Dictionary Definition
kinematics n : the branch of mechanics concerned
with motion without reference to force or mass
User Contributed Dictionary
English
Translations
 Finnish: kinematiikka
 Slovak: kinematika
See also
Extensive Definition
Kinematics (Greek
κινειν,kinein, to move) is a branch of dynamics
which describes the motion
of objects without consideration of the circumstances leading to
the motion. In contrast, kinetics
is concerned with the forces and interactions that produce or
affect the motion. The simplest application of kinematics is to
point particle motion (translational
kinematics or linear kinematics). The description of rotation
(rotational
kinematics or angular kinematics) is more complicated. The
state of a generic rigid body may be described by combining both
translational and rotational kinematics (rigidbody
kinematics). A more complicated case is the kinematics of a
system of rigid bodies, possibly linked together by mechanical
joints. The kinematic
description of fluid flow is even more complicated, and not
generally thought of in the context of kinematics.
Translational motion
Translational or curvilinear kinematics is the
description of the motion in space of a point along a trajectory.
This path can be linear, or curved as seen with projectile motion.
There are three basic concepts that are required for understanding
translational motion:
Displacement (denoted by r below )is the shortest
distance between two points: the origin and the displaced point.
The origin is (0,0) on a coordinate
system that is defined by the observer. Because displacement
has both magnitude (length) and direction, it is a vector whose initial point is the
origin and terminal point is the displaced point.
Velocity (denoted by υ below) is the rate of
change in displacement with respect to time; that is the
displacement of a point changes with time. Velocity is also a
vector. For a constant velocity, every unit of time adds the length
of the velocity vector (in the same direction) to the displacement
of the moving point. Instantaneous velocity (the velocity at an
instant of time) is defined as

 \boldsymbol v = \frac \ ,
Acceleration (denoted by a below) is the rate of
change in velocity with respect to time. Acceleration is also a
vector. As with velocity if acceleration is constant, for every
unit of time the length of the acceleration vector (in the same
direction) is added to the velocity. If the change in velocity (a
vector) is known, the acceleration is parallel to it. Instantaneous
acceleration (the acceleration at an instant of time) is defined
as:

 \boldsymbol a = \frac \ ,

 \boldsymbol a = \frac \ ,
Integral relations
The above definitions can be inverted by integration to find: \boldsymbol(t) =\boldsymbol_0+\ \int_0^t \ dt' \ \boldsymbol (t')
 \boldsymbol(t) =\boldsymbol_0+\ \int_0^t \ dt' \ \boldsymbol (t')

 =\boldsymbol_0+\boldsymbol_0\ t +\ \int_0^t \ dt' \int_0^ \ dt \ \boldsymbol (t)
 =\boldsymbol_0+\boldsymbol_0\ t +\ \int_0^t \ dt' \left(tt'\right) \ \boldsymbol (t') \ ,
Constant acceleration
When acceleration is constant both in direction and in magnitude, the point is said to be undergoing uniformly accelerated motion. In this case, the above equations can be simplified:: \boldsymbol v = \int_0^ \boldsymbol a dt = \boldsymbol_0 + \boldsymbol a t Those who are familiar with calculus may recognize this as an initial value problem. Because acceleration ( a ) is a constant, integrating it with respect to time (t) gives a change in velocity. Adding this to the initial velocity ( υ0) gives the final velocity ( υ ).: \boldsymbol r = \int_0^t \boldsymbol v dt = \int_0^t \left( \boldsymbol v_0 + \boldsymbol a t \right) dt = \boldsymbol v_0 t + \frac \boldsymbol a t^2 Using the above formula, we can substitute for υ to arrive at this equation, where r is displacement.: \boldsymbol r = \frac t By using the definition of an average, this equation states that average velocity times time equals displacement. Using Eq. (1) to find υ−υ0 and multiplying by Eq. (3) we find a connection between the final velocity at time t and the displacement at that time: \boldsymbol t = \left( \boldsymbol v  \boldsymbol _0 \right)\boldsymbol \frac t \ ,
 v^2= v_0^2 + 2 a s\ ,
Relative velocity
To describe the motion of object A with respect
to object B, when we know how each is moving with respect to a
reference object O, we can use vector algebra. Choose an origin for
reference, and let the positions of objects A, B, and O be denoted
by rA, rB, and rO. Then the position of A relative to the reference
object O is
 \boldsymbol_ = \boldsymbol_  \boldsymbol_ \,\!
