Complete Elementary Solution to Some Optimum Trajectory Problems

A complete elementary and direct method is given for solving a class of optimum trajectory and design problems which arise in rocketry. The solution of problems which involve an optimum rela­ tion among final fuel consumption, position and time is reduced to elementary algebra. It is as­ sumed that aerodynamic forces are negligible and gravity is constant or can be expressed as a func­ tion of time. The problem of intercepting a ballistic missile with minimum fuel consumption is studied in some detail as a typical problem. For this problem a hyperbola can be plotted on a pre­ pared graph and the answers read. Solutions for the lower bound of fuel consumption, corre­ sponding to infinite thrust can be read also. It is assumed that enough of the initial conditions are given so that the problem is properly posed. A more general class including the Mayer reciprocal problems can also be handled, provided the velocity at one end is not involved in the end conditions. The method can be applied to initial design problems wherein it is desired to make an estimate of thrust and fuel requirements for a rocket which is to effect some specified mission, such as to gain a given altitude or intercept a specified type of target. A COMPLETE and elementary method is given in this paper for solving a class of problems which arises in the study of optimum performance of a rocket and in associated design problems. The method reduces the solution of prob­ lems, such as that of intercepting a target following a known path above the atmosphere, in a uniform gravitational field, to elementary algebra. For many problems a prepared graph can be used to reduce the work of solving to the plot­ ting of a simple curve and reading off values associated with the solution. The method is based on the following simplifying assump­ tions: The gravitational field is a known function of time, aerodynamic forces are negligible, the rocket is treated as a point mass, and the path of the target point is a known function of time. The method is carried through in some detail for the case of a rocket whose maximum thrust is constant, and for a ballistic target. The general type of problems which it treats are: Initial conditions are assumed given; it is desired to intercept a target following a known path in minimum time, or with minimum fuel consumption, or to maximize the range attained by a rocket with a given amount of fuel. Most of the problems which are Ma}^er reciprocal [(1) p. 574] 2 can be solved more or less directly, and some problems wherein the initial values are not all given. It is well known [see (4) or (8) for elementary proofs ] that under these assumptions, thrust must be fixed in direction and as large as possible during an initial period for maximum performance. With these assumptions, a single grid for dis­ placement can be drawn up in dimensionless form for all rock­ ets whose thrust is similar (defined in the text) during its period of application. On this graph the displacement of the target point may be plotted in the proper coordinate set, and from the relation between the two curves it can be deter­ mined at what times, if any, interception is possible, in­ cluding the earliest and latest times, and the time which cor­ responds to minimum fuel consumption. For a ballistic