Linear motion

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Linear motion, also called rectilinear motion, is one-dimensional
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 ...
along a
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
, and can therefore be described mathematically using only one spatial
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
. The linear motion can be of two types: uniform linear motion, with constant
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 is a ...
(zero
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 the ...
); and non-uniform linear motion, with variable velocity (non-zero acceleration). The motion of a
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
(a point-like object) along a line can be described by its position $x$, which varies with $t$ (time). An example of linear motion is an athlete running a
100-meter dash The 100 metres, or 100-meter dash, is a sprint race in track and field competitions. The shortest common outdoor running distance, the dash is one of the most popular and prestigious events in the sport of athletics. It has been contest ...
along a straight track. Linear motion is the most basic of all motion. According to
Newton's first law of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
, objects that do not experience any
net force Net Force may refer to: * Net force, the overall force acting on an object * ''NetForce'' (film), a 1999 American television film * Tom Clancy's Net Force, a novel series * Tom Clancy's Net Force Explorers Tom Clancy's Net Force Explorers or Ne ...
will continue to move in a straight line with a constant velocity until they are subjected to a net force. Under everyday circumstances, external forces such as
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
and
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
can cause an object to change the direction of its motion, so that its motion cannot be described as linear. One may compare linear motion to general motion. In general motion, a particle's position and velocity are described by vectors, which have a magnitude and direction. In linear motion, the directions of all the vectors describing the system are equal and constant which means the objects move along the same axis and do not change direction. The analysis of such systems may therefore be simplified by neglecting the direction components of the vectors involved and dealing only with the
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
.

# Displacement

The motion in which all the particles of a body move through the same distance in the same time is called translatory motion. There are two types of translatory motions: rectilinear motion; curvilinear motion. Since linear motion is a motion in a single dimension, the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
traveled by an object in particular direction is the same as
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
. The SI unit of displacement is the
metre The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
. If $x_1$ is the initial position of an object and $x_2$ is the final position, then mathematically the displacement is given by: $\Delta x = x_2 - x_1$ The equivalent of displacement in
rotational motion Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
is the angular displacement $\theta$ measured in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s. The displacement of an object cannot be greater than the distance because it is also a distance but the shortest one. Consider a person travelling to work daily. Overall displacement when he returns home is zero, since the person ends up back where he started, but the distance travelled is clearly not zero.

# Velocity

Velocity refers to a displacement in one direction with respect to an interval of time. It is defined as the rate of change of displacement over change in time. Velocity is a vectorial quantity, representing a direction and a magnitude of movement. The magnitude of a velocity is called speed. The SI unit of speed is $\text\cdot \text^,$ that is
metre per second The metre per second is the unit of both speed (a scalar (physics), scalar quantity) and velocity (a Vector (mathematics and physics), vector quantity, which has direction and magnitude) in the International System of Units (SI), equal to the sp ...
.

## Average velocity

The average velocity of a moving body is its total displacement divided by the total time needed to reach a body from the initial point to the final point. It is an estimated velocity for a distance to travel. Mathematically, it is given by: $\mathbf_\text = \frac = \frac$ where: *$t_1$ is the time at which the object was at position $\mathbf_1$ and *$t_2$ is the time at which the object was at position $\mathbf_2$ The magnitude of the average velocity $\left, \mathbf_\text\$ is called an average speed.

## Instantaneous velocity

In contrast to an average velocity, referring to the overall motion in a finite time interval, the instantaneous velocity of an object describes the state of motion at a specific point in time. It is defined by letting the length of the time interval $\Delta t$ tend to zero, that is, the velocity is the time derivative of the displacement as a function of time. $\mathbf = \lim_ \frac = \frac .$ The magnitude of the instantaneous velocity $, \mathbf,$ is called the instantaneous speed.

# Acceleration

Acceleration is defined as the rate of change of velocity with respect to time. Acceleration is the second derivative of displacement i.e. acceleration can be found by differentiating position with respect to time twice or differentiating velocity with respect to time once. The SI unit of acceleration is $\mathrm$ or
metre per second squared The metre per second squared is the unit of acceleration in the International System of Units (SI). As a derived unit, it is composed from the SI base units of length, the metre, and time, the second. Its symbol is written in several forms as m/s ...
. If $\mathbf_\text$ is the average acceleration and $\Delta \mathbf = \mathbf_2 - \mathbf_1$ is the change in velocity over the time interval $\Delta t$ then mathematically, $\mathbf_\text = \frac = \frac$ The instantaneous acceleration is the limit, as $\Delta t$ approaches zero, of the ratio $\Delta \mathbf$ and $\Delta t$, i.e., $\mathbf = \lim_ \frac = \frac = \frac$

# Jerk

The rate of change of acceleration, the third derivative of displacement is known as jerk. The SI unit of jerk is $\mathrm$. In the UK jerk is also known as jolt.

