of the projectile which is this. And we are trying to for what simplest possible projectile. Now what are the formulas for this projectile? Which is what? Level. Same level. Same level. So this formula is not values. You are not going to gain anything if you just remember the final result. It up. So it's a constant equation case. So I can use equations of motion. I can use equations of motion. So those three equations of motion, v is equal to u plus 18. We write down these equations of motion. What do you think the obvious x and y is your axis choice? For your plan. So this is my x axis and that is my both three equations along x axis and then along y axis. Understood? Okay. So can you tell me what is the initial velocity of x axis? Component. The component is u cos theta initial velocity of x axis. Along y axis what it is? U sin theta. Acceleration along it is 0. Effusion along y axis is saying upward is positive. Okay. So that is why effusion along y axis is minus g. Equations along y axis. Just write down three equations along y axis. Please write down. Okay. It's very systematic study. All you do is get a small length. What's that? What? Displacement along y axis is different. Displacement along y and displacement along x for x. Simple, right? Effusion x for x axis. Effusion y for x axis. Effusion along y axis. From x axis. What did you think? Final velocity of x axis. Final velocity of x axis. This is how I will write x is what? U cos theta. A axis. 0. A is 0. So this is a very, very important finding that along x axis the velocity is constant. It doesn't depend on time because there is no explanation to change the velocity. Velocity along x axis will not be changed. Velocity along x axis in a projectile under gravity will be same. When I say x axis, I mean to say horizontal direction. You can choose your axis like this and inclined also. Okay. This is one of the important finding. What about y axis? How do you write velocity along y axis equals to? U sin theta minus u sin theta minus g into total time of like t. I don't know what is t. Any doubt? U cos theta into is this and what? Plus? 0. Seriously? 0. So time of like u square sin theta by g. All of you understood this. u cos theta. There is sin theta because here suppose theta. So it will be what? Sin theta cos theta because cos sin theta cos theta by g. Yes or no? So for a particular sum of angular positions is 9 meters. Then for a given initial velocity range is same. Range is same. R is same. Understood? Now just one small thing. Can you find out the edge max? To find edge max what you should do? Is edge max displacement around? Which is what? U sin theta by g. Half it is for x. Let's do it quickly and tell me what is the answer. Time unit is used as equal to u t plus half by d square between a and c. Times is what? U sin theta by g. U sin theta is velocity of y axis. T is what? U sin theta by minus half times g into t square which is? U square sin square theta by g square. One of the g you get is displacement you are getting. Now question that suppose it is not symmetrical. Suppose it is like this. You might throw a projectile at that and then I am trying to find the maximum height. You have to get the height. Actually a better way to solve this question. Velocity between a and u square sin square eta a is minus g. S is s y so s y will come out to 2 g. Safety of derived like that. This is much better way to solve because this method you can use everywhere. So next class we will continue with the projectile different kinds of projectile motions.