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Projectile motion is a form of
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
experienced by an object or particle (a
projectile A projectile is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance. Although any objects in motion through space are projectiles, they are commonly found i ...
) that is projected in a gravitational field, such as from
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
's
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
, and moves along a curved path under the action of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
only. In the particular case of projectile motion of Earth, most calculations assume the effects of
air resistance In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
are passive and negligible. The curved path of objects in projectile motion was shown by
Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
to be a
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...
, but may also be a straight line in the special case when it is thrown directly upwards. The study of such motions is called
ballistics Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing a ...
, and such a trajectory is a
ballistic trajectory Projectile motion is a form of motion experienced by an object or particle (a projectile) that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path under the action of gravity only. In the part ...
. The only force of mathematical significance that is actively exerted on the object is gravity, which acts downward, thus imparting to the object a downward
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
towards the Earth’s
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
. Because of the object's
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
, no external force is needed to maintain the horizontal velocity
component Circuit Component may refer to: •Are devices that perform functions when they are connected in a circuit.   In engineering, science, and technology Generic systems * System components, an entity with discrete structure, such as an assem ...
of the object's motion. Taking other forces into account, such as
aerodynamic drag In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding ...
or internal propulsion (such as in a
rocket A rocket (from it, rocchetto, , bobbin/spool) is a vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entir ...
), requires additional analysis. A
ballistic missile A ballistic missile is a type of missile that uses projectile motion to deliver warheads on a target. These weapons are guided only during relatively brief periods—most of the flight is unpowered. Short-range ballistic missiles stay within t ...
is a
missile In military terminology, a missile is a guided airborne ranged weapon capable of self-propelled flight usually by a jet engine or rocket motor. Missiles are thus also called guided missiles or guided rockets (when a previously unguided rocket ...
only guided during the relatively brief initial powered phase of flight, and whose remaining course is governed by the laws of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
.
Ballistics Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing a ...
() is the science of dynamics that deals with the flight, behavior and effects of projectiles, especially
bullet A bullet is a kinetic projectile, a component of firearm ammunition that is shot from a gun barrel. Bullets are made of a variety of materials, such as copper, lead, steel, polymer, rubber and even wax. Bullets are made in various shapes and co ...
s,
unguided bomb An unguided bomb, also known as a free-fall bomb, gravity bomb, dumb bomb, or iron bomb, is a conventional or nuclear aircraft-delivered bomb that does not contain a guidance system and hence simply follows a ballistic trajectory. This describe ...
s,
rocket A rocket (from it, rocchetto, , bobbin/spool) is a vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entir ...
s, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance. The elementary equations of ballistics neglect nearly every factor except for initial velocity and an assumed constant gravitational acceleration. Practical solutions of a ballistics problem often require considerations of air resistance, cross winds, target motion, varying acceleration due to gravity, and in such problems as launching a rocket from one point on the Earth to another, the rotation of the Earth. Detailed mathematical solutions of practical problems typically do not have closed-form solutions, and therefore require numerical methods to address.


Kinematic quantities

In projectile motion, the horizontal motion and the vertical motion are independent of each other; that is, neither motion affects the other. This is the principle of ''compound motion'' established by
Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
in 1638, and used by him to prove the parabolic form of projectile motion. A ballistic trajectory is a parabola with homogeneous acceleration, such as in a space ship with constant acceleration in absence of other forces. On Earth the acceleration changes magnitude with altitude and direction with latitude/longitude. This causes an elliptic trajectory, which is very close to a parabola on a small scale. However, if an object was thrown and the Earth was suddenly replaced with a
black hole A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...
of equal mass, it would become obvious that the ballistic trajectory is part of an elliptic
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
around that black hole, and not a parabola that extends to infinity. At higher speeds the trajectory can also be circular, parabolic or
hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
(unless distorted by other objects like the Moon or the Sun). In this article a homogeneous acceleration is assumed.


Acceleration

Since there is only acceleration in the vertical direction, the velocity in the horizontal direction is constant, being equal to \mathbf_0 \cos\theta . The vertical motion of the projectile is the motion of a particle during its free fall. Here the acceleration is constant, being equal to g. The components of the acceleration are: : a_x = 0 , : a_y = -g .


