Kinematics (from Greek κινεῖν, kinein, to move) is the branch of classical mechanics that describes the motion of objects without consideration of the causes leading to the motion.[1][2][3][4]
“It is natural to begin this discussion by considering the various possible types of motion in themselves, leaving out of account for a time the causes to which the initiation of motion may be ascribed; this preliminary enquiry constitutes the science of Kinematics. — ET Whittaker[4]”Kinematics is not to be confused with another branch of classical mechanics: analytical dynamics (the study of the relationship between the motion of objects and its causes), sometimes subdivided into kinetics (the study of the relation between external forces and motion) and statics (the study of the relations in a system at equilibrium). Kinematicsalso differs from dynamics as used in modern-day physics to describe time-evolution of a system.
The term kinematics is less common today than in the past, but still has a role in physics.[5] (See analytical dynamics for more detail on usage). The term "kinematics" also finds use in biomechanics and animal locomotion.[6]
The simplest application of kinematics is for particle motion, translational or rotational. The next level of complexity is introduced by the introduction of rigid bodies, which are collections of particles having time invariant distances amongst themselves. Rigid bodies might undergo translation and rotation or a combination of both. A more complicated case is the kinematics of a system of rigid bodies, possibly linked together by mechanical joints.
Some kinds of computational fluid dynamics use a (complicated) particle-based kinematic model of the fluid, but fluid flow is generally thought of using a continuum mechanics model rather than kinematics.
“It is natural to begin this discussion by considering the various possible types of motion in themselves, leaving out of account for a time the causes to which the initiation of motion may be ascribed; this preliminary enquiry constitutes the science of Kinematics. — ET Whittaker[4]”Kinematics is not to be confused with another branch of classical mechanics: analytical dynamics (the study of the relationship between the motion of objects and its causes), sometimes subdivided into kinetics (the study of the relation between external forces and motion) and statics (the study of the relations in a system at equilibrium). Kinematicsalso differs from dynamics as used in modern-day physics to describe time-evolution of a system.
The term kinematics is less common today than in the past, but still has a role in physics.[5] (See analytical dynamics for more detail on usage). The term "kinematics" also finds use in biomechanics and animal locomotion.[6]
The simplest application of kinematics is for particle motion, translational or rotational. The next level of complexity is introduced by the introduction of rigid bodies, which are collections of particles having time invariant distances amongst themselves. Rigid bodies might undergo translation and rotation or a combination of both. A more complicated case is the kinematics of a system of rigid bodies, possibly linked together by mechanical joints.
Some kinds of computational fluid dynamics use a (complicated) particle-based kinematic model of the fluid, but fluid flow is generally thought of using a continuum mechanics model rather than kinematics.
Linear motionSee also: Mechanics of planar particle motionLinear or translational kinematics[7][8] is the description of the motion in space of a point along a line, also known as trajectory or path.[note 1]This path can be either straight (rectilinear) or curved (curvilinear).
[edit]Position, displacement, and distanceThe position of a point in space is its location relative to a chosen origin. It is a vector quantity, expressing both the distance of the point from the origin and its direction from the origin. In three-dimensions, the position of point A can be expressed with a three-dimensional vector
where xA, yA, and zA are the Cartesian coordinates of the point.
Displacement is a vector describing the difference in position between two points. If point A has position rA = (xA,yA,zA) and point B has position rB = (xB,yB,zB), the displacement rAB of B from A is given by
The distance traveled is always greater than or equal to the displacement.In physics, distance is a distinct quantity from either position or displacement. It is ascalar quantity, describing the length of the path between two points along which the particle has travelled. The straight-line distance |r| of point A from the origin can be found from the position r by
When considering the motion of a particle over time, distance is the length of the particle's path; displacement is the change from its initial position to its final position. For example, a race car traversing a 10 km closed loop from start to finish travels a distance of 10 km; its displacement, however, is zero because it arrives back at its initial position.
