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. Trajectory in quantum mechanics is not defined due to Heisenberg uncertainty principle that position and momentum can not be measured simultaneously.

In classical mechanics, the mass might be a projectile or a satellite.^{[2]} For example, it can be an orbit — the path of a planet, asteroid, or comet as it travels around a central mass.

# Physics of trajectories

A familiar example of a trajectory is the path of a projectile, such as a thrown ball or rock.

One of the remarkable achievements of Newtonian mechanics was the derivation of the laws of Kepler. 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.^{[1]} 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 forces 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.

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

# Examples

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.

The **range**, *R*, is the greatest distance the object travels along the x-axis in the I sector. The **initial velocity**, *vi*, is the speed at which said object is launched from the point of origin. The **initial angle**, *θi*, is the angle at which said object is released. The *g* is the respective gravitational pull on the object within a null-medium.

The **height**, *h*, is the greatest parabolic height said object reaches within its trajectory

giving the range as

This equation can be rearranged to find the angle for a required range

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.

# Catching balls

If a projectile, such as a baseball or cricket ball, travels in a parabolic path, with negligible air resistance, and if a player is positioned so as to catch it as it descends, he sees its angle of elevation increasing continuously throughout its flight.