
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 mea ...
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 surf ...
's
surface, 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 str ...
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 ...
are passive and negligible. The curved path of objects in projectile motion was shown by
Galileo 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 ...
, and such a trajectory is a
ballistic trajectory. 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. 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 o ...
, no external force is needed to maintain the horizontal velocity
component of the object's motion. Taking other forces into account, such as
aerodynamic drag 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 entire ...
), 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 ...
is a
missile
In military terminology, a missile is a missile guidance, 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 ...
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 classical ...
.
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 ...
() 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 ...
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 descri ...
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 entire ...
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
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
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 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 defo ...
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 a ...
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
. 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:
:
,
:
.
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 ...
, which can be expressed as the sum of horizontal and vertical components as follows:
:
.
The components
and
can be found if the initial launch angle,
, is known:
:
,
:
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:
:
,
:
.
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):
:
.
Displacement

At any time
, the projectile's horizontal and vertical
displacement are:
:
,
:
.
The magnitude of the displacement is:
:
.
Consider the equations,
:
.
If
t is eliminated between these two equations the following equation is obtained:
:
.
Since
g,
θ, and
v0 are constants, the above equation is of the form
:
,
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:
:
.
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
:
.
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
. This expression can be obtained by transforming the Cartesian equation as stated above by
and
.
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.
:
After the flight, the projectile returns to the horizontal axis (x-axis), so
.
:
:
:
:
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:
:
As above, this expression can be reduced to
:
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:
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
, that is,
:
.
Time to reach the maximum height(h):
:
.
For the vertical displacement of the maximum height of projectile:
:
:
The maximum reachable height is obtained for ''θ''=90°:
:
Relation between horizontal range and maximum height
The relation between the range
d on the horizontal plane and the maximum height
h reached at
is:
:
:
:
×
:
.
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 (
).
:
.
Time to reach ground:
:
.
From the horizontal displacement the maximum distance of projectile:
:
,
so
:
.
Note that
d has its maximum value when
:
,
which necessarily corresponds to
:
,
or
:
.

The total horizontal distance
(d) traveled.
:
When the surface is flat (initial height of the object is zero), the distance traveled:
:
Thus the maximum distance is obtained if
θ is 45 degrees. This distance is:
:
Application of the work energy theorem
According to the
work-energy theorem
In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stre ...
the vertical component of velocity is:
:
.
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.
:
There are two solutions:
:
(shallow trajectory)
and
:
(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:
:
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
.
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
, and we find
:
This gives
:
If we denote the angle whose tangent is by , then
:
:
:
:
This implies
:
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:
:
where
is the initial velocity,
is the launch angle and
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:
:
.
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:
. The speed-dependence of the friction force is linear (
) at very low speeds (
Stokes drag) and quadratic (
) at larger speeds (
Newton drag).
The transition between these behaviours is determined by the
Reynolds number, which depends on speed, object size and
kinematic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the intern ...
of the medium. For Reynolds numbers below about 1000, the dependence is linear, above it becomes quadratic. In air, which has a
kinematic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the intern ...
around
, this means that the drag force becomes quadratic in ''v'' when the product of speed and diameter is more than about
, which is typically the case for projectiles.
* Stokes drag:
(for
)
* Newton drag:
(for
)
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:
Trajectory of a projectile with Stokes drag
Stokes drag, where
, only applies at very low speed in air, and is thus not the typical case for projectiles. However, the linear dependence of
on
causes a very simple differential equation of motion
:
in which the two cartesian components become completely independent, and thus easier to solve.
Here,
,
and
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
. For the derivation only the case where
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 Scientific law, 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 re ...
, both in the x and y directions.
In the x direction
and in the y direction
.
This implies that:
(1),
and
(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, a ...
, thus the steps leading to a unique solution for
vx and, subsequently,
x will not be enumerated. Given the initial conditions
(where
vx0 is understood to be the x component of the initial velocity) and
for
:
(1a)
:
(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
and
when
.
(2)
(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 calc ...
.
(2c)
(2d)
(2e)
(2f)
(2g)
And by integration we find:
(3)
Solving for our initial conditions:
(2h)
(3a)
With a bit of algebra to simplify (3a):
:
(3b)
The total time of the journey in the presence of air resistance (more specifically, when
) can be calculated by the same strategy as above, namely, we solve the equation
. While in the case of zero air resistance this equation can be solved elementarily, here we shall need the
Lambert W function
In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential func ...
. The equation
is of the form
, and such an equation can be transformed into an equation solvable by the
function (see an example of such a transformation
here). Some algebra shows that the total time of flight, in closed form, is given as
:
.
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 ...
, for the case of
Reynolds numbers above about 1000 is Newton drag with a drag force proportional to the speed squared,
. In air, which has a
kinematic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the intern ...
around
, this means that the product of speed and diameter must be more than about
.
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 (physics), drag). This is the steady gain in speed caused exclusively by the force of gravitational attract ...
*
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 ...
is given by the following
drag formula,
::
::Where:
::*''F
D'' 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 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 gravit ...
in free-fall as
and the characteristic settling time constant
.
*Near-horizontal motion: In case the motion is almost horizontal,
, such as a flying bullet, the vertical velocity component has very little influence on the horizontal motion. In this case:
::
::
::
: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:
::
::
::
:Here
::
::
::
:and
::
:where
is the initial upward velocity at
and the initial position is
.
:A projectile can not rise longer than
vertically before it reaches the peak.
*Vertical motion downward:
::
::
::
:After a time
, the projectile reaches almost terminal velocity
.
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 equations ...
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 contras ...
, for instance by applying a
reduction to a first-order system. The equation to be solved is
:
.
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 o ...
.
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. ...
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 quantit ...
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 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 or ...
.
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, deno ...
is taken as R, and g as the standard surface gravity. Let the launch velocity relative to the first cosmic velocity.
Total range d between launch and impact:
:
Maximum range of a projectile for optimum launch angle ():
: with , the first 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 ...
Maximum height of a projectile above the planetary surface:
:
Maximum height of a projectile for vertical launch ():
: with , the second cosmic velocity
Time of flight:
:
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 (Ve ...
Notes
References
{{DEFAULTSORT:Projectile Motion
Mechanics