Trajectory

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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.

In discrete mathematics, the term trajectory denotes the sequence [(f^k(x))_}] of values which one gets by iterated application of a mapping [f] to an element [x] of its source.

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(t)} = -\nabla V(\vec(t)) ] with [\vec = (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.

Examples

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 acceleration of gravity. This gives the equations of motion

[ \frac = \frac = 0]

[ \frac = - 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 [/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

[p(t)\;] be the position of the projectile, expressed as a vector

[t\;] be the time into the flight of the projectile,

[v_h \;] be initial the horizontal velocity (which is constant)

[v_v \;] be the initial vertical velocity upwards.

The path of the projectile is known to be a parabola so

[p(t) = ( A t, 0 , a t^2 + b t + c )\,]

where [A,\,a,\,b,\,c] are parameters to be found. The first and second derivatives of [p] are:

[p'(t) = ( A , 0 , 2 a t + b ),\quad p''(t) = ( 0 , 0 , 2 a ).]

At [t=0]

[p(0)=0,\ p'(0)=(v_h,0,v_v),\ p''(0)=(0,0,-g)]

so [A = v_h,\ a = -g/2,\ b = v_v,\ c = 0]. Giving eqn of parabola as

[p(t) = (v_h t,0,v_v t - g t^2/2)\,\qquad](Equation I: trajectory of parabola).

Range and height

The range [R] of the projectile is found when the [z]-component of [p] is zero, that is when

[0 = v_v t - g t^2/2 = t \left( v_v - g t/2\right)\,]

which has solutions at [t=0] and [t = 2 v_v /g] (the hang-time of the projectile). The range is then [R = 2 v_h v_v/g.\,]

From the symmetry of the parabola the maximum height occurs at the halfway point [t=v_v/g] at position

[p(v_v/g)=(v_h v_v/g,0,v_v^2/(2g))\,]

This can also be derived by finding when the [z]-component of [p'] is zero.

Angle of elevation

In terms of angle of elevation [\theta] and initial speed [v]:

[v_h=v \cos \theta,\quad v_v=v \sin \theta \;]

giving the range as

[R= 2 v^2 \cos(\theta) \sin(\theta) / g = v^2 \sin(2\theta) / g\,.]

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

[ = \frac 1 2 \sin^ \left( \over } \right) ] (Equation II: angle of projectile launch)

Note that the sine function is such that there are two solutions for [\theta] for a given range [d_h]. Physically, this corresponds to a direct shot versus a mortar shot up and over obstacles to the target.

The angle [\theta] giving the maximum range can be found by considering the derivative or [R] with respect to [\theta] and setting it to zero.

[= cos(2\theta)=0]

which has a non trivial solutions at [\theta=\pi/2=45^\circ]. The maximum range is then [R_ = v^2/g\,]. At this angle [sin(\pi/2)=1/\sqrt] so the maximum height obtained is [].

To find the angle giving the maximum height for a given speed calculate the derivative of the maximum height [H=v sin(\theta) /(2g)] with respect to [\theta], that is [=v cos(\theta) /(2g)] which is zero when [\theta=\pi=90^\circ]. So the maximum height [H_=] is obtain when the projectile is fired straight up.

Uphill/downhill in uniform gravity in a vacuum