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A radial hyperbolic trajectory is a non-periodic trajectory on a straight line where the relative speed of the two objects always exceeds the escape velocity. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1.
Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola , as can be seen when graphed on a Minkowski diagram whose coordinates represent a suitable inertial (non-accelerated) frame.
Using the inverse hyperbolic function artanh, the rapidity w corresponding to velocity v is w = artanh(v/c) where c is the speed of light. For low speeds, by the small-angle approximation, w is approximately v / c. Since in relativity any velocity v is constrained to the interval −c < v < c the ratio v / c satisfies −1 < v / c < 1.
Hyperbolic motions are often taken from inversive geometry: these are mappings composed of reflections in a line or a circle (or in a hyperplane or a hypersphere for hyperbolic spaces of more than two dimensions). To distinguish the hyperbolic motions, a particular line or circle is taken as the absolute.
The hyperbolic excess velocity is related to the specific orbital energy or characteristic energy by ...
The hyperbolic functions arise from the difference between the squares of the time and spatial coordinates in the spacetime interval, rather than a sum. The geometric significance of the hyperbolic functions can be visualized by taking x = 0 or ct = 0 in the transformations.
The relationship between arrival time and offset is hyperbolic and it is the principal criterion that a geophysicist uses to decide whether an event is a reflection or not. [2] It is distinguished from dip moveout (DMO), the systematic change in arrival time due to a dipping layer.
If the body has a velocity greater than escape velocity then its path will form a hyperbolic trajectory and it will have an excess hyperbolic velocity, equivalent to the extra energy the body has. A relatively small extra delta- v above that needed to accelerate to the escape speed can result in a relatively large speed at infinity.