NationMaster - Encyclopedia: Trajectory

Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e.g. Poincaré map). A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... A pocket watch, a device used to measure time Two distinct views exist on the meaning of time. ... In mathematics, in the theory of dynamical systems, a PoincarÃ© map or PoincarÃ© section is the intersection of a trajectory of something which moves periodically (or quasi-periodically, or chaotically), in a space of at least three dimensions, with a transversal hypersurface of one fewer dimension. ...

Colloquially, a trajectory is the path in space followed by a body. Such a body can be a projectile, for example. Strictly speaking trajectory refers only to that portion of the path during which the body undergoes a transient movement between practically stationary or repetitive motions or until the body eventually stops moving. In a wider sense it also includes the meaning of orbit - the path of a planet, an asteroid or comet, for example. A trajectory can be described mathematically by the geometry of the path or as the position of the object over time. Alex polson A projectile is any object sent through space by the application of a force. ... The eight planets and three dwarf planets of the Solar System. ... 253 Mathilde, a C-type asteroid. ... Comet Hale-Bopp A comet is a small body in the solar system that orbits the Sun and (at least occasionally) exhibits a coma (or atmosphere) and/or a tail â€” both primarily from the effects of solar radiation upon the comets nucleus, which itself is a minor body composed...

A familiar example is the path of a thrown object such as a ball or a rock. In a greatly simplified model the object moves only under the influence of a uniform homogenous gravitational force field. This can be a good approximation for a rock that is thrown for short distances for example, at the surface of the moon. In this simple approximation the trajectory takes the shape of a parabola. Generally, when determining trajectories it may be necessary to account for nonuniform gravitational forces, air resistance (drag and aerodynamics). This is the focus of the discipline of ballistics. Homogeneous is an adjective that has several meanings. ... Originally a term coined by Michael Faraday to provide an intuitive paradigm, but theoretical construct (in the Kuhnian sense), for the behavior of electromagnetic fields, the term force field refers to the lines of force one object (the source object) exerts on another object or a collection of other objects. ... Adjective lunar Bulk silicate composition (estimated wt%) SiO2 44. ... It has been suggested that Analyzing the parabola be merged into this article or section. ... An object falling through a gas or liquid experiences a force in direction opposite to its motion. ... This article is about the branch of Physics. ... Ballistics (gr. ...

In discrete mathematics, the term trajectory denotes the sequence $(f^k(x))_{k in mathbb{N}}$ of values which one gets by iterated application of a mapping $f$ to an element $x$ of its source. Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ...

The word trajectory is also often used metaphorically, for instance, to describe an individual's career. In language, a metaphor (from the Greek: metapherin rhetorical trope) is defined as a direct comparison between two or more seemingly unrelated subjects. ...

1 Physics of trajectories
2 Examples
3 See also
4 External Links

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. It has been suggested that this article or section be merged with Classical mechanics. ... Johannes Keplers primary contributions to astronomy/astrophysics were the three laws of planetary motion. ... The Sun is the star at the centre of the Solar System. ... Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... The ellipse and some of its mathematical properties. ... It has been suggested that Analyzing the parabola be merged into this article or section. ... A planet (from the Greek πλανήτης, planetes or wanderers) is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. ... Comet Hale-Bopp, showing a white dust tail and blue gas tail (February 1997) A comet is a small astronomical object similar to an asteroid but composed largely of ice. ... In physics, force is an influence that may cause a body to accelerate. ... The plasma in the solar wind meeting the heliopause For the British comic, see Solar Wind (comic). ... Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic radiation. ...

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. Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand Nature. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... Reason is a term used in philosophy and other human sciences to refer to the faculty of the human mind that creates and operates with abstract concepts. ... A phenomenon (plural: phenomena) is an observable event, especially something special (literally something that can be seen from the Greek word phainomenon = observable). ...

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. Unsolved problems in physics: What causes anything to have mass? Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... In physics, a potential is a scalar quantity that can be used to analyze the effects of complicated vectorial forces and similar quantities by means of simple conservation laws. ... The principle of inertia is one of the fundamental laws of classical physics which are used to describe the motion of matter and how it is affected by applied forces. ...

The motion of the particle is described by the second-order differential equation An illustration of a 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. Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ...

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 It has been suggested that Analyzing the parabola be merged into this article or section. ... The nominal acceleration due to gravity at sea level on the Earths surface, also known as standard gravity, is defined as exactly 9. ... In advanced physics, equations of motion usually refer to the Euler-Lagrange equations, differential equations derived from the Lagrangian. ...

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

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

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. Galileo Galilei (15 February 1564 â€“ 8 January 1642) was an Italian physicist, astronomer, and philosopher who was closely associated with the scientific revolution. ... The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times, beginning with the Renaissance. ... Look up Vacuum in Wiktionary, the free dictionary. ... Evangelista Torricelli, portrait by an unknown artist. ... Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ...

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 $2 v h v v / g$ , 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. Look up range in Wiktionary, the free dictionary. ...

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

$0 = v_v t - g t^2/2 = t left( v_v - g t/2right),$

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 $θ$ and initial speed $v$ :

$v_h=v cos theta,quad v_v=v sin theta ;$

giving the range as

$R= 2 v^2 cos(theta) sin(theta) / g = v^2 sin(2theta) / g,.$

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

${ theta } = frac 1 2 sin^{-1} left( { {g R} over { v^2 } } right)$ (Equation II: angle of projectile launch)

Note that the sine function is such that there are two solutions for $θ$ for a given range $d h$ . Physically, this corresponds to a direct shot versus a mortar shot up and over obstacles to the target. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... US soldier loading a M224 60-mm mortar. ...

