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Path of a moving object

A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously. The mass might be a projectile or a satellite. For example, it can be an orbit — the path of a planet, asteroid, or comet as it travels around a central mass. In control theory, a trajectory is a time ordered set of states of a dynamical system (see e. g. Poincaré map). In discrete mathematics, a trajectory is a sequence of values calculated by the iterated application of a mapping to an element of its source.

The ideal case of motion of a projectile in a uniform gravitational field in the absence of other forces (such as air drag) was first investigated by Galileo Galilei. To neglect the action of the atmosphere in shaping a trajectory would have been considered a futile hypothesis by practical minded investigators all through the Middle Ages in Europe. Nevertheless, by anticipating the existence of the vacuum, later to be demonstrated on Earth by his collaborator Evangelista Torricelli, Galileo was able to initiate the future science of mechanics. In a near vacuum, as it turns out for instance on the Moon, his simplified parabolic trajectory proves essentially correct. In the analysis that follows, we derive the equation of motion of a projectile as measured from an inertial frame at rest with respect to the ground. Associated with the frame is a right hand coordinate system with its origin at the point of launch of the projectile. The axis is tangent to the ground, and the axis is perpendicular to it ( parallel to the gravitational field lines ). Let be the acceleration of gravity. Relative to the flat terrain, let the initial horizontal speed be and the initial vertical speed be It will also be shown that the range is , and the maximum altitude is The maximum range for a given initial speed is obtained when , i. e. the initial angle is 45. This range is , and the maximum altitude at the maximum range is .

If instead of a uniform downwards gravitational force we consider two bodies orbiting with the mutual gravitation between them, we obtain Kepler's laws of planetary motion. The derivation of these was one of the major works of Isaac Newton and provided much of the motivation for the development of differential calculus.