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Encyclopedia > Kinematics

Kinematics (Greek κινειν,kinein, to move) is a branch of mechanics which describes the motion of objects without the consideration of the masses or forces that bring about the motion. In contrast, dynamics is concerned with the forces and interactions that produce or affect the motion. For other uses, see Mechanic (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... This article or section is in need of attention from an expert on the subject. ... In physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. ... In physics, dynamics is the branch of classical mechanics that is concerned with the effects of forces on the motion of objects. ... Interaction is a kind of action that occurs as two or more objects have an effect upon one another. ...

Kinematics studies how the position of an object changes with time. Position is measured with respect to a set of coordinates. Velocity is the rate of change of position. Acceleration is the rate of change of velocity. Velocity and Acceleration are the two principal quantities which describe how position changes. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In physics, velocity is defined as the rate of change of displacement or the rate of displacement. ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ...

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 (rigid-body 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. A good example of kinematics in real life can be found at [1] In classical mechanics, translational kinematics is the science of describing the motion of a point particle, possibly under constraints. ... In geometry a rotation representation expresses the orientation of an object (or coordinate frame) relative to a coordinate reference frame. ... In classical mechanics, rigid-body kinematics is the science of describing the translational and rotational motion of a rigid body, possibly under constraints. ... This article is about a joint in zootomical anatomy. ...

## Translational motion

Translational (or linear) kinematics is the description of the motion in space of a point along a trajectory (which can be rectilinear or curved) and involves the definition and use of the following three quantities:

(Linear) Position: (Linear) Velocity: (Linear) Acceleration: (to be written)

### Relative velocity

Main article: Relative velocity

To describe the motion of object A with respect to object O, when we know how each is moving with respect to object B, we use the following equation involving vectors and vector addition: Relative velocity is a measurement of velocity between two objects moving in different frames of reference. ...

$r_{A/O} = r_{B/O} + r_{A/B} ,!$

The above relative motion equation states that the motion of A relative to O is equal to the motion of B relative to O plus the motion of A relative to B.

For example, let Ann move with velocity VA and let Bob move with velocity VB, each velocity given with respect to the ground. To find how fast Ann is moving relative to Bob (we call this velocity VA / B), the equation above gives:

$V_{A} = V_{B} + V_{A/B} ,! .$

To find VA / B we simply rearrange this equation to obtain:

$V_{A/B} = V_{A} -V_{B} ,! .$

At velocities comparable to the speed of light, these equations of relative motions are found through Einstein's theory of special relativity rather than the above equation of relative motion. â€œLightspeedâ€ redirects here. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ...

### Equations of uniformly accelerated motion

An object moving with constant acceleration is said to be undergoing uniformly accelerated motion (UAM). Its motion can be described with four simple algebraic equations:

$,x_f - x_i = v_i t + frac{1}{2} at^2 qquad x_f - x_i = frac{1}{2} (v_f + v_i)t$
$,v_f = v_i + a t qquad v_f^2 = v_i^2 + 2 a (x_f - x_i)$

where vi and vf are the initial and final velocities, xi and xf are the initial and final positions on a reference axis, a is the constant acceleration, and t is the timespan between the initial and final positions.

### Example: Rectilinear (1D) motion

An object is fired upwards, reaches its apex, and then begins its descent under a constant acceleration of -9.81 m/s2.

Consider an object which is fired directly upwards and falls back to the ground so that its trajectory is contained in a straight line. If we adopt the convention that the upward direction is the positive direction, the object experiences a constant acceleration of approximately -9.81 m/s2. Therefore, its motion can be modeled with the equations governing uniformly accelerated motion. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

There are several interesting kinematic questions we can ask about the particles motion: How long will it be airborne? What altitude will it reach before it begins to fall? What will its final velocity be when it reaches the ground? For the sake of example, assume the object has an initial velocity of +50 m/s.

#### How long will it be airborne?

To answer this question, we apply the formula

$x_f - x_i = v_i t + frac{1}{2} at^2.$

Since the question asks for the length of time between the object leaving the ground and hitting the ground on its fall, the displacement is zero.

$0 = v_i t + frac{1}{2} at^2 = t(v_i + frac{1}{2} at)$

We find two solutions for it. The trivial solution says the time is zero; this is actually also true, it is the first moment the displacement is zero: just when it starts motion. However, the solution of interest is

$t = -frac{2v_i}{a} = -frac{2*50}{-9.81} = 10.2 s$

#### What altitude will it reach before it begins to fall?

In this case, we use the fact that the object has a velocity of zero at the apex of its trajectory. Therefore, the applicable equation is:

$v_f^2 = v_i^2 + 2 a (x_f - x_i)$

If the origin of our coordinate system is at the ground, then xi is zero. Then we solve for xf and substitute known values:

$x_f = frac{v_f^2 - v_i^2}{2 a} + x_i = frac{0-50^2}{2*-9.81}+0 = 127.55 m$

#### What will its final velocity be when it reaches the ground?

To answer this question, we use the fact that the object has an initial velocity of zero at the apex before it begins its descent. We can use the same equation we used for the last question, using the value of 127.55 m for xi.

