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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.

1 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

(see also gradientMultivariate calculus In vector calculus, gradient is a vector-valued operator that acts on a scalar field. The gradient of a scalar field is a vector field called gradient field which shows its rate and direction of change. For example, consider a room.)

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 sectionIn mathematics, a conic section (or just conic is a curved locus of points, formed by intersecting a cone with a plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properts on some plane of motion.

1.1 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 and the initial vertical speed is then the rangeThe range of a vehicle is the maximum distance it can cover without needing to be re fueled or re charged. The range of a gun or missile is the maximum distance it can be fired over and still hit its target. The actual distance that the missile can travel is and the altitude . The maximum range for a given total initial speed is when , i.e. the angle is 45 degrees. Then the range is , and the altitude a quarter of that.

2 See also

• Orbit
• Equation of motionIn advanced physics, equations of motion usually refer to the Euler-Lagrange equations, differential equations derived from the Lagrangian. The following article is about elementary physics only. Linear equations of motion In kinematics, four equations of