If the position of the particle is known as a function of time (r = r(t)), the distance d it travels from time t1 to time t2 can be found through calculus by
[edit]Velocity and speedVelocity is the measure of the rate of change in displacement with respect to time; that is, how the displacement of a point changes with time. Velocity also is a vector. Instantaneous velocity (the velocity at an instant of time) is defined through differential calculus as
where dr is an infinitesimally small displacement and dt is an infinitesimally small length of time.[note 2] As per its definition in the derivative form, velocity can be said to be the time rate of change of displacement. Further, as dr is tangential to the actual path, so is the velocity. Average velocity (velocity over a length of time) is defined as
where Δr is the change in displacement and Δt is the interval of time over which displacement changes. As Δt becomes smaller and smaller, the value of v approaches the value of v.
The speed of an object is the magnitude |v| of its velocity. It is a scalar quantity.
[edit]AccelerationAcceleration is the vector quantity describing the rate of change with time of velocity. Instantaneous acceleration (the acceleration at an instant of time) is defined as:
where dv is an infinitesimally small change in velocity and dt is an infinitesimally small length of time. Average acceleration (acceleration over a length of time) is defined as:
where Δv is the change in velocity and Δt is the interval of time over which velocity changes. As Δt becomes smaller and smaller, a → a.
If acceleration is constant in magnitude and direction, 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.
[edit]Types of motionThere are two types of motion in general: uniform and non-uniform. Uniform motion implies constant velocity in a straight line. Non-uniform motion implies acceleration. If the acceleration changes in time, the rate of change of acceleration is called the jerk.
[edit]Integral relationsThe above definitions can be inverted by mathematical integration to find:
where the double integration is reduced to one integration by interchanging the order of integration, and subscript 0 signifies evaluation at t = 0(initial values).
[edit]Kinematics of constant accelerationMany physical situations can be modeled as constant-acceleration processes, such as projectile motion.
Integrating acceleration a with respect to time t gives the change in velocity. When acceleration is constant both in direction and in magnitude, the point is said to be undergoing uniformly accelerated motion. In this case, the integral relations can be simplified:
Additional relations between displacement, velocity, acceleration, and time can be derived. Since a = (v − v0)/t,
By using the definition of an average, this equation states that when the acceleration is constant average velocity times time equals displacement.
A relationship without explicit time dependence may also be derived for one-dimensional motion. By setting r0 = 0 and noting that at = v − v0,
where · denotes the dot product. Dividing the t on both sides and carrying out the dot-products:
In the case of straight-line motion, r is parallel to a, and r has magnitude equal to the path length s = r − r0 at time t. Then
This relation is useful when time is not known explicitly.
[edit]Relative velocityMain article: Relative velocityTo 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
Consequently, the position of A relative to B is
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 re-arranged:
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 relations between displacements become relations between velocities by simple time-differentiation, and a second differentiation makes them apply to accelerations.
For example, let Ann move with velocity relative to the reference (we drop the O subscript for convenience) and let Bob move with velocity , each velocity given with respect to the ground (point O). To find how fast Ann is moving relative to Bob (we call this velocity ), the equation above gives:
To find we simply rearrange this equation to obtain:
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.
[show]Example: Rectilinear (1D) motion[show]Example: Projectile (2D) motion[edit]Rotational motionMain article: Circular motionFigure 1: The angular velocity vector Ω points up for counterclockwise rotation and down for clockwise rotation, as specified by the right-hand rule. Angular position θ(t) changes with time at a rate ω(t) = dθ/dt.Rotational or angular kinematics is the description of the rotation of an object.[9] 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. The z-axis has been chosen for convenience.
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⊥(t) on a plane perpendicular to the axis of rotation. Then theangular position of that point is the angle θ from a reference axis (typically the positive x-axis) to the vector r⊥(t) in a known rotation sense (typically given by the right-hand rule).
Angular velocity: The angular velocity ω is the rate at which the angular position θ changes with respect to time t:
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 right-hand rule.