Velocity

Let the projectile be launched with an initial
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
\mathbf(0) \equiv \mathbf_0 , which can be expressed as the sum of horizontal and vertical components as follows: : \mathbf_0 = v_\mathbf + v_\mathbf . The components v_ and v_ can be found if the initial launch angle, \theta , is known: : v_ = v_0\cos(\theta), : v_ = v_0\sin(\theta) The horizontal component of the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of the object remains unchanged throughout the motion. The vertical component of the velocity changes linearly, because the acceleration due to gravity is constant. The accelerations in the x and y directions can be integrated to solve for the components of velocity at any time t, as follows: : v_x = v_0 \cos(\theta) , : v_y = v_0 \sin(\theta) - gt . The magnitude of the velocity (under the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
, also known as the triangle law): : v = \sqrt .


Displacement

At any time t , the projectile's horizontal and vertical displacement are: : x = v_0 t \cos(\theta) , : y = v_0 t \sin(\theta) - \fracgt^2 . The magnitude of the displacement is: : \Delta r=\sqrt . Consider the equations, : x = v_0 t \cos(\theta) , y = v_0 t\sin(\theta) - \fracgt^2 . If t is eliminated between these two equations the following equation is obtained: : y = \tan(\theta) \cdot x-\frac \cdot x^2 . Since g, θ, and v0 are constants, the above equation is of the form : y=ax+bx^2 , in which a and b are constants. This is the equation of a parabola, so the path is parabolic. The axis of the parabola is vertical. If the projectile's position (x,y) and launch angle (θ or α) are known, the initial velocity can be found solving for v0 in the aforementioned parabolic equation: : v_0 = \sqrt .


Displacement in polar coordinates

The parabolic trajectory of a projectile can also be expressed in polar coordinates instead of Cartesian coordinates. In this case, the position has the general formula : r( \phi ) = \frac \left(\tan\theta\sec\phi -\tan\phi\sec\phi \right) . In this equation, the origin is the midpoint of the horizontal range of the projectile, and if the ground is flat, the parabolic arc is plotted in the range 0 \leq \phi \leq \pi . This expression can be obtained by transforming the Cartesian equation as stated above by y = r \sin\phi and x = r \cos\phi .


Properties of the trajectory


Time of flight or total time of the whole journey

The total time t for which the projectile remains in the air is called the time of flight. : y = v_0 t \sin(\theta) - \fracgt^2 After the flight, the projectile returns to the horizontal axis (x-axis), so y=0 . : 0 = v_0 t \sin(\theta) - \fracgt^2 : v_0 t \sin(\theta) = \fracgt^2 : v_0 \sin(\theta) = \fracgt : t = \frac Note that we have neglected air resistance on the projectile. If the starting point is at height y0 with respect to the point of impact, the time of flight is: : t = \frac = \frac As above, this expression can be reduced to : t = \frac = \frac = \frac = \frac = \frac = \frac if θ is 45° and y0 is 0.


Time of flight to the target's position

As shown above in the Displacement section, the horizontal and vertical velocity of a projectile are independent of each other. Because of this, we can find the time to reach a target using the displacement formula for the horizontal velocity: x = v_0 t \cos(\theta) \frac=v_0\cos(\theta) t=\frac This equation will give the total time ''t'' the projectile must travel for to reach the target's horizontal displacement, neglecting air resistance.


Maximum height of projectile

The greatest height that the object will reach is known as the peak of the object's motion. The increase in height will last until v_y=0 , that is, : 0=v_0 \sin(\theta) - gt_h . Time to reach the maximum height(h): : t_h = \frac . For the vertical displacement of the maximum height of projectile: : h = v_0 t_h \sin(\theta) - \frac gt^2_h : h = \frac The maximum reachable height is obtained for ''θ''=90°: : h_ = \frac


Relation between horizontal range and maximum height

The relation between the range d on the horizontal plane and the maximum height h reached at \frac is: : h = \frac h = \frac : d = \frac : \frac = \frac × \frac : \frac = \frac h = \frac .


Maximum distance of projectile

The range and the maximum height of the projectile does not depend upon its mass. Hence range and maximum height are equal for all bodies that are thrown with the same velocity and direction.. The horizontal range d of the projectile is the horizontal distance it has traveled when it returns to its initial height (y=0). : 0 = v_0 t_d \sin(\theta) - \fracgt_d^2 . Time to reach ground: : t_d = \frac . From the horizontal displacement the maximum distance of projectile: : d = v_0 t_d \cos(\theta) , so : d = \frac\sin(2\theta) . Note that d has its maximum value when : \sin 2\theta=1 , which necessarily corresponds to : 2\theta=90^\circ , or : \theta=45^\circ . The total horizontal distance (d) traveled. : d = \frac \left( v \sin \theta + \sqrt \right) When the surface is flat (initial height of the object is zero), the distance traveled: : d = \frac Thus the maximum distance is obtained if θ is 45 degrees. This distance is: : d_ = \frac


Application of the work energy theorem

According to the work-energy theorem the vertical component of velocity is: : v_y^2 = (v_0 \sin \theta)^2-2gy . These formulae ignore aerodynamic drag and also assume that the landing area is at uniform height 0.