$v_f = sqrt{v_i^2 + 2 a (x_f - x_i)} = sqrt{0^2 + 2 (-9.81) (0 - 127.55)} = 50 m/s$

We find that the final and initial speeds are equal, a result which agrees with conservation of energy. In physics, the conservation of energy states that the total amount of energy in an isolated system remains constant, although it may change forms, e. ...

### Example: Projectile (2D) motion

An object fired at an angle θ from the ground follows a parabolic trajectory.

Suppose that an object is not fired vertically but is fired at an angle θ from the ground. The object will then follow a parabolic trajectory, and its horizontal motion can be modeled independently of its vertical motion. Assume that the object is fired at an initial velocity of 50 m/s and 30 degrees from the horizontal. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

#### How far will it travel before hitting the ground?

The object experiences an acceleration of -9.81 m/s2 in the vertical direction and no acceleration in the horizontal direction. Therefore, the horizontal displacement is

$Delta x = x_f - x_i = v_i cos theta t + frac{1}{2} at^2 = v_i cos theta t$

In order to solve this equation, we must find t. This can be done by analyzing the motion in the vertical direction. If we impose that the vertical displacement is zero, we can use the same procedure we did for rectilinear motion to find t.

$0 = v_i sin theta t + frac{1}{2} at^2 = t(v_i sin theta + frac{1}{2} at)$

We now solve for t and substitute this expression into the original expression for horizontal displacement. (Note the use of the trigonometric identity 2sinθcosθ = sin2θ) In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

$Delta x = v_i cos theta left(frac{-2 v_i sin theta}{a}right) = -frac{v_i^2 sin 2theta}{a} = 220.70 m$

## Rotational motion

The angular velocity vector points up for counterclockwise rotation and down for clockwise rotation, as specified by the right-hand rule.

Rotational kinematics is the description of the rotation of an object and involves the definition and use of the following three quantities: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The left-handed orientation is shown on the left, and the right-handed on the right. ...

Angular position: If a vector is defined as the oriented distance from the axis of rotation to a point on an object, the angular position of that point is the oriented angle &theta from a reference axis (e.g. the positive x-semiaxis) to that vector. An oriented angle is an angle swept about a known rotation axis and in a known rotation sense. In two-dimensional kinematics (the description of planar motion), the rotation axis is normal to the reference frame and can be represented by a rotation point (or center), and the rotation sense is represented by the sign of the angle (typically, a positive sign means counterclockwise sense). Angular displacement can be regarded as a relative position. It is represented by the oriented angle swept by the above mentioned point (or vector), from an angular position to another.

Angular velocity: The magnitude of the angular velocity ω is the rate at which the angular position θ changes with respect to time t:

$mathbf{omega} = frac {mathrm{d}theta}{mathrm{d}t}$

Angular acceleration: The magnitude of the angular acceleration α is the rate at which the angular velocity ω changes with respect to time t:

$mathbf{alpha} = frac {mathrm{d}mathbf{omega}}{mathrm{d}t}$

The equations of translational kinematics can easily be extended to planar rotational kinematics with simple variable exchanges:

$,!theta_f - theta_i = omega_i t + frac{1}{2} alpha t^2 qquad theta_f - theta_i = frac{1}{2} (omega_f + omega_i)t$
$,!omega_f = omega_i + alpha t qquad alpha = frac{omega_f - omega_i}{t} qquad omega_f^2 = omega_i^2 + 2 alpha (theta_f - theta_i)$

.

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.

## 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. A constraint is a limitation of possibilities. ... A central force acting on an object is one whose magnitude depends only on the scalar distance r of the object from the origin and whose direction is along the position vector from the origin to the object. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

### Fixed rectangular coordinates

In 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 is a unit vector in the x direction, j is a unit vector in the y direction, and k is a unit vector in the z direction. In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

The position vector, s (or r), the velocity vector, v, and the acceleration vector, a are expressed using rectangular coordinates in the following way: Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ...

