The role of work and mechanical energy in classical relativity has been a subject of renewed interest in this publication. Here we present a problem that illustrates the relationship between impulse and kinetic energy for a rocket-powered object that can also change its gravitational potential energy. The same introductory physics principles lead to a remarkable result when applied to the mechanics of spaceflight—the Oberth effect—whereby a small impulse can cause a large change in a rocket’s orbital energy without violating any conservation laws.
References
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We desire both the largest possible initial speed v and exhaust speed vex (also referred to as the fuel’s specific impulse in m/s or N.s/kg) in order to provide the largest boost to the rocket, according to Eqs. (1) and (2). Also see
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The gravitational potential energy of the rocket and fuel should properly be assigned to the systems of planet + rocket and planet + fuel, respectively. For brevity, we refer to the mechanical energy of the “rocket” or “fuel” alone. If we assume a near-infinite mass for the planet, we can neglect any changes in its motion as a result of its interactions with the rocket or fuel.
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Perhaps easier to visualize (by reversing the arrows in Fig. 3), the time-reverse version of this maneuver is an inelastic head-on collision between an orbiting fuel mass and an incoming rocket, whereby the combined object ends up with a mechanical energy less than that of the colliding objects by an amount ΔEchem.
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The Oberth effect is distinct from a gravity-assist “slingshot” maneuver, which describes an elastic momentum and energy exchange between an unpowered spacecraft and a planet when both are under the gravitational influence of a third body, typically the Sun. See
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Online at http://orbit.medphys.ucl.ac.uk.
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To obtain free educational licenses for Systems Tool Kit (STK), visit http://www.agi.com/education.
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Online at http://www.kerbalspaceprogram.com. See also
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See the online supplements at TPT Online, http://dx.doi.org/10.1119/1.5126818 under the Supplemental tab.
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2019
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