Angle of reach

The "angle of reach" is the angle (θ) at which a projectile must be launched in order to go a distance d, given the initial velocity v. : \sin(2\theta) = \frac There are two solutions: : \theta = \frac \arcsin \left( \frac \right) (shallow trajectory) and : \theta = \frac \arccos \left( \frac \right) (steep trajectory)


Angle θ required to hit coordinate (x, y)

To hit a target at range x and altitude y when fired from (0,0) and with initial speed v the required angle(s) of launch θ are: : \theta = \arctan The two roots of the equation correspond to the two possible launch angles, so long as they aren't imaginary, in which case the initial speed is not great enough to reach the point (x,y) selected. This formula allows one to find the angle of launch needed without the restriction of y=0 . One can also ask what launch angle allows the lowest possible launch velocity. This occurs when the two solutions above are equal, implying that the quantity under the square root sign is zero. This requires solving a quadratic equation for v^2 , and we find : v^2/g=y+\sqrt. This gives : \theta=\arctan\left(y/x+\sqrt\right). If we denote the angle whose tangent is by , then : \tan\theta=\frac : \tan(\pi/2-\theta)=\frac : \cos^2(\pi/2-\theta)=\frac 12(\sin\alpha+1) : 2\cos^2(\pi/2-\theta)-1=\cos(\pi/2-\alpha) This implies : \theta = \pi/2 - \frac 12(\pi/2-\alpha). In other words, the launch should be at the angle halfway between the target and Zenith (vector opposite to Gravity)


Total Path Length of the Trajectory

The length of the parabolic arc traced by a projectile L, given that the height of launch and landing is the same and that there is no air resistance, is given by the formula: :L = \frac \left( 2\sin\theta + \cos^2\theta\cdot\ln \frac \right) = \frac \left( \sin\theta + \cos^2\theta\cdot\tanh^(\sin\theta) \right) where v_0 is the initial velocity, \theta is the launch angle and g is the acceleration due to gravity as a positive value. The expression can be obtained by evaluating the arc length integral for the height-distance parabola between the bounds ''initial'' and ''final'' displacements (i.e. between 0 and the horizontal range of the projectile) such that: :L = \int_^ \sqrt\,\mathrmx = \int_^ \sqrt\,\mathrmx.


Trajectory of a projectile with air resistance

Air resistance creates a force that (for symmetric projectiles) is always directed against the direction of motion in the surrounding medium and has a magnitude that depends on the absolute speed: \mathbf = -f(v)\cdot\mathbf. The speed-dependence of the friction force is linear (f(v) \propto v) at very low speeds (
Stokes drag In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by so ...
) and quadratic (f(v) \propto v^2) at larger speeds ( Newton drag). The transition between these behaviours is determined by the
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
, which depends on speed, object size and kinematic viscosity of the medium. For Reynolds numbers below about 1000, the dependence is linear, above it becomes quadratic. In air, which has a kinematic viscosity around 0.15\,\mathrm, this means that the drag force becomes quadratic in ''v'' when the product of speed and diameter is more than about 0.015\,\mathrm, which is typically the case for projectiles. * Stokes drag: \mathbf = -k_\cdot\mathbf\qquad (for Re \lesssim 1000) * Newton drag: \mathbf = -k\,, \mathbf, \cdot\mathbf\qquad (for Re \gtrsim 1000) The
free body diagram A free body diagram consists of a diagrammatic representation of a single body or a subsystem of bodies isolated from its surroundings showing all the forces acting on it. In physics and engineering, a free body diagram (FBD; also called a force ...
on the right is for a projectile that experiences air resistance and the effects of gravity. Here, air resistance is assumed to be in the direction opposite of the projectile's velocity: \mathbf = -f(v)\cdot\mathbf