Angular acceleration: The magnitude of the angular acceleration α is the rate at which the angular velocity ω changes with respect to time t:
The equations of translational kinematics can easily be extended to planar rotational kinematics with simple variable exchanges:
Here θi and θf are, respectively, the initial and final angular positions, ωi and ωf are, respectively, the initial and final angular velocities, and α 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.
[edit]Point object in circular motionSee also: Rigid body and OrientationFigure 2: Velocity and acceleration for nonuniform circular motion: the velocity vector is tangential to the orbit, but the acceleration vector is not radially inward because of its tangential component aθ that increases the rate of rotation: dω/dt = |aθ|/R.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:
where uR is a unit vector pointing outward from the axis of rotation toward the periphery of the circle of motion, located at a radius R from the axis.
Linear velocity. The velocity of the object is then
The magnitude of the unit vector uR (by definition) is fixed, so its time dependence is entirely due to its rotation with the radius to the object, that is,
where uθ is a unit vector perpendicular to uR pointing in the direction of rotation, ω(t) is the (possibly time varying) angular rate of rotation, and the symbol × denotes the vector cross product. The velocity is then:
The velocity therefore is tangential to the circular orbit of the object, pointing in the direction of rotation, and increasing in time if ω increases in time.
Linear acceleration. In the same manner, the acceleration of the object is defined as:
[edit]Position, displacement, and distanceThe position of a point in space is its location relative to a chosen origin. It is a vector quantity, expressing both the distance of the point from the origin and its direction from the origin. In three-dimensions, the position of point A can be expressed with a three-dimensional vector
where xA, yA, and zA are the Cartesian coordinates of the point.
Displacement is a vector describing the difference in position between two points. If point A has position rA = (xA,yA,zA) and point B has position rB = (xB,yB,zB), the displacement rAB of B from A is given by
The distance traveled is always greater than or equal to the displacement.In physics, distance is a distinct quantity from either position or displacement. It is ascalar quantity, describing the length of the path between two points along which the particle has travelled. The straight-line distance |r| of point A from the origin can be found from the position r by
When considering the motion of a particle over time, distance is the length of the particle's path; displacement is the change from its initial position to its final position. For example, a race car traversing a 10 km closed loop from start to finish travels a distance of 10 km; its displacement, however, is zero because it arrives back at its initial position.
If the position of the particle is known as a function of time (r = r(t)), the distance d it travels from time t1 to time t2 can be found through calculus by
[edit]Velocity and speedVelocity is the measure of the rate of change in displacement with respect to time; that is, how the displacement of a point changes with time. Velocity also is a vector. Instantaneous velocity (the velocity at an instant of time) is defined through differential calculus as
where dr is an infinitesimally small displacement and dt is an infinitesimally small length of time.[note 2] As per its definition in the derivative form, velocity can be said to be the time rate of change of displacement. Further, as dr is tangential to the actual path, so is the velocity. Average velocity (velocity over a length of time) is defined as
where Δr is the change in displacement and Δt is the interval of time over which displacement changes. As Δt becomes smaller and smaller, the value of v approaches the value of v.
The speed of an object is the magnitude |v| of its velocity. It is a scalar quantity.
[edit]AccelerationAcceleration is the vector quantity describing the rate of change with time of velocity. Instantaneous acceleration (the acceleration at an instant of time) is defined as:
where dv is an infinitesimally small change in velocity and dt is an infinitesimally small length of time. Average acceleration (acceleration over a length of time) is defined as:
where Δv is the change in velocity and Δt is the interval of time over which velocity changes. As Δt becomes smaller and smaller, a → a.
If acceleration is constant in magnitude and direction, 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.
[edit]Types of motionThere are two types of motion in general: uniform and non-uniform. Uniform motion implies constant velocity in a straight line. Non-uniform motion implies acceleration. If the acceleration changes in time, the rate of change of acceleration is called the jerk.
[edit]Integral relationsThe above definitions can be inverted by mathematical integration to find:
where the double integration is reduced to one integration by interchanging the order of integration, and subscript 0 signifies evaluation at t = 0(initial values).