$vec s = x vec i + y vec j + z vec k , !$

$vec v = dot {s} = dot {x} vec {i} + dot {y} vec {j} + dot {z} vec {k} , !$

$vec a = ddot {s} = ddot {x} vec {i} + ddot {y} vec {j} + ddot {z} vec {k} , !$

Note: $dot {x} = frac{mathrm{d}x}{mathrm{d}t}$ , $ddot {x} = frac{mathrm{d}^2x}{mathrm{d}t^2}$

### Two dimensional rotating reference frame

This coordinate system only expresses planar motion.

This system of coordinates 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 non rotating, the origin of these coordinates can rotate and translate - often following a particle on a body that is being studied. In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, a basis or set of generators is a collection of objects that can be systematically combined to produce a larger collection of objects. ...

#### 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, we must take the derivatives of the unit vectors into account when taking the derivative of any of these vectors. If the coordinate frame is rotating at a rate of ω in the counterclockwise direction (that's ωk using the right hand rule) then the derivatives of the unit vectors are as follows: The right hand rule is also an algorithm used to solve Mazes In mathematics and physics, the right-hand rule is a convention for determining relative directions of certain vectors. ...

$dot{vec i} = omega vec k times vec i = omega vec j$

$dot{vec j} = omega vec k times vec j = - omega vec i$

#### 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. A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ...

##### Position

Position is straightforward:

$vec s = x vec j$

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:

$vec v = frac{mathrm{d}vec s}{mathrm{d}t} = frac{mathrm{d} (x vec i)}{mathrm{d}t} + frac{mathrm{d} (y vec j)}{mathrm{d}t}$

By the product rule, this is: In mathematics, the product rule of calculus, also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

$vec v = dot x vec i + x dot{vec i} + dot y vec j + y dot{vec j}$

Which from the identities above we know to be:

$vec v = dot x vec i + x omega vec j + dot y vec j - y omega vec i = (dot x - y omega) vec i + (dot y + x omega) vec j$

or equivalently

$vec v = (dot x vec i + dot y vec j) + (y dot{vec j} + x dot{vec i}) = vec v_{rel} + vec omega times vec r$

where $vec v_{rel}$ is the velocity of the particle relative to the coordinate system.

##### Acceleration

Acceleration is the time derivative of velocity.

We know that:

$vec a = frac{mathrm{d} vec v}{mathrm{d}t} = frac{mathrm{d} vec v_{rel}}{mathrm{d}t} + frac{mathrm{d} (vec omega times vec r)}{mathrm{d}t}$

Consider the $frac{mathrm{d} vec v_{rel}}{mathrm{d}t}$ part. $vec v_{rel}$ has two parts we want to find the derivative of: the relative change in velocity ($vec a_{rel}$), and the change in the coordinate frame ($omega times vec v_{rel}$).

$frac{mathrm{d} vec v_{rel}}{mathrm{d}t} = vec a_{rel} + omega times vec v_{rel}$

Next, consider $frac{mathrm{d} (vec omega times vec r)}{mathrm{d}t}$. Using the chain rule:

$frac{mathrm{d} (vec omega times vec r)}{mathrm{d}t} = dot{vec omega} times vec r + vec omega times dot{vec r}$

$dot{vec r}$ we know from above:

$frac{mathrm{d} (vec omega times vec r)}{mathrm{d}t} = dot{vec omega} times vec r + vec omega times (vec omega times vec r) + vec omega times vec v_{rel}$

So all together:

$vec a = vec a_{rel} + omega times vec v_{rel} + dot{vec omega} times vec r + vec omega times (vec omega times vec r) + vec omega times vec v_{rel}$

And collecting terms:

$vec a = vec a_{rel} + 2(omega times vec v_{rel}) + dot{vec omega} times vec r + vec omega times (vec omega times vec r)$

(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, : An open surface with X-, Y-, and Z-contours shown. ... In physics, velocity is defined as the rate of change of displacement or the rate of displacement. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... For the cross product in algebraic topology, see KÃ¼nneth theorem. ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ...

$v_G(t) = omega times r_{G/O} ,!$

For the case of an object that does not tip or turn, this reduces to v = R ω .

### 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.

Look up kinematics in
Wiktionary, the free dictionary.

Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... Inverse kinematics is the process of determining the movement of interconnected segments of a body or model. ... This article or section is in need of attention from an expert on the subject. ...

Results from FactBites:

 Kinematics - Wikipedia, the free encyclopedia (959 words) In physics, kinematics is the branch of mechanics concerned with the motions of objects without being concerned with the forces that cause the motion. Because of its relative simplicity, kinematics is usually taught before dynamics or the concept of a force is introduced. One fundamental equation in kinematics is the equation for the derivative of a vector described in a rotating frame of reference.
More results at FactBites »

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