Trajectory of a projectile with Stokes drag

Stokes drag, where \mathbf \propto \mathbf, only applies at very low speed in air, and is thus not the typical case for projectiles. However, the linear dependence of F_\mathrm on v causes a very simple differential equation of motion :\frac\beginv_x \\ v_y\end = \begin-\mu\,v_x \\ -g-\mu\,v_y\end in which the two cartesian components become completely independent, and thus easier to solve. Here, v_0,v_x and v_y will be used to denote the initial velocity, the velocity along the direction of x and the velocity along the direction of y, respectively. The mass of the projectile will be denoted by m, and \mu:=k/m. For the derivation only the case where 0^o \le \theta \le 180^o is considered. Again, the projectile is fired from the origin (0,0). The relationships that represent the motion of the particle are derived by
Newton's Second Law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in mo ...
, both in the x and y directions. In the x direction \Sigma F = -kv_x = ma_x and in the y direction \Sigma F = -kv_y - mg = ma_y . This implies that: a_x = -\mu v_x = \frac (1), and a_y = -\mu v_y - g = \frac (2) Solving (1) is an elementary
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
, thus the steps leading to a unique solution for vx and, subsequently, x will not be enumerated. Given the initial conditions v_x = v_ (where vx0 is understood to be the x component of the initial velocity) and x=0 for t=0 : v_x = v_ e^ (1a) : x(t) = \frac\left(1-e^\right) (1b) While (1) is solved much in the same way, (2) is of distinct interest because of its non-homogeneous nature. Hence, we will be extensively solving (2). Note that in this case the initial conditions are used v_y=v_ and y=0 when t=0 . \frac = -\mu v_y - g (2) \frac + \mu v_y = - g (2a) This first order, linear, non-homogeneous differential equation may be solved a number of ways; however, in this instance, it will be quicker to approach the solution via an
integrating factor In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calcul ...
e^ .

e^(\frac + \mu v_y) = e^(-g) (2c) (e^v_y)^\prime = e^(-g) (2d) \int = e^v_y = \int (2e) e^v_y = \frac e^(-g) + C (2f) v_y = \frac + Ce^ (2g)

And by integration we find:

y = -\fract - \frac(v_ + \frac)e^ + C (3) Solving for our initial conditions:

v_y(t) = -\frac + (v_ + \frac)e^ (2h) y(t) = -\fract - \frac(v_ + \frac)e^ + \frac(v_ + \frac) (3a)

With a bit of algebra to simplify (3a): : y(t) = -\fract + \frac\left(v_ + \frac\right)\left(1 - e^\right) (3b) The total time of the journey in the presence of air resistance (more specifically, when F_=-kv) can be calculated by the same strategy as above, namely, we solve the equation y(t)=0. While in the case of zero air resistance this equation can be solved elementarily, here we shall need the Lambert W function. The equation y(t)= -\fract + \frac(v_ + \frac)(1 - e^) = 0 is of the form c_1t+c_2+c_3e^=0, and such an equation can be transformed into an equation solvable by the W function (see an example of such a transformation
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a ...
). Some algebra shows that the total time of flight, in closed form, is given as :t=\frac\left(1+\fracv_+W\left(-\left(1+\frac v_\right)e^\right)\right).


Trajectory of a projectile with Newton drag

The most typical case of
air resistance In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
, for the case of
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
s above about 1000 is Newton drag with a drag force proportional to the speed squared, F_ = -k v^2. In air, which has a kinematic viscosity around 0.15\,\mathrm, this means that the product of speed and diameter must be more than about 0.015\,\mathrm. Unfortunately, the equations of motion can ''not'' be easily solved analytically for this case. Therefore, a numerical solution will be examined. The following assumptions are made: * Constant
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodie ...
*
Air resistance In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
is given by the following drag formula, ::\mathbf = -\tfrac c \rho A\, v\,\mathbf ::Where: ::*''FD'' is the drag force ::*''c'' is the
drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag e ...
::*''ρ'' is the
air density The density of air or atmospheric density, denoted '' ρ'', is the mass per unit volume of Earth's atmosphere. Air density, like air pressure, decreases with increasing altitude. It also changes with variation in atmospheric pressure, temperature a ...
::*''A'' is the
cross sectional area In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. ...
of the projectile ::*''μ'' = ''k''/''m'' = ''cρA''/(2''m'')


