trajectory

Other Definitions
trajectory (dict)

A trajectory is an imagined trace of positions followed by an object moving through space. Some common examples of trajectories: (i) the path taken by a falling body, and (ii) the orbit of a spacecraft. A particular trajectory may be described mathematically either by the geometry of the entire trajectory (i.e. the set of all positions taken by the object), or as the position of the object as function of time. A familiar example is a projectile launched under the influence of only a uniform gravitational force field. A rock thrown on the practically airless surface of the Moon is a good approximation. In this case, the trajectory takes the shape of a parabola, provided the rock is not thrown too far. More generally, the precise trajectory of a projectile requires taking into account nonuniform gravitational forces and other forces such as drag and wind. This is the focus of the discipline of ballistics. A projectile, such as a baseball, when thrown through the air, is influenced by both gravity and aerodynamics. More generally, trajectory refers to the ordered set of intermediate states assumed by a dynamical system as a result of time evolution. The word trajectory is also often used metaphorically, for instance, to describe an individual's career.

Physics of trajectories

One of the remarkable achievements of Newtonian mechanics was the derivation of the laws of Kepler, in the case of the gravitational field of a single point mass (representing the Sun). The trajectory is a conic section, like an ellipse or a parabola. This agrees with the observed orbits of planets and comets, 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, in fact, also initiated by Newton, in his youth). Over the centuries, countless scientists 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

m

, moving in a potential field

V

. Physically speaking, mass represents inertia, and the field

V

represents external forces, of a particular kind known as "conservative". That is, given

V

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

m \frac{d^2 \vec{x}(t)}{dt^2} = -\nabla V(\vec{x}(t))

with

\vec{x} = (x, y, z)

On the right-hand side, the force is given in terms of

\nabla V

, the gradient of the potential, taken at positions along the trajectory. This is the mathematical form of Newton's second law of motion: mass times acceleration equals force, for such situations.

Example: Uniform gravity, no drag or wind

The case of uniform gravity, disregarding drag and wind, yields a trajectory which is a parabola. To model this, one chooses

V = m g z

, where

g

(gee) is the so-called acceleration of gravity. This gives the equations of motion

\frac{d^2 x}{dt^2} = \frac{d^2 y}{dt^2} = 0

\frac{d^2 z}{dt^2} = - g

Simplifications are made for the sake of studying the basics. The actual situation, at least on the surface of Earth, is considerably more complicated than this example would suggest, when it comes to computing actual trajectories. By deliberately introducing such simplifications, into the study of the given situation, one does, in fact, approach the problem in a way that has proved exceedingly useful in physics. The present example is one of those originally investigated by Galileo Galilei. To neglect the action of the atmosphere, in shaping a trajectory, would (at best) 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. And in a near vacuum, as it turns out for instance on the Moon, his simplified parabolic trajectory proves essentially correct. Relative to a flat terrain, let the initial horizontal speed be

v_h

, and the initial vertical speed be

v_v

. It will be shown that, the range is

2v_h v_v/g

, and the maximum altitude is

{v_v^2}/2g

. The maximum range, for a given total initial speed

v

, is obtained when

v_h=v_v

, i.e. the initial angle is 45 degrees. This range is

v^2/g

, and the maximum altitude at the maximum range is a quarter of that.

Derivation

The equations of motion may be used to calculate the characteristics of the trajectory. Let:

t \;

be the time into the flight of the projectile

d_h(t) \;

be the horizonal displacement at time t

d_v(t) = z \;

be the vertical displacement at time t

v_h \;

be the horizonal velocity (which is constant)

v_v \;

be the initial vertical velocity upwards

v \;

be the initial speed

v_v(t) \;

be the vertical velocity at time t

Along the horizontal dimension,

v_h

is a constant and thus by the equations of motion,

d_h=v_h t \;

(Equation 1)

The vertical distance, or altitude follows the equations of motion for constant negative acceleration

g

d_v(t)=v_vt- ) 2v_v)

(derivative of square root)

={1 \over g} (2 \sqrt{v^2-{v_v}^2} - \frac{2{v_v}^2}{\sqrt{v^2-{v_v}^2}})

(simplify second term)

Set to zero and solve for

v

0={1 \over g} (2 \sqrt{v^2-{v_v}^2} - \frac{2{v_v}^2}{\sqrt{v^2-{v_v}^2}})

\sqrt{v^2-{v_v}^2}=\frac

v^2-{v_v}^2=v_v^2

v^2=2v_v^2

(Equation 9)

Thus maximum range occurs when

v^2

2v_v^2

and this can be substituted back into equation 8:

v_h=\sqrt{v^2-v_v^2}=\sqrt{v^2-v_v^2}=\sqrt{2v_v^2-v_v^2}=v_v

Thus the maximum range occurs when

v_h=v_v

. The actual maximum range may now be calculated by substituting

v_h=v_v

and equation 9 into equation 5:

d_h =  { {2 v_h v_v} \over g } = { {2 v_v v_v} \over g } = { {2 v_v^2} \over g } = {v^2 \over g}

Maximum range in polar coordinates

The same conclusion may be drawn by starting with equation 5a.

\frac{d}{d\theta} d_h = \frac{d}{d\theta} {2a}=\frac{-(v_v/v_h) \pm \sqrt{(v_v/v_h)^2 - 0} }{2(-g/2v_h^2)} = \frac{-v_v/v_h \pm v_v/v_h}{-g/v_h^2}=0, \frac{2{v_v}{v_h}}{g}

This is the same result as equation 5 above. In polar coordinates and using the trigonometric identity

2 \sin \theta \cos \theta = \sin 2 \theta

, the intersections are:

d_h=\frac{-b \pm \sqrt {b^2-4ac\ }}{2a}=\frac{-(\sin \theta / \cos \theta) \pm \sqrt{(\sin \theta / \cos \theta)^2 - 0 }}{2(-g/2v^2 \cos^2 \theta)} =0, \frac{2 \sin \theta / \cos \theta}{g/v^2 \cos^2 \theta} = 0, \frac{2 v^2 \sin \theta \cos \theta}{g} = 0, \frac{v^2 \sin 2 \theta}{g} This is the same result as in equation 5a above. Similarly, the vertex of the parabola is the maximum altitude for a given range.

Trajectory

Physics of trajectories

Example: Uniform gravity, no drag or wind

Derivation

Maximum range in polar coordinates

See also