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Linear motion

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Type of motion in which the path of the moving object is a straight line

For the class of linkages, seestraight line mechanism.

Classical mechanics
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${\textbf {F}}={\frac {d\mathbf {p} }{dt}}$ Second law of motion
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Linear motion, also calledrectilinear motion,^[1] is one-dimensionalmotion along astraight line, and can therefore be described mathematically using only one spatialdimension. The linear motion can be of two types:uniform linear motion, with constantvelocity (zeroacceleration); andnon-uniform linear motion, with variable velocity (non-zero acceleration). The motion of aparticle (a point-like object) along a line can be described by its position $x {\displaystyle x}$ , whichvaries with $t {\displaystyle t}$ (time). An example of linear motion is an athlete running a100-meter dash along a straight track.^[2]

Linear motion is the most basic of all motion. According toNewton's first law of motion, objects that do not experience anynet force 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 asgravity andfriction can cause an object to change the direction of its motion, so that its motion cannot be described as linear.^[3]

One may compare linear motion to general motion. In general motion, a particle's position and velocity are described byvectors, 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 themagnitude.^[2]

Background

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Displacement

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Main article:Displacement (vector)

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, thedistance traveled by an object in particular direction is the same asdisplacement.^[4] TheSI unit of displacement is themetre.^[5]^[6] 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 inrotational motion is theangular displacement $\theta$ measured inradians.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

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Main articles:Velocity andSpeed

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.^[7] Velocity is a vector quantity, representing a direction and a magnitude of movement. The magnitude of a velocity is called speed. The SI unit of speed is ${\text{m}}\cdot {\text{s}}^{-1},$ that ismetre per second.^[6]

Average velocity

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Theaverage velocity of a moving body is its total displacement divided by the total time needed to travel from the initial point to the final point. It is an estimated velocity for a distance to travel. Mathematically, it is given by:^[8]^[9]

$\mathbf {v} _{\text{avg}}={\frac {\Delta \mathbf {x} }{\Delta t}}={\frac {\mathbf {x} _{2}-\mathbf {x} _{1}}{t_{2}-t_{1}}}$

where:

$t_{1}$ is the time at which the object was at position $\mathbf {x} _{1}$ and
$t_{2}$ is the time at which the object was at position $\mathbf {x} _{2}$

The magnitude of the average velocity $\left|\mathbf {v} _{\text{avg}}\right|$ is called an average speed.

Instantaneous velocity

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In contrast to an average velocity, referring to the overall motion in a finite time interval, theinstantaneous 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.

$\mathbf {v} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {x} }{\Delta t}}={\frac {d\mathbf {x} }{dt}}.$

The magnitude of the instantaneous velocity $|\mathbf {v} |$ is called the instantaneous speed. The instantaneous velocity equation comes from finding the limit as t approaches 0 of the average velocity. The instantaneous velocity shows the position function with respect to time. From the instantaneous velocity the instantaneous speed can be derived by getting the magnitude of the instantaneous velocity.

Acceleration

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Main article: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.^[10] The SI unit of acceleration is $\mathrm {m\cdot s^{-2}}$ ormetre per second squared.^[6]

If $\mathbf {a} _{\text{avg}}$ is the average acceleration and $\Delta \mathbf {v} =\mathbf {v} _{2}-\mathbf {v} _{1}$ is the change in velocity over the time interval $\Delta t$ then mathematically, $\mathbf {a} _{\text{avg}}={\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {\mathbf {v} _{2}-\mathbf {v} _{1}}{t_{2}-t_{1}}}$

The instantaneous acceleration is the limit, as $\Delta t$ approaches zero, of the ratio $\Delta \mathbf {v}$ and $\Delta t$ , i.e., $\mathbf {a} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {x} }{dt^{2}}}$

Jerk

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Main article:Jerk (physics)

The rate of change of acceleration, the third derivative of displacement is known as jerk.^[11] The SI unit of jerk is $\mathrm {m\cdot s^{-3}}$ . In the UK jerk is also referred to as jolt.

Jounce

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Main article:Jounce

The rate of change of jerk, the fourth derivative of displacement is known as jounce.^[11] The SI unit of jounce is $\mathrm {m\cdot s^{-4}}$ which can be pronounced asmetres per quartic second.

