MATHEMATICAL THEORY OF ROCKET FLIGHT BY J. BARKLEY ROSSER, PH. D. Professor of Mathematics at Cornell University Formerly, Chief, Theoretical Ballistics Section Alleyany Ballistics Laboratory ROBERT R. NEWTON, PH. D. Member of Technical Staff, Bell Telephone Laboratories, Inc., Murray Hill, N. J. Formerly, Research Associate Allegany Ballistics Laboratory GEORGE L. GROSS, PH. D. Research Engineer in Applied Mathematics, Grumman Aircraft Engineering Corporation Beth page, N. Y. Formerly, Research Associate A lleyany Ballistics Laboratory Office of Scientific Research and Development National Defense Research Committee NKW YORK AND LONDON MCGRAW-HILL BOOK COMPANY, INC. 1947 MATHEMATICAL THEORY OF ROCKET FLIGHT PRINTED IN THE UNITED STATES OF AMERICA PREFACE This is the official final report to the Office of Scientific Research and Development concerning the work done on the exterior ballistics of fin-stabilized rocket projectiles under the supervision of Section H of Division 3 of the National Defense Research Committee at the Allegany Ballistics Laboratory during 1944 and 1945, when the laboratory was operated by The George Washington University under contract OEMsr-273 with the Office of Scientific Research and Devel opment. As such, its official title is Final Report No. B2.2 of the Allegany Ballistics Laboratory, OSRD 5878. After the removal of secrecy restrictions on this report, a consider able amount of expository material was added. It is our hope that thereby the report has been made readable for anyone interested in the flight of rockets. Two slightly different types of readers are antici pated. One is the trained scientist who has had no previous experience with rockets. Theother is the person with little scientific training who is interested in what makes a rocket go. The first type of reader should be able to comprehend the report in its entirety. For the benefit of the second type of reader, who will wish to skip the more mathematical portions, wo have attempted to supply simple explana tions at the beginnings of most sections telling what is to be accom plished in those sections. It is our hope that a reader can, if so minded, skip most of the mathematics and still be able to form a general idea of rocket flight. Although this is a report of the work done at Allegany Ballistics Laboratory, it must not be supposed that all the material in the report originated there. We have been most fortunate in receiving the whole hearted cooperation and assistance of scientists in other laboratories. Many of them, notably the English scientists, were well advanced in the theory before we even began. Without the fine start given us by these other workers, this report could certainly not have been written. However, we were fortunate enough to discover two means of avoiding certain difficulties of the theory. The first is that of using some dynamical laws especially suited to rockets in deriving the equations of motion, and the second is that of using some mathematical functions especially suited to rockets in solving the equations of motion. The explanation and illustration of these simplifying devices take up a considerable portion of the report, although for completeness we have included material not involving them. vi PREFACE In attempting to acknowledge the contributions of other workers, we are in a difficult position. Approximately a hundred reports by otherworkers were useful in one way or another in the preparatf on of this report. However, most of them are still bound by military secrecy, so that only the few cited in our meager list of bibliographical references can be mentioned here. Many figures are copied from these unmentioiied reports. Sizable portions of our report, such as Chap. II and Appendix 1, lean very heavily on certain of these unmentioned reports, but no specific credit is given...