Editing Trajectory (section)

== Physics of trajectories ==
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A familiar example of a trajectory is the path of a projectile, such as a thrown ball or rock. In a significantly simplified model, the object moves only under the influence of a uniform gravitational [[Force field (physics)|force field]]. This can be a good approximation for a rock that is thrown for short distances, for example at the surface of the [[Moon]]. In this simple approximation, the trajectory takes the shape of a [[parabola]]. Generally when determining trajectories, it may be necessary to account for nonuniform gravitational forces and air resistance ([[drag (physics)|drag]] and [[aerodynamics]]). This is the focus of the discipline of [[ballistics]].

One of the remarkable achievements of [[Newtonian mechanics]] was the derivation of [[Kepler's laws of planetary motion]]. In the gravitational field of a point mass or a spherically-symmetrical extended mass (such as the [[Sun]]), the trajectory of a moving object is a [[conic section]], usually an [[ellipse]] or a [[hyperbola]].{{efn|It is theoretically possible for an orbit to be a radial straight line, a circle, or a parabola. These are limiting cases which have zero probability of occurring in reality.}} This agrees with the observed orbits of [[planets]], [[comets]], and artificial spacecraft to a reasonably good approximation, although if a comet passes close to the Sun, then it is also influenced by other [[force]]s such as the [[solar wind]] and [[radiation pressure]], which modify the orbit and cause the comet to eject material into space.

Newton's theory later developed into the branch of [[theoretical physics]] known as [[classical mechanics]]. It employs the mathematics of [[differential calculus]] (which was also initiated by Newton in his youth). Over the centuries, countless scientists have contributed to the development of these two disciplines. Classical mechanics became a most prominent demonstration of the power of rational thought, i.e. [[reason]], in science as well as technology. It helps to understand and predict an enormous range of [[phenomena]]; trajectories are but one example.

Consider a particle of [[mass]] <math>m</math>, moving in a [[Gravitational potential|potential field]] <math>V</math>. In physical terms, mass represents [[inertia]], and the field <math>V</math> represents external forces of a particular kind known as "conservative". Given <math>V</math> at every relevant position, there is a way to infer the associated force that would act at that position, say from gravity. Not all forces can be expressed in this way, however.

The motion of the particle is described by the second-order [[differential equation]]

:<math> m \frac{\mathrm{d}^2 \vec{x}(t)}{\mathrm{d}t^2} = -\nabla V(\vec{x}(t)) \text{ with } \vec{x}=(x,y,z).</math>

On the right-hand side, the force is given in terms of <math>\nabla V</math>, the [[gradient]] of the potential, taken at positions along the trajectory. This is the mathematical form of Newton's [[Newton's second law|second law of motion]]: force equals mass times acceleration, for such situations.