An elliptic property of parabolic trajectories (original) (raw)
Parabolic Motion, 2024
This paper is based on object path on thrown and that leads to the bending of space and how it occur.
A Hidden Circle in the Paths of a Family of Projectiles
viXra, 2018
It is already known that an ellipse passes through the apexes of a family of the parabolic paths of projectiles shot from a point with constant speed but different angles of projection. In this article we describe a method to show that a circle passes through the focii of such a family of projectile paths whose center is the point of projection. In this method, we don’t use vectors, or calculus or Newton’s laws of motion.
ON PARABOLA SOME NEW GEOMETRICAL PROPERTIES
Parabola is one of the conic sections obtained by the intersections of circular cones by planes. A parabola is defined such that the set of all points in the plane are equidistant from a given line (Directrix) and a given fixed point (Focus). Mr. Menaechmus (c.375-325 BC), a pupil of Eudoxus, tutor to Alexander the Great, and a friend of Plato, is credited with the discovery of the conics. The parabola has many important applications like parabolic antenna, parabolic microphone, automobile headlight reflectors, the design of ballistic missiles and etc. The reflective and refractive properties have many advantages and frequently used in physics. Similarly, its geometrical properties are widely used in engineering and many other areas. Some new properties of the parabola have been developed now and defined in this article with necessary analytic geometric equations and illustrated with an example.
As an aid to teachers and students who are learning to apply Geometric Algebra to high-school-level physics, we provide this second installment in our guide guide to Hestenes's treatment of constant-acceleration motion. Specifically, we present a more-detailed version of Hestenes's solution to the problem of finding the time and distance at which a projectile will reach a specific point along a given line of sight. We begin by reviewing the GA ideas that we will use, and finish by verifying the solution via a GeoGebra worksheet.
Exploring Projectile Motion: A Comprehensive Analysis of the Physics of Projectile Trajectory
Salma A. Ali, 2024
This article presents a detailed examination of projectile motion, exploring its various types and their fundamental role within the broader framework of classical mechanics. Projectile motion, characterized by the curved trajectories of objects under the influence of gravity, is a fundamental concept in classical mechanics and objects in free flight. This study delves into both human-propelled and mechanically-launched projectiles—such as thrown objects, arrows, bullets, and artillery shells—exploring how kinematic equations, air resistance, and gravitational forces govern their behavior. By integrating the analysis of motion with key theoretical physics principles, including energy conservation, Newton’s laws, and the influence of external forces, this article highlights the interdependence of projectile motion with notable physics problems. The discussion extends to the applications of these principles in diverse fields such as engineering, military technology, and sports, offering a holistic view of how the theoretical framework of physics guides real-world problem-solving. This comprehensive examination illustrates the broad relevance of projectile motion in both academic and practical contexts, positioning it as a cornerstone for understanding complex systems in physics.
Wind-influenced projectile motion
We solved the wind-influenced projectile motion problem with the same initial and final heights and obtained exact analytical expressions for the shape of the trajectory, range, maximum height, time of flight, time of ascent, and time of descent with the help of the Lambert W function. It turns out that the range and maximum horizontal displacement are not always equal. When launched at a critical angle, the projectile will return to its starting position. It turns out that a launch angle of 90° maximizes the time of flight, time of ascent, time of descent, and maximum height and that the launch angle corresponding to maximum range can be obtained by solving a transcendental equation. Finally, we expressed in a parametric equation the locus of points corresponding to maximum heights for projectiles launched from the ground with the same initial speed in all directions. We used the results to estimate how much a moderate wind can modify a golf ball's range and suggested other possible applications.
The Fifth Section, the Semi Parabolic Curves, when the Focus equals the Vertex
Mağallaẗ al-Kitāb li-l-ʿulūm al-ṣirfaẗ, 2024
This article introduces a unique case study involving open curves of parabolic form situated within two-dimensional spaces. It presents a new form of a two-dimensional curve achieved by repositioning the focal point to coincide with the vertex position, resulting in what is termed a Semi-Parabolic Curve (SPC) where the focal point acts as the vertex referred to as the SPC head point. In essence, the SPC represents the path traced by a point on a plane, where its distance from a fixed point (the focus), is always greater than or equal to its distance from a fixed straight line (the directrix). Furthermore, the article provides the coordinate equations that govern the points along these curves. With the potential to pave the way for exploring additional geometric aspects relevant to this class of curves, and to enabling comparative analyses across diverse mathematical and geometric domains, particularly in three-dimensional contexts in the future.
TB Projectile motion in the 'Language' of orbital motion
We consider the orbit of projectiles launched with arbitrary speeds from the Earth's surface. This is a generalization of Newton's discussion about the transition from parabolic to circular orbits, when the launch speed approaches the value ν = √ gR E . We find the range for arbitrary launch speeds and angles, and calculate the eccentricity of the elliptical orbits.