Formulation

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Main article:Equations of motion

In case of constant acceleration, the fourphysical quantities acceleration, velocity, time and displacement can be related by using theequations of motion.^[12]^[13]^[14]

\mathbf {v} _{\text{f}}=\mathbf {v} _{\text{i}}+\mathbf {a} t

\mathbf {d} =\mathbf {v} _{\text{i}}t+{\frac {1}{2}}\mathbf {a} t^{2}

\mathbf {v} _{\text{f}}^{2}=\mathbf {v} _{\text{i}}^{2}+2\mathbf {ad}

\mathbf {d} ={\frac {t}{2}}\left(\mathbf {v} _{\text{f}}+\mathbf {v} _{\text{i}}\right)

Here,

$\mathbf {v} _{\text{i}}$ is the initial velocity
$\mathbf {v} _{\text{f}}$ is the final velocity
$\mathbf {a}$ is acceleration
$\mathbf {d}$ is displacement
$t {\displaystyle t}$ is time

These relationships can be demonstrated graphically. Thegradient 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.

Comparison to rotational motion

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The following table refers to rotation of arigid body about a fixed axis: $\mathbf {s}$ isarc length, $\mathbf {r}$ is the distance from the axis to any point, and $\mathbf {a} _{\mathbf {t} }$ is thetangential acceleration, which is the component of the acceleration that isparallel to the motion. In contrast, thecentripetal acceleration, $\mathbf {a} _{\mathbf {c} }=v^{2}/r=\omega ^{2}r$ , isperpendicular to the motion. The component of the force parallel to the motion, or equivalently,perpendicular to the line connecting thepoint of application to the axis is $\mathbf {F} _{\perp }$ . The sum is over $j {\displaystyle j}$ from $1 {\displaystyle 1}$ to $N {\displaystyle N}$ particles and/or points of application.

Analogy between Linear Motion and Rotational motion^[15]
Linear motion	Rotational motion	Defining equation
Displacement = $\mathbf {x}$	Angular displacement = $\theta$	$\theta =\mathbf {s} /\mathbf {r}$
Velocity = $\mathbf {v}$	Angular velocity = $\omega$	$\omega =\mathbf {v} /\mathbf {r}$
Acceleration = $\mathbf {a}$	Angular acceleration = $\alpha$	$\alpha =\mathbf {a_{\mathbf {t} }} /\mathbf {r}$
Mass = $\mathbf {m}$	Moment of Inertia = $\mathbf {I}$	${\textstyle \mathbf {I} =\sum _{j}\mathbf {m} _{j}\mathbf {r} _{j}^{2}}$
Force = $\mathbf {F} =\mathbf {m} \mathbf {a}$	Torque = $\tau =\mathbf {I} \alpha$	${\textstyle \tau =\sum _{j}\mathbf {r} _{j}\mathbf {F} _{\perp j}}$
Momentum= $\mathbf {p} =\mathbf {m} \mathbf {v}$	Angular momentum= $\mathbf {L} =\mathbf {I} \omega$	${\textstyle \mathbf {L} =\sum _{j}\mathbf {r} _{j}\mathbf {p} _{j}}$
Kinetic energy = ${\textstyle {\frac {1}{2}}\mathbf {m} \mathbf {v} ^{2}}$	Kinetic energy = ${\textstyle {\frac {1}{2}}\mathbf {I} \omega ^{2}}$	${\textstyle {\frac {1}{2}}\sum _{j}\mathbf {m} _{j}\mathbf {v} _{j}^{2}={\frac {1}{2}}\sum _{j}\mathbf {m} _{j}\mathbf {r} _{j}^{2}\omega ^{2}}$

The following table shows the analogy in derived SI units:

Classical mechanics SI units

Dimensions	1	L	L²	Dimensions	1	θ	θ²
Linear/translational quantities				Angular/rotational quantities
T	time:t s	absement:A m s		T	time:t s
1		distance:d,position:r,s,x,displacement m	area:A m²	1		angle:θ,angular displacement:θ rad	solid angle:Ω rad², sr
T⁻¹	frequency:f s⁻¹,Hz	speed:v,velocity:v m s⁻¹	kinematic viscosity:ν, specific angular momentum: h m² s⁻¹	T⁻¹	frequency:f,rotational speed:n,rotational velocity:n s⁻¹,Hz	angular speed:ω,angular velocity:ω rad s⁻¹
T⁻²		acceleration:a m s⁻²		T⁻²	rotational acceleration s⁻²	angular acceleration:α rad s⁻²
T⁻³		jerk:j m s⁻³		T⁻³		angular jerk:ζ rad s⁻³
M	mass:m kg	weighted position:M ⟨x⟩ = ∑mx	moment of inertia: I kg m²	ML
MT⁻¹	Mass flow rate: ${\dot {m}}$ kg s⁻¹	momentum:p,impulse:J kg m s⁻¹,N s	action:𝒮,actergy:ℵ kg m² s⁻¹,J s	MLT⁻¹		angular momentum:L,angular impulse:ΔL kg m rad s⁻¹
MT⁻²		force:F,weight:F_g kg m s⁻²,N	energy:E,work:W,Lagