 Hello and welcome. In this segment, we will consider the ascent mission, its definition and some features that will help us understand some of the fundamental aspects of the rocket technology. So, let us first try and understand what ascent mission is all about. In general, we will find that ascent mission, which is nothing but the task of transporting an object of interest from earth to a designated spot in space and imparted sufficient energy so that it can carry out the designated tasks. So, this ascent mission is characterized by events that occur at specific time interval. These are usually related to operation of various stages as well as other considerations that impact the performance. In most cases, the events are time referenced in order to ensure that each segment performs as designed and that there is a way by which you could verify the performance, whether it has happened in the specified time. So, I want to show you a typical ascent mission sequence that is for a Delta II rocket. The figure is taken courtesy the space vehicle Encyclopedia, but what I would like you to look at in this picture is the various events, their sequence and their respective time stamp. So, at the beginning we have the lift off, then as you can see there is a motor button out with a time stamp of 63 seconds. If I go to another point, there is a stage 2 ignition at 274 seconds, there is stage 3 ignition at 1396 seconds and then the final the spacecraft separation at 1596 seconds. Also there is a small table on the bottom right corner which gives additional information about this particular operation in the form of two additional parameters, that is the velocity at each of those locations and the acceleration which the vehicle is undergoing at different time instants. Now, with this you could immediately realize that an ascent mission is reasonably complex scenario and will require a great amount of effort in analysis and design. The complexity comes from the fact that there is a certain connection between the trajectory information that is the velocity altitude inclination with the time, which obviously means that there is a specific trajectory that one needs to achieve. So, which means there is a requirement that is group of parameters which are in the trajectory must be achieved at a specific time and which obviously means that you must be able to predetermine what that segment should be so that this particular connection can be established. Another point which is not very clearly seen in this picture, but it is very much there that as the trajectory evolves over a period of time, there are changes in the force system which have in mechanics related implication as well. Just to give you a perspective, when you are closer to the earth, atmosphere is dense, when you are in the higher atmosphere, you have large velocities. So, there is also a change in the way in which the atmosphere will act, the gravity will act. So, obviously, you need to appropriately take care of these aspects and ensure that a desired trajectory is realized. Typically, in the context of ascent mission, it is found useful to create appropriate segments. And within each segment, a specific type of model is used which is applicable for that segment. And then these segments are arranged in the correct sequence so that when a mission is performed in that sequence, the desired trajectory is realized. Understand this further and also to do this task in a reasonable manner. A typical ascent trajectory is broken into two broad phases which are directly driven by the nature of forces as well as the maneuvers which are required to be performed in those segments and is shown in the picture below. So, it is the same picture that we saw earlier, but now we have also introduced the various segments which will involve different force systems and hence different types of mechanics based models to capture the physics. So, very close to earth, you have vertical lift-off. Then from up to 50 kilometers, you have the atmospheric phase because typically for all state space missions, we assume that beyond 40 kilometers, the atmosphere possibly will not exist. And then of course, we have the major boost phase where a large amount of mechanical energy is imparted by burning large amount of propellants. And once the sufficient velocity is acquired, you start performing applicable trajectory maneuvers and then reach a point where possibly you want to release the payload which is a spacecraft or satellite after which it is assumed that the spacecraft will perform the task as designed or as required. So, it is with these segments that we are now more concerned. So, let us look at these segments in little more detail. So, let us first look at the lift-off and atmospheric phase and see what are the mechanics related implications of this particular set of phases. The lift-off is of course an extremely critical phase particularly because you are having an unstable configuration that you are trying to balance a rocket on its tail and the toppling due to gravity is the major issue. And from mechanics point of view, it is generally modeled as the thrust which is the forward force greater than the weight so that it will lift off and start moving vertically. There is a requirement that you must ensure the vertical attitude during this phase. So, once clear of launch tower, boosters and first stage engines propel the vehicle through dense atmosphere. In this phase, it is found that gravity and thrust are the dominant forces. Aerodynamic drag also becomes significant because atmospheric density is higher even though the velocities may not be very high. Because the density is higher, aerodynamic drag becomes a significant force. In the next phase where you have the boost which means you are out of the atmosphere by enlarge and you are inclining yourself. Of course, once you are out of the atmosphere, larger velocity increments are possible. So, you burn a larger amount of propellant, generate large amount of thrust so that you can achieve large accelerations and large velocity increments. And towards the end of the boost phase, maneuvers are carried out to rotate the velocity vector in order to achieve the desired terminal inclination and this is where you will require some kind of maneuver or control over the trajectory. Of course, beyond the 40 or 50 kilometer altitude, generally we assume that the motion is in vacuum. Even though strictly speaking, it is not because there will be some amount of atmospheric density which will continue even though it may not be in the continuum format. There may be dissociation of gases, particularly once you reach about 100 kilometer altitude where you have ionosphere and you cross it. There are charged particles with great speed which can have some impact. So, in many realistic cases up to about 75 kilometers, we still assume that some amount of drag may be quite small, still applicable for more realistic and accurate solutions. But for initial configuration design, we generally find that beyond 40 kilometers, assuming the atmosphere to be absent gives a fairly good estimate of the launch vehicle, performance and its trajectory. Then of course, we reach the terminal phase is the most crucial as the spacecraft is going to separate from the launch vehicle. So, it is a physical breakage, the connections are broken. So, you must be sure that the initial conditions required for the spacecraft are achieved by the terminal phase and that the separation itself is a clean operation. Of course, in this phase the thrust is generally very small and the only force which is going to have a mechanic's implication is the gravitational force. But then in this phase as you would see in both the pictures that we have introduced, the distance traveled over Earth can be very large and that now you may need to bring in a suitable geometric model for Earth's surface for modeling the kinematics and the associated dynamics. So, as you would have noted as the spacecraft is released at the end, it will obviously operate in vacuum. But for the launch vehicle, right from the lift off to the injection point in different segments, different four systems are applicable and hence you also need to appropriately look at the mathematical models which are applicable in each of those segments. And that brings us to this important idea of ascent mission modeling. Let us look at how we are going to capture this so that we can achieve the objective of the trajectory and also achieve the configuration of a rocket which will make it happen. So, this whole process of modeling of ascent mission involves three components. The first, we need to identify the applicable forces in that segment. Then we need to identify the desired motion parameters that are of interest from the trajectory perspective. And the physical laws which are going to govern the motion in that segment. While there can be many forces as we have seen in the preceding discussion, typically forces due to propellant burning, gravity and atmosphere are going to dominate the motion and the trajectory of the launch vehicle. Further, we will find that the motion variables of interest are position and velocity vectors of the center of mass which sometimes are split into the velocity, the altitude and the flight path angle with respect to a suitable frame of reference. So, let us now introduce the idea of basic governing equations of motion for the launch vehicle as it moves through the atmosphere from lift off to the terminal point. Of course, to do that, let us first go back to our Newton's law and formulate the equation of motion of center of mass in an inertial frame which all of us know can be written as follows. This is the statement of Newton's second law that rate of change of momentum is of course extremely simple representation and it is of course written in the vectorial form with arrow at the top of the variables where the position vector x is nothing but the integral of the velocity vector and is the corresponding kinematic equation. So, the first is the dynamic equation, the second one is the kinematic equation which means in order for us to generate the information for position vector and the velocity vector we must have the force vector and we must solve this first order vector differential equation. So, the first step would be to define force system so that we can solve the first order vector differential equations to generate the velocity vector and then use the velocity vector to generate the position vector. So, this is the basic requirement from the mechanics perspective as far as this particular exercise is concerned. Now, how are we going to express the velocity vector? So, let us look at the elements of the velocity vector and we realize that this is not a very simple expression. Of course, we have not introduced a coordinate system but assuming that there is a coordinate system which is referenced to another inertial coordinate system then V naught represents the velocity of the origin of the chosen coordinate system in relation to the inertial coordinate system. VB is the velocity of the vehicle in the chosen coordinate system and omega represents the rotational velocity of the chosen coordinate system and RB represents the position of the vehicle in the chosen rotating frame of reference. So, you realize that in a general context the expression for V would contain all these three terms. Of course, with certain simplifications and assumptions we might be in a position to also simplify the expression for velocity but that we will see when we start looking at some other details. Of course, it is just mentioned at this stage that for most launch missions the launch point is commonly taken as the origin of a suitable reference frame so that it becomes easier from the interpretation of the trajectory point of view. Next, let us look at the choice of coordinate axis. In general, we find that the Cartesian coordinate system that is x, y and z are convenient and also used to represent forces and motion variables. However, in cases where motion is confined to a plane which will happen in most of the ascent missions we can either use x, y or polar coordinates are theta or even curvy linear coordinates s and n to represent the motion of the rocket center of mass starting from lift off to its terminal point and then the third point about the mathematical model are the forces. So, ascent mission starts from earth surface and ends at a prefixed point where payload is released. Of course, we have already seen from our earlier two conceptual pictures that the motion profile is generally highly curvy linear. Of course, we have also noted the fact that the launch vehicle is going to experience large accelerations because of the propulsive forces and will also be acted upon by the gravity and the aerodynamic forces. Thus, we need accurate mathematical models for gravity, propulsive and aerodynamic forces to set up the governing equations. So, to summarize ascent mission while composed of time sequence events can be defined in terms of trajectory segments based on nature of forces and motion. Newton's law is employed to formulate equations in the coordinate system that is located at the launch site. And finally, we also note that forces that influence the ascent trajectory are propulsive, gravitational and aerodynamic in nature. With that, we conclude our discussion on the basic features and philosophy of an ascent mission from a mechanics perspective. We will now look at each of those elements one by one and the first element that we will look at will be how to synthesize the forces so that we can solve the first order differential equation to generate the velocity vector. So, with that, we will close this segment and we will again meet in the next segment. Till then, bye and thank you.