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Why Are Rockets Needed?
Published in Travis S. Taylor, Introduction to Rocket Science and Engineering, 2017
We have yet to discuss in detail the hyperbolic conic section, which represents orbits with eccentricities greater than 1. The vis viva equation is a little different for such orbits because they have excess energy at infinity and, therefore, have a slightly different solution. In our discussion of the elliptical orbits in Section 2.3.5, we showed from the energy in Equation 2.62 that the energy of the system is a constant and is balanced between kinetic and potential energy. Also, when the apoapsis is at infinity, the ellipse is a parabola, and both the kinetic and potential energy at that point are zero. For a hyperbolic orbit, this is not the case. There is excess kinetic energy at infinity. This results in the vis viva equation for a hyperbolic trajectory to be
Mathematical formulas from the sciences
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Vis-viva equation (elliptic orbits) v2=G(m1+m2)2r1a $ v^{2} = G(m_{1} + m_{2} )\left( {\begin{array}{*{20}c} \frac{2}{r} & \frac{1}{a} \\ \end{array} } \right) $
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
If the satellite is in an elliptical orbit, with the semimajor axis of the ellipse given by a, the velocity at any point is given by the “vis-viva” equation: v=GM2r-1a
Using visualisations to develop skills in astrodynamics
Published in European Journal of Engineering Education, 2020
Lucinda Berthoud, Jonathan Walsh
The University of Bristol has delivered a Space Systems course module as part of the 4-year Aerospace Engineering ‘Integrated Masters’ degree (Bachelor and Master’s rolled into one course) for many years. It is a compulsory course unit in the second year of Aerospace Engineering and is optional for students from the Engineering Design course. The cohort studied was 148 students. It is worth 10 credits out of 120 credits for the year and originally comprised of 24 hours of lectures with 3 examples sheets. Of the 24 hours of lectures, 7 hours are used to cover a theoretical introduction to orbits, which is of particular interest here. This includes Kepler’s and Newton’s laws (and proving Kepler’s laws from Newton); conic sections; 3D reference systems; orbital elements; ground tracks and different types of orbits; 2-body motion; the Kepler equation and the vis-viva equation; out-of-plane manoeuvres; Hohmann transfers; basic rendezvous principles. The other lectures cover various aspects of spacecraft design such as power, propulsion, attitude and orbit control, etc.