# Jounce

The rate of change of jerk, the fourth derivative of displacement is known as jounce. The SI unit of jounce is $\mathrm$ which can be pronounced as ''metres per quartic second''.

# Equations of kinematics

In case of constant acceleration, the four
physical quantities A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
acceleration, velocity, time and displacement can be related by using the
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 (Ver ...
$\mathbf = \mathbf + \mathbf t$ $\mathbf = \mathbf \mathbf + \begin\frac\end \mathbf \mathbf^2$ $^2 = ^2 + 2 \mathbf$ $\mathbf = \tfrac \left(\mathbf + \mathbf\right) t$ here, *$\mathbf$ is the initial velocity *$\mathbf$ is the final velocity *$\mathbf$ is the acceleration *$\mathbf$ is the displacement *$t$ is the time These relationships can be demonstrated graphically. The
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of a line on a displacement time graph represents the velocity. The gradient of the velocity time graph gives the acceleration while the area under the velocity time graph gives the displacement. The area under a graph of acceleration versus time is equal to the change in velocity.

# Analogy with circular motion

The following table refers to rotation of a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
about a fixed axis: $\mathbf s$ is arclength, $\mathbf r$ is the distance from the axis to any point, and $\mathbf_\mathbf$ is the
tangential acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Accelerations are Euclidean vector, vector quantities (in that they have Magnitude (mathematics), magnitude and Direction ...
, which is the component of the acceleration that is ''parallel'' to the motion. In contrast, the
centripetal A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved trajectory, path. Its direction is always orthogonality, orthogonal to the motion of the body and towards the fixed po ...
acceleration, $\mathbf_\mathbf=v^2/r=\omega^2 r$, is ''perpendicular'' to the motion. The component of the force parallel to the motion, or equivalently, ''perpendicular'' to the line connecting the point of application to the axis is $\mathbf_\perp$. The sum is over $\mathbf j$ from $1$ to $N$ particles and/or points of application. The following table shows the analogy in derived SI units:

*
Angular motion In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of r ...
*
Centripetal force A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
*
Inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
*
Linear actuator A linear actuator is an actuator that creates motion in a straight line, in contrast to the circular motion of a conventional electric motor. Linear actuators are used in machine tools and industrial machinery, in computer Peripheral, periphera ...
*
Linear bearing A linear-motion bearing or linear slide is a bearing designed to provide free motion in one direction. There are many different types of linear motion bearings. Motorized linear slides such as machine slides, X-Y tables, roller tables and som ...
*
Linear motor A linear motor is an electric motor that has had its stator and rotor "unrolled", thus, instead of producing a torque (rotation), it produces a linear force along its length. However, linear motors are not necessarily straight. Characteristicall ...
*
Mechanics of planar particle motion This article describes a particle in planar motionSee for example, , when observed from non-inertial reference frames.''Fictitious forces'' (also known as a ''pseudo forces'', ''inertial forces'' or ''d'Alembert forces''), exist for observers i ...
* Motion graphs and derivatives *
Reciprocating motion Reciprocating motion, also called reciprocation, is a repetitive up-and-down or back-and-forth linear motion. It is found in a wide range of mechanisms, including reciprocating engines and pumps. The two opposite motions that comprise a single r ...
*
Rectilinear propagation Rectilinear propagation describes the tendency of electromagnetic waves (light) to travel in a straight line. Light does not deviate when travelling through a homogeneous medium, which has the same refractive index throughout; otherwise, light su ...
* Uniformly accelerated linear motion

# References

* Resnick, Robert and Halliday, David (1966), ''Physics'', Chapter 3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 * Tipler P.A., Mosca G., "Physics for Scientists and Engineers", Chapter 2 (5th edition), W. H. Freeman and company: New York and Basing stoke, 2003.