trajectory n : the path followed by an object moving through space [syn: flight]
- The path of a body as it travels through space.
- The ordered set of intermediate states assumed by a dynamical system as a result of time evolution.
- Metaphorically, a course of development, such as that of a war or career.
path of a body
ordered set of intermediate states
Trajectory is the path a moving object follows through space. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit - the path of a planet, an asteroid or a comet as it travels around a central mass. A trajectory can be described mathematically either by the geometry of the path, or as the position of the object over time.
In control theory a trajectory is a time-ordered set of states of a dynamical system (see e.g. Poincaré map). In discrete mathematics, a trajectory is a sequence (f^k(x))_ of values calculated by the 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 trajectoriesA familiar example of a trajectory is the path of a projectile such as a thrown ball or 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.
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 = -\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.
Uniform gravity, no drag or windThe 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 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.
The equations of motion may be used to calculate the characteristics of the trajectory.
- 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 the initial horizontal velocity (which is constant)
- v_v \; be the initial vertical velocity upwards.
- t\; be the time into the flight of the projectile,
- p(t) = ( A t, 0 , a t^2 + b t + c )\,
- p'(t) = ( A , 0 , 2 a t + b ),\quad p(t) = ( 0 , 0 , 2 a ).
- p(0)= (0, 0, 0)\ p'(0)=(v_h,0,v_v),\ p(0)=(0,0,-g)
- A = v_h,\ a = -g/2,\ b = v_v,\ c = 0.
- p(t) = (v_h t,0,v_v t - g t^2/2)\,\qquad (Equation I: trajectory of parabola).
Range and heightThe 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)\,
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))\,
Angle of elevationIn terms of angle of elevation \theta and initial speed v:
- v_h=v \cos \theta,\quad v_v=v \sin \theta \;
- R= 2 v^2 \cos(\theta) \sin(\theta) / g = v^2 \sin(2\theta) / g\,.
- = \frac 1 2 \sin^ \left( \right) (Equation II: angle of projectile launch)
- = \cos(2\theta)=0
To find the angle giving the maximum height for a given speed calculate the derivative of the maximum height H=v^2 sin(\theta) /(2g) with respect to \theta, that is =v^2 \cos(\theta) /(2g) which is zero when \theta=\pi=180^\circ. So the maximum height H_= is obtain when the projectile is fired straight up. The equation of the trajectory of a projectile fired in uniform gravity in a vacuum on Earth in Cartesian coordinates is
where v0 is the initial speed, h is the height the projectile is fired from, and g is the acceleration due to gravity).
Uphill/downhill in uniform gravity in a vacuumGiven a hill angle \alpha and launch angle \theta 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:
- \frac =(1-\cot \theta \tan \alpha)\sec \alpha (Equation 11)
In this equation, downhill occurs when \alpha is between 0 and -90 degrees. For this range of \alpha we know: \tan(-\alpha)=-\tan \alpha and \sec ( - \alpha ) = \sec \alpha. Thus for this range of \alpha, R_s/R=(1+\tan \theta \tan \alpha)\sec \alpha . 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" \theta_:
- 1=(1-\tan \theta \tan \alpha)\sec \alpha \quad \;
- \theta_=\arctan((1-\csc \alpha)\cot \alpha) \quad \;
Equation 11 may also be used to develop the "rifleman's rule" for small values of \alpha and \theta (i.e. close to horizontal firing, which is the case for many firearm situations). For small values, both \tan \alpha and \tan \theta 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:
- \frac =(1-0)\sec \alpha
- R=R_s \cos \alpha \ "Rifleman's rule"
Derivation based on equations of a parabolaThe 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=-\fracx^2 + \frac x
- x=\frac\left(\frac-m\right) (Solving above x)
- y=mx=m \frac \left(\frac-m\right)
- =\frac \sqrt
- =\frac \left(\frac-m\right) \sqrt
- =\frac \sqrt
Now \alpha is defined as the angle of the hill, so by definition of tangent, m=\tan \alpha. This can be substituted into the equation for R_s:
- R_s=\frac \left(\frac-\tan \alpha\right) \sqrt
- R_s=R(1-\tan\theta\tan\alpha)\sec\alpha \;
Orbiting objectsIf 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.
- Projectile Motion Flash Applet
- Trajectory calculator
- An interactive simulation on projectile motion
- Projectile Motion Simulator, java applet
- Projectile calculation in MS Excel – calculation of the projectile position after a given time, the maximum height reached and the range of the projectile. The projectile path is plotted on an Excel chart and all cell formulae are shown in mathematical notation.
- Parabolic Projectile Motion: Shooting a Harmless Tranquilizer Dart at a Falling Monkey by Roberto Castilla-Meléndez, Roxana Ramírez-Herrera, and José Luis Gómez-Muñoz, The Wolfram Demonstrations Project.
- Trajectory, ScienceWorld.
trajectory in Catalan: Trajectòria
trajectory in German: Trajektorie (Physik)
trajectory in Estonian: Trajektoor
trajectory in Spanish: Trayectoria
trajectory in French: Trajectoire
trajectory in Italian: Traiettoria
trajectory in Lithuanian: Trajektorija
trajectory in Japanese: 弾道
trajectory in Polish: Trajektoria
trajectory in Portuguese: Trajetória
trajectory in Russian: Траектория материальной точки
trajectory in Slovak: Trajektória
trajectory in Ukrainian: Траекторія
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