[edit]Kinematics of constant accelerationMany physical situations can be modeled as constant-acceleration processes, such as projectile motion.
Integrating acceleration a with respect to time t gives the change in velocity. When acceleration is constant both in direction and in magnitude, the point is said to be undergoing uniformly accelerated motion. In this case, the integral relations can be simplified:
Additional relations between displacement, velocity, acceleration, and time can be derived. Since a = (v − v0)/t,
By using the definition of an average, this equation states that when the acceleration is constant average velocity times time equals displacement.
A relationship without explicit time dependence may also be derived for one-dimensional motion. By setting r0 = 0 and noting that at = v − v0,
where · denotes the dot product. Dividing the t on both sides and carrying out the dot-products:
In the case of straight-line motion, r is parallel to a, and r has magnitude equal to the path length s = r − r0 at time t. Then
This relation is useful when time is not known explicitly.
[edit]Relative velocityMain article: Relative velocityTo 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
Consequently, the position of A relative to B is
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 re-arranged:
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 relations between displacements become relations between velocities by simple time-differentiation, and a second differentiation makes them apply to accelerations.
For example, let Ann move with velocity relative to the reference (we drop the O subscript for convenience) and let Bob move with velocity , each velocity given with respect to the ground (point O). To find how fast Ann is moving relative to Bob (we call this velocity ), the equation above gives:
To find we simply rearrange this equation to obtain:
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.
[show]Example: Rectilinear (1D) motion[show]Example: Projectile (2D) motion[edit]Rotational motionMain article: Circular motionFigure 1: The angular velocity vector Ω points up for counterclockwise rotation and down for clockwise rotation, as specified by the right-hand rule. Angular position θ(t) changes with time at a rate ω(t) = dθ/dt.Rotational or angular kinematics is the description of the rotation of an object.[9] 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. The z-axis has been chosen for convenience.
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⊥(t) on a plane perpendicular to the axis of rotation. Then theangular position of that point is the angle θ from a reference axis (typically the positive x-axis) to the vector r⊥(t) in a known rotation sense (typically given by the right-hand rule).
Angular velocity: The angular velocity ω is the rate at which the angular position θ changes with respect to time t:
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 right-hand rule.
Angular acceleration: The magnitude of the angular acceleration α is the rate at which the angular velocity ω changes with respect to time t:
The equations of translational kinematics can easily be extended to planar rotational kinematics with simple variable exchanges:
Here θi and θf are, respectively, the initial and final angular positions, ωi and ωf are, respectively, the initial and final angular velocities, and α 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.
[edit]Point object in circular motionSee also: Rigid body and OrientationFigure 2: Velocity and acceleration for nonuniform circular motion: the velocity vector is tangential to the orbit, but the acceleration vector is not radially inward because of its tangential component aθ that increases the rate of rotation: dω/dt = |aθ|/R.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:
where uR is a unit vector pointing outward from the axis of rotation toward the periphery of the circle of motion, located at a radius R from the axis.
Linear velocity. The velocity of the object is then
The magnitude of the unit vector uR (by definition) is fixed, so its time dependence is entirely due to its rotation with the radius to the object, that is,
where uθ is a unit vector perpendicular to uR pointing in the direction of rotation, ω(t) is the (possibly time varying) angular rate of rotation, and the symbol × denotes the vector cross product. The velocity is then:
The velocity therefore is tangential to the circular orbit of the object, pointing in the direction of rotation, and increasing in time if ω increases in time.
Linear acceleration. In the same manner, the acceleration of the object is defined as:
Coordinate systemsSee also: Generalized coordinates, Curvilinear coordinates, Orthogonal coordinates, and Frenet-Serret formulasIn 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 bepolar coordinates. Polar coordinates are extended into three dimensions with either the spherical polar or cylindrical polar coordinate systems. These are most useful in systems exhibiting spherical or cylindrical symmetry respectively.
