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For example, at a place where escape speed is 11.2 km/s, the addition of 0.4 km/s yields a hyperbolic excess speed of 3.02 km/s. = This is an example of the Oberth effect. The converse is also true - a body does not need to be slowed by much compared to its hyperbolic excess speed (e.g. by atmospheric drag near periapsis) for velocity to fall ...
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a) and (b, 1/b) on the rectangular hyperbola xy = 1, or the corresponding region when this hyperbola is re-scaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola.
The linear eccentricity of an ellipse or hyperbola, denoted c (or sometimes f or e), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a : that is, e = c a {\displaystyle e={\frac {c}{a}}} (lacking a center, the linear eccentricity for ...
The following are also concurrent: (1) the circle that is centered at the hyperbola's center and that passes through the hyperbola's vertices; (2) either directrix; and (3) either of the asymptotes. [22] Since both the transverse axis and the conjugate axis are axes of symmetry, the symmetry group of a hyperbola is the Klein four-group.
In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field.A central force is a force (possibly negative) that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center.
x a is the distance from the foot of the perpendicular on the x-axis to the origin (regarded as positive on one side and negative on the other); y a is the distance from the foot of the perpendicular on the y-axis to the origin. Circles about the origin in hyperbolic axial coordinates.
For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, a given mass M of gas with changing volume will have variable density δ = M / V , and the ideal gas law may be written P = k T δ so that an isobaric process traces a hyperbola in the ...
If the circles k and q are orthogonal, then a straight line passing through the center O of k and intersecting q, does so at inverse points with respect to k. Given a triangle OAB in which O is the center of a circle k, and points A' and B' inverses of A and B with respect to k, then
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