Special cases

Even though the general case of a projectile with Newton drag cannot be solved analytically, some special cases can. Here we denote the
terminal velocity Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid ( air is the most common example). It occurs when the sum of the drag force (''Fd'') and the buoyancy is equal to the downward force of grav ...
in free-fall as v_\infty=\sqrt and the characteristic settling time constant t_f=1/\sqrt. *Near-horizontal motion: In case the motion is almost horizontal, , v_x, \gg, v_y, , such as a flying bullet, the vertical velocity component has very little influence on the horizontal motion. In this case: ::\dot_x(t) = -\mu\,v_x^2(t) ::v_x(t) = \frac ::x(t) = \frac\ln(1+\mu\,v_\cdot t) :The same pattern applies for motion with friction along a line in any direction, when gravity is negligible. It also applies when vertical motion is prevented, such as for a moving car with its engine off. *Vertical motion upward: ::\dot_y(t) = -g-\mu\,v_y^2(t) ::v_y(t) = v_\infty \tan\frac ::y(t) = y_ + \frac\ln\left(\cos\frac\right) :Here ::v_\infty \equiv \sqrt, ::t_f \equiv \frac, ::t_ \equiv t_f \arctan = \frac \arctan, :and ::y_ \equiv -\frac\ln = \frac\ln :where v_ is the initial upward velocity at t = 0 and the initial position is y(0) = 0. :A projectile can not rise longer than t_\mathrm=\fract_f vertically before it reaches the peak. *Vertical motion downward: ::\dot_y(t) = -g+\mu\,v_y^2(t) ::v_y(t) = -v_\infty \tanh\frac ::y(t) = y_ - \frac\ln\left(\cosh\frac\right) :After a time t_f, the projectile reaches almost terminal velocity -v_\infty.


Numerical solution

A projectile motion with drag can be computed generically by
numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equatio ...
of the
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
, for instance by applying a reduction to a first-order system. The equation to be solved is :\frac\beginx \\ y \\ v_x \\ v_y\end = \beginv_x \\ v_y \\ -\mu\,v_x\sqrt \\ -g-\mu\,v_y\sqrt\end. This approach also allows to add the effects of speed-dependent drag coefficient, altitude-dependent air density and position-dependent gravity field.


Lofted trajectory

A special case of a ballistic trajectory for a rocket is a lofted trajectory, a trajectory with an
apogee An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any el ...
greater than the minimum-energy trajectory to the same range. In other words, the rocket travels higher and by doing so it uses more energy to get to the same landing point. This may be done for various reasons such as increasing distance to the horizon to give greater viewing/communication range or for changing the angle with which a missile will impact on landing. Lofted trajectories are sometimes used in both missile rocketry and in
spaceflight Spaceflight (or space flight) is an application of astronautics to fly spacecraft into or through outer space, either with or without humans on board. Most spaceflight is uncrewed and conducted mainly with spacecraft such as satellites in ...
.Ballistic Missile Defense, Glossary, v. 3.0
US Department of Defense The United States Department of Defense (DoD, USDOD or DOD) is an executive branch department of the federal government charged with coordinating and supervising all agencies and functions of the government directly related to national sec ...
, June 1997.


Projectile motion on a planetary scale

When a projectile without air resistance travels a range that is significant compared to the earth's radius (above ≈100 km), the
curvature of the earth Spherical Earth or Earth's curvature refers to the approximation of figure of the Earth as a sphere. The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of Greek philosophers. I ...
and the non-uniform
Earth's gravity The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation). It is a vector qua ...
have to be considered. This is for example the case with spacecraft or intercontinental projectiles. The trajectory then generalizes from a parabola to a Kepler-
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
with one focus at the center of the earth. The projectile motion then follows
Kepler's laws of planetary motion In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orb ...
. The trajectories' parameters have to be adapted from the values of a uniform gravity field stated above. The
earth radius Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, den ...
is taken as R, and g as the standard surface gravity. Let \tilde v:=v/\sqrt the launch velocity relative to the first cosmic velocity. Total range d between launch and impact: : d = \frac \Big/ \sqrt Maximum range of a projectile for optimum launch angle (\theta=\tfrac12\arccos\left(\tilde v^2/(2-\tilde v^2)\right)): : d_ = \frac \big/ \left(1-\tfrac12\tilde v^2\right)       with v<\sqrt, the first cosmic velocity Maximum height of a projectile above the planetary surface: : h = \frac \Big/ \left(1-\tilde v^2+\sqrt\right) Maximum height of a projectile for vertical launch (\theta=90^\circ): : h_ = \frac \big/ \left(1-\tfrac12\tilde v^2\right)       with v<\sqrt, the
second cosmic velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically s ...
Time of flight: : t = \frac \cdot \frac \left(1 + \frac\arcsin\frac\right)


See also

*
Equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...


Notes


References

{{DEFAULTSORT:Projectile Motion Mechanics