[edit]Fixed rectangular coordinatesIn this coordinate system, vectors are expressed as an addition of vectors in the x, y, and z direction from a non-rotating origin. Usually i, j, kare unit vectors in the x-, y-, and z-directions.
The position vector, r (or s), the velocity vector, v, and the acceleration vector, a are expressed using rectangular coordinates in the following way:
Note: ,
[edit]Two dimensional rotating reference frameSee also: Centripetal forceThis coordinate system expresses only planar motion. It is based on three orthogonal unit vectors: the vector i, and the vector j which form abasis 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 non-rotating, the origin of these coordinates can rotate and translate - often following a particle on a body that is being studied.
[edit]Derivatives of unit vectorsThe 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:
[edit]Position, velocity, and accelerationGiven these identities, we can now figure out how to represent the position, velocity, and acceleration vectors of a particle using this reference frame.
[edit]PositionPosition is straightforward:
It is just the distance from the origin in the direction of each of the unit vectors.
[edit]VelocityVelocity is the time derivative of position:
By the product rule, this is:
Which from the identities above we know to be:
or equivalently
where vrel is the velocity of the particle relative to the rotating coordinate system.
[edit]AccelerationFirst to fourth derivatives of positionAcceleration is the time derivative of velocity.
We know that:
Consider the part. has two parts we want to find the derivative of: the relative change in velocity (), and the change in the coordinate frame
().
Next, consider . Using the chain rule:
from above:So all together:
And collecting terms:[10]
[edit]Kinematic constraintsThis section requires expansion.A kinematic constraint is any condition relating properties of a dynamic system that must hold true at all times. Below are some common examples:
[edit]Rolling without slippingAn 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,
.For the case of an object that does not tip or turn, this reduces to v = R ω.
[edit]Inextensible cordThis 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 is the total length, and accordingly the time derivative of this sum is zero. See Kelvin and Tait[11][12] and Fogiel.[13] A dynamic problem of this type is the pendulum. Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord.[14] An equilibrium problem (not kinematic) of this type is the catenary.[15]
[edit]Fixed rectangular coordinatesIn this coordinate system, vectors are expressed as an addition of vectors in the x, y, and z direction from a non-rotating origin. Usually i, j, kare unit vectors in the x-, y-, and z-directions.
The position vector, r (or s), the velocity vector, v, and the acceleration vector, a are expressed using rectangular coordinates in the following way:
Note: ,
[edit]Two dimensional rotating reference frameSee also: Centripetal forceThis coordinate system expresses only planar motion. It is based on three orthogonal unit vectors: the vector i, and the vector j which form abasis 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 non-rotating, the origin of these coordinates can rotate and translate - often following a particle on a body that is being studied.
[edit]Derivatives of unit vectorsThe 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:
[edit]Position, velocity, and accelerationGiven these identities, we can now figure out how to represent the position, velocity, and acceleration vectors of a particle using this reference frame.
[edit]PositionPosition is straightforward:
It is just the distance from the origin in the direction of each of the unit vectors.
[edit]VelocityVelocity is the time derivative of position:
By the product rule, this is:
Which from the identities above we know to be:
or equivalently
where vrel is the velocity of the particle relative to the rotating coordinate system.
[edit]AccelerationFirst to fourth derivatives of positionAcceleration is the time derivative of velocity.
We know that:
Consider the part. has two parts we want to find the derivative of: the relative change in velocity (), and the change in the coordinate frame
().
Next, consider . Using the chain rule:
from above:So all together:
And collecting terms:[10]
[edit]Kinematic constraintsThis section requires expansion.A kinematic constraint is any condition relating properties of a dynamic system that must hold true at all times. Below are some common examples:
[edit]Rolling without slippingAn 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,
.For the case of an object that does not tip or turn, this reduces to v = R ω.
[edit]Inextensible cordThis 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 is the total length, and accordingly the time derivative of this sum is zero. See Kelvin and Tait[11][12] and Fogiel.[13] A dynamic problem of this type is the pendulum. Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord.[14] An equilibrium problem (not kinematic) of this type is the catenary.[15]