The angle $θ$ giving the maximum range can be found by considering the derivative or $R$ with respect to $θ$ and setting it to zero.

${dRover dtheta}={2v^2over g} cos(2theta)=0$

which has a non trivial solutions at $theta=pi/2=45^circ$ . The maximum range is then $R_{max} = v^2/g,$ . At this angle $sin(pi/2)=1/sqrt{2}$ so the maximum height obtained is ${v^2 over 4g}$ .

To find the angle giving the maximum height for a given speed calculate the derivative of the maximum height $H = v s i n (θ) / (2 g)$ with respect to $θ$ , that is ${dHover dtheta}=v cos(theta) /(2g)$ which is zero when $theta=pi=90^circ$ . So the maximum height $H_{max}={vover 2g}$ is obtain when the projectile is fired straight up.

Uphill/downhill in uniform gravity in a vacuum

The factual accuracy of this section is disputed.
Please see the relevant discussion on the talk page

Given a hill angle $α$ and launch angle $θ$ as before, it can be shown that the range along the hill $R s$ forms a ratio with the original range $R$ along the imaginary horizontal, such that: Image File history File links Circle-question-red. ...

$frac{R_s} {R}=(1-cot theta tan alpha)sec alpha$ (Equation 11)

In this equation, downhill occurs when $α$ is between 0 and -90 degrees. For this range of $α$ we know: $tan( - α) = - tanα$ and $sec( - α) = secα$ . Thus for this range of $α$ , $R s / R = (1 + tanθtanα)secα$ . Thus $R s / R$ is a positive value meaning the range downhill is always further than along level terrain. This makes perfect sense as it is expected that gravity will assist the projectile, giving it greater range.

While the same equation applies to projectiles fired uphill, the interpretation is more complex as sometimes the uphill range may be shorter or longer than the equivalent range along level terrain. Equation 11 may be set to $R s / R = 1$ (i.e. the slant range is equal to the level terrain range) and solving for the "critical angle" $θ c r$ :

$1=(1-tan theta tan alpha)sec alpha quad ;$

$theta_{cr}=arctan((1-csc alpha)cot alpha) quad ;$

Equation 11 may also be used to develop the "rifleman's rule" for small values of $α$ and $θ$ (i.e. close to horizontal firing, which is the case for many firearm situations). For small values, both $tanα$ and $tanθ$ have a small value and thus when multiplied together (as in equation 11), the result is almost zero. Thus equation 11 may be approximated as: Riflemans rule is a rule of thumb that allows a rifleman to accurately fire a rifle that has been calibrated for horizontal targets at uphill or downhill targets. ...

$frac{R_s} {R}=(1-0)sec alpha$

And solving for level terrain range, $R$

$R=R_s cos alpha$ "Rifleman's rule"

Thus if the shooter attempts to hit the level distance R, s/he will actually hit the slant target. "In other words, pretend that the inclined target is at a horizontal distance equal to the slant range distance multiplied by the cosine of the inclination angle, and aim as if the target were really at that horizontal position."[1]

Derivation based on equations of a parabola

The intersect of the projectile trajectory with a hill may most easily be derived using the trajectory in parabolic form in Cartesian coordinates (Equation 10) intersecting the hill of slope $m$ in standard linear form at coordinates $(x, y)$ :

$y=mx+b ;$ (Equation 12) where in this case,

y = d v

x = d h

and

b = 0

Substituting the value of $d v = m d h$ into Equation 10:

$m x=-frac{g}{2v^2{cos}^2 theta}x^2 + frac{sin theta}{cos theta} x$

$x=frac{2v^2cos^2theta}{g}left(frac{sin theta}{cos theta}-mright)$ (Solving above x)

This value of x may be substituted back into the linear equation 12 to get the corresponding y coordinate at the intercept:

$y=mx=m frac{2v^2cos^2theta}{g} left(frac{sin theta}{cos theta}-mright)$

Now the slant range $R s$ is the distance of the intercept from the origin, which is just the hypotenuse of x and y: For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

$R_s=sqrt{x^2+y^2}=sqrt{left(frac{2v^2cos^2theta}{g}left(frac{sin theta}{cos theta}-mright)right)^2+left(m frac{2v^2cos^2theta}{g} left(frac{sin theta}{cos theta}-mright)right)^2}$

$=frac{2v^2cos^2theta}{g} sqrt{left(frac{sin theta}{cos theta}-mright)^2+m^2 left(frac{sin theta}{cos theta}-mright)^2}$

$=frac{2v^2cos^2theta}{g} left(frac{sin theta}{cos theta}-mright) sqrt{1+m^2}$

Now $α$ is defined as the angle of the hill, so by definition of tangent, $m = tanα$ . This can be substituted into the equation for $R s$ : In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...

$R_s=frac{2v^2cos^2theta}{g} left(frac{sin theta}{cos theta}-tan alpharight) sqrt{1+tan^2 alpha}$

Now this can be refactored and the trigonometric identity for $sec alpha = sqrt {1 + tan^2 alpha}$ may be used: In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

$R_s=frac{2v^2costhetasintheta}{g}left(1-frac{sintheta}{costheta}tanalpharight)secalpha$

Now the flat range $R = v 2 sin2θ / g = 2 v 2 sinθcosθ / g$ by the previously used trigonometric identity and $sinθ / cosθ = t a n θ$ so: In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

$R_s=R(1-tanthetatanalpha)secalpha ;$

$frac{R_s}{R}=(1-tanthetatanalpha)secalpha$

Orbiting objects

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 Newton and provided much of the motivation for the development of differential calculus. This article does not cite its references or sources. ... The newton (symbol: N) is the SI unit of force. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ...

External Links

Categories: Accuracy disputes | Ballistics | Mechanics

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