 \boldsymbol_ = \boldsymbol_A  \boldsymbol_B = \boldsymbol_A  \boldsymbol_O  \left(\boldsymbol_B\boldsymbol_O\right) = \boldsymbol_\boldsymbol_ \ .
The above relative equation states that the
motion of A relative to B is equal to the motion of A relative to O
minus the motion of B relative to O. It may be easier to visualize
this result if the terms are rearranged:
 \boldsymbol_ = \boldsymbol_ + \boldsymbol_ \ ,
or, in words, the motion of A relative to the
reference is that of B plus the relative motion of A with respect
to B. These relation between displacements become relations between
velocities by simple timedifferentiation, and a second
differentiation makes them apply to accelerations.
For example, let Ann move with velocity
\boldsymbol_ relative to the reference (we drop the O subscript for
convenience) and let Bob move with velocity \boldsymbol_, each
velocity given with respect to the ground (point O). To find how
fast Ann is moving relative to Bob (we call this velocity
\boldsymbol_), the equation above gives:
 \boldsymbol_ = \boldsymbol_ + \boldsymbol_ \,\! .
To find \boldsymbol_ we simply rearrange this
equation to obtain:
 \boldsymbol_ = \boldsymbol_ \boldsymbol_ \,\! .
At velocities comparable to the speed of
light, these equations are not valid. They are replaced by
equations derived from Einstein's theory of
special relativity.
Rotational motion
Rotational kinematics is the description of the
rotation of an object. The description of rotation requires some
method for describing orientation, for example, the Euler
angles. In what follows attention is restricted to simple
rotation about an axis of fixed orientation for convenience chosen
as the zaxis..
Description of rotation then involves these three
quantities:
Angular position: The oriented distance from a
selected origin on the rotational axis to a point of an object is a
vector r ( t ) locating the point. The vector r ( t ) has some
projection (or, equivalently, some component) r\perp ( t ) on a
plane perpendicular to the axis of rotation. Then the angular
position of that point is the angle θ from a reference axis
(typically the positive xaxis) to the vector r\perp ( t ) in a
known rotation sense (typically given by the righthand
rule).
Angular velocity: The angular velocity ω is the
rate at which the angular position θ changes with respect to time
t:
 \mathbf = \frac
The angular velocity is represented in Figure 1
by a vector Ω pointing along the axis of rotation with magnitude ω
and sense determined by the direction of rotation as given by the
righthand rule.
Angular acceleration: The magnitude of the
angular acceleration \alpha is the rate at which the angular
velocity \omega changes with respect to time t:
 \mathbf = \frac
The equations of translational kinematics can
easily be extended to planar rotational kinematics with simple
variable exchanges:
 \,\!\theta_f  \theta_i = \omega_i t + \frac \alpha t^2 \qquad \theta_f  \theta_i = \frac (\omega_f + \omega_i)t
 \,\!\omega_f = \omega_i + \alpha t \qquad \alpha = \frac \qquad \omega_f^2 = \omega_i^2 + 2 \alpha (\theta_f  \theta_i)\ .
Here \,\!\theta_i and \,\!\theta_f are,
respectively, the initial and final angular positions, \,\!\omega_i
and \,\!\omega_f are, respectively, the initial and final angular
velocities, and \,\!\alpha is the constant angular acceleration.
