Trajectory
In ordinary language, a trajectory is the path followed by a body moving through space, for instance, the path taken by a falling body or the orbit of a planet. One may either consider just the geometry of the path (i.e. the set of positions), or also the position as function of time. E.g., of Kepler's laws of planetary motion the first is concerned with the geometry only and the second with the position as function of time.
In the case of a projectile launched under the influence of a constant force field (modeling the effect of gravity over very short distances), the trajectory can be described by a (possibly degenerate) parabola. More generally, the movement of projectiles requires taking into account gravitational forces which are not constant and possibly other forces such as drag and wind. This is the focus of the discipline of ballistics.
In a wider sense, trajectory refers to the ordered set of intermediate states assumed by a dynamical system as a a result of time evolution. It is also used metaphorically, for instance, to describe an individual's career.
Physics of trajectories
Consider a particle of mass m moving in a potential field; the motion of the particle is governed by the second-order differential equation
- <math> \mathbf{m} \frac{d^2 \vec{x}(t)}{dt^2} = -\operatorname{grad} V(\vec{x}(t)) <math>
(see also gradient)
One of the remarkable achievements of Newtonian mechanics was the derivation of the fact that in the case of the gravitational field of a point mass, the trajectories must be conic sections on some plane of motion.
Constant gravity, no drag or wind
In the simplified case of constant gravity, and no drag or wind, on a flat terrain, the trajectory is a parabola. If the horizontal speed is <math>v_h<math> and the initial vertical speed is <math>v_v<math> then the range is <math>2v_h v_v/g<math> and the altitude <math>{v_v^2}\over{2g}<math>. The maximum range for a given total initial speed <math>v<math> is when <math>v_h=v_v<math>, i.e. the angle is 45 degrees. Then the range is <math>v^2/g<math>, and the altitude a quarter of that.
See also
de:Trajektorie et:Trajektoor