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
of values calculated by the iterated application of a mapping
to an element of its source.
The word trajectory is also often used metaphorically, for instance, to describe an individual's career.
Physics of trajectories
A 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
). 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 , moving in a potential field . Physically speaking, mass represents inertia, and the field represents external forces, of a particular kind known as "conservative". That is, given 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
On the right-hand side, the force is given in terms of , 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 wind
The case of uniform gravity, disregarding drag and wind, yields a trajectory which is a parabola. To model this, one chooses , where is the acceleration of gravity. This gives the equations of motion
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 , and the initial vertical speed be . It will be shown that, the range is , and the maximum altitude is . The maximum range, for a given total initial speed , is obtained when , i.e. the initial angle is 45 degrees. This range is , 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.
- be the position of the projectile, expressed as a vector
- be the time into the flight of the projectile,
- be the initial horizontal velocity (which is constant)
- be the initial vertical velocity upwards.
The path of the projectile is known to be a parabola so
are parameters to be found. The first and second derivatives of
This yields the formula for a parabolic trajectory:
- (Equation I: trajectory of parabola).
Range and height
of the projectile is found when the
is zero, that is when
which has solutions at
(the hang-time of the projectile
The range is then
From the symmetry of the parabola the maximum height occurs at the halfway point at position
This can also be derived by finding when the
Angle of elevation
In terms of angle of elevation
and initial speed
giving the range as
This equation can be rearranged to find the angle for a required range
- (Equation II: angle of projectile launch)
Note that the sine
function is such that there are two solutions for
for a given range
. Physically, this corresponds to a direct shot versus a mortar
shot up and over obstacles to the target.
giving the maximum range can be found by considering the derivative or
with respect to
and setting it to zero.
which has a non trivial solutions at
The maximum range is then
. At this angle
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 with respect to , that is
which is zero when . So the maximum height 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 vacuum
Given a hill angle
and launch angle
as before, it can be shown that the range along the hill
forms a ratio with the original range
along the imaginary horizontal, such that:
- (Equation 11)
In this equation, downhill occurs when is between 0 and -90 degrees. For this range of we know: and . Thus for this range of ,
. Thus is a positive value meaning the range downhill is always further than along level terrain. The lower level of terrain causes the projectile to remain in the air longer, allowing it to travel further horizontally before hitting the ground.
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 (i.e. the slant range is equal to the level terrain range) and solving for the "critical angle" :
Equation 11 may also be used to develop the "rifleman's rule" for small values of and (i.e. close to horizontal firing, which is the case for many firearm situations). For small values, both and 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:
And solving for level terrain range,
- "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."
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
in standard linear form at coordinates
- (Equation 12) where in this case, , and
Substituting the value of into Equation 10:
- (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:
Now the slant range
is the distance of the intercept from the origin, which is just the hypotenuse
of x and y:
Now is defined as the angle of the hill, so by definition of tangent, . This can be substituted into the equation for :
Now this can be refactored and the trigonometric identity
may be used:
Now the flat range
by the previously used trigonometric identity
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 Isaac Newton
and provided much of the motivation for the development of differential calculus