Although position in space and velocity in space are both true
vectors (in terms of their properties under rotation), as is
angular velocity, angle itself is not a true vector.
Point object in circular motion
This example deals with a "point" object, by which is meant that complications due to rotation of the body itself about its own center of mass are ignored.Displacement. An object in circular motion is
located at a position r ( t ) given by:
 \boldsymbol (t) = R \mathbf_R (t) \ ,
Linear velocity. The velocity of the object is
then
 \boldsymbol (t) =\frac \boldsymbol (t) = R \frac\mathbf_R (t) \ .
 \frac\mathbf_R (t) = \boldsymbol \mathbf_R = \omega (t) \mathbf_ \ ,
 \boldsymbol (t) = R\omega (t) \mathbf_ \ .
Linear acceleration. In the same manner, the
acceleration of the object is defined as:
 \boldsymbol (t) = \frac \boldsymbol (t) = R\frac\omega\mathbf_ \

 =\mathbf_ R\frac\omega + R\omega\frac\mathbf_
 =\mathbf_ R\frac\omega + R\omega \boldsymbol\mathbf_
 =\mathbf_ R\frac\omega  \mathbf_\omega^2R\
 =\mathbf_ + \mathbf_R \ ,
Coordinate systems
In any given situation, the most useful coordinates may be determined by constraints on the motion, or by the geometrical nature of the force causing or affecting the motion. Thus, to describe the motion of a bead constrained to move along a circular hoop, the most useful coordinate may be its angle on the hoop. Similarly, to describe the motion of a particle acted upon by a central force, the most useful coordinates may be polar coordinates.Fixed rectangular coordinates
In this coordinate system, vectors are expressed
as an addition of vectors in the x, y, and z direction from a
nonrotating origin. Usually i, j, k are unit vectors
in the x, y, and zdirections.
The position vector, r (or s), the velocity
vector, v, and the acceleration vector, a are
expressed using rectangular coordinates in the following way:
 \boldsymbol = x\ \hat + y \ \hat + z \ \hat \, \!
 \boldsymbol = \dot = \dot \ \hat + \dot \ \hat + \dot \ \hat \, \!
 \boldsymbol = \ddot = \ddot \ \hat + \ddot \ \hat + \ddot \ \hat \, \!
Note: \dot = \frac , \ddot = \frac
Two dimensional rotating reference frame
seealso Centripetal force This coordinate system expresses only planar motion. It is based on three orthogonal unit vectors: the vector i, and the vector j which form a basis for the plane in which the objects we are considering reside, and k about which rotation occurs. Unlike rectangular coordinates, which are measured relative to an origin that is fixed and nonrotating, the origin of these coordinates can rotate and translate  often following a particle on a body that is being studied.Derivatives of unit vectors
The position, velocity, and acceleration vectors of a given point can be expressed using these coordinate systems, but we have to be a bit more careful than we do with fixed frames of reference. Since the frame of reference is rotating, the unit vectors also rotate, and this rotation must be taken into account when taking the derivative of any of these vectors. If the coordinate frame is rotating at angular rate ω in the counterclockwise direction (that is, Ω = ω k using the right hand rule) then the derivatives of the unit vectors are as follows: \dot = \omega \hat \times \hat = \omega\hat
 \dot = \omega \hat \times \hat =  \omega \hat
Position, velocity, and acceleration
Given these identities, we can now figure out how to represent the position, velocity, and acceleration vectors of a particle using this reference frame.Position
Position is straightforward: \boldsymbol = x \ \hat + y \ \hat
It is just the distance from the origin in the
direction of each of the unit vectors.
Velocity
Velocity is the time derivative of position: \boldsymbol = \frac = \frac + \frac
By the product
rule, this is:
 \boldsymbol = \dot x \ \hat + x \dot + \dot y \ \hat + y \dot
Which from the identities above we know to
be:
 \boldsymbol = \dot x \ \hat + x \omega \ \hat + \dot y \ \hat  y \omega \ \hat = (\dot x  y \omega) \ \hat + (\dot y + x \omega) \ \hat
or equivalently
 \boldsymbol= (\dot x \ \hat + \dot y \ \hat ) + (y \dot + x \dot) = \boldsymbol_ + \boldsymbol \times \boldsymbol
where vrel is the velocity of the particle
relative to the rotating coordinate system.
Acceleration
Acceleration is the time derivative of velocity.We know that:
 \boldsymbol = \frac \boldsymbol
Consider the \stackrel \boldsymbol_ part.
\boldsymbol_ has two parts we want to find the derivative of: the
relative change in velocity (\boldsymbol_), and the change in the
coordinate frame (\boldsymbol \times \boldsymbol_).
 \frac = \boldsymbol_ + \boldsymbol \times \boldsymbol_
Next, consider \stackrel (\boldsymbol
\times\boldsymbol). Using the chain rule:
 \frac = \dot \times \boldsymbol + \boldsymbol \times \dot
 \dot=\boldsymbol=\boldsymbol_ + \boldsymbol \times \boldsymbol from above:
 \frac =
So all together:
 \boldsymbol = \boldsymbol_ + \boldsymbol \times \boldsymbol_ +
And collecting terms:
 \boldsymbol = \boldsymbol_ + 2(\boldsymbol \times \boldsymbol_) +
Three dimensional rotating coordinate frame
(to be written)Kinematic constraints
A kinematic constraint is any condition relating properties of a dynamic system that must hold true at all times. Below are some common examples:Rolling without slipping
An object that rolls against a surface without slipping obeys
the condition that the velocity of its center of
mass is equal to the cross
product of its angular
velocity with a vector from the point of contact to the center
of mass,
 \boldsymbol_G(t) = \boldsymbol \times \boldsymbol_.
Inextensible cord
This is the case where bodies are connected by some cord that remains in tension and cannot change length. The constraint is that the sum of all components of the cord, however they are defined, is the total length, and the time derivative of this sum is zero.References and notes
See also
External links
 Java applet of 1D kinematics
 Flash animated tutorial for 1D kinematics
 Physclips: Mechanics with animations and video clips from the University of New South Wales
 KINEMATICS FOR HIGH SCHOOL AND IIT JEE LEVEL
 KMODDL: Kinematic Models for Design Digital Library, Cornell University Library
kinematics in Arabic: علم الحركة
kinematics in Bosnian: Kinematika
kinematics in Catalan: Cinemàtica
kinematics in Czech: Kinematika
kinematics in Danish: Kinematik
kinematics in German: Kinematik
kinematics in Modern Greek (1453):
Κινηματική
kinematics in Spanish: Cinemática
kinematics in Persian: جنبششناسی
kinematics in French: Cinématique
kinematics in Galician: Cinemática
kinematics in Croatian: Kinematika
kinematics in Indonesian: Kinematika
kinematics in Italian: Cinematica
kinematics in Hebrew: קינמטיקה
kinematics in Georgian: კინემატიკა
kinematics in Lithuanian: Kinematika
kinematics in Hungarian: Kinematika
kinematics in Malay (macrolanguage):
Kinematik
kinematics in Dutch: Kinematica
kinematics in Japanese: 運動学
kinematics in Polish: Kinematyka
kinematics in Portuguese: Cinemática
kinematics in Russian: Кинематика точки
kinematics in Albanian: Kinematika
kinematics in Slovak: Kinematika
kinematics in Slovenian: Kinematika
kinematics in Finnish: Kinematiikka
kinematics in Swedish: Kinematik
kinematics in Tamil: அசைவு விபரியல்
kinematics in Turkish: Kinematik
kinematics in Ukrainian: Кінематика
kinematics in Urdu: جنبشیات
kinematics in Yiddish: קינעמאטיק
kinematics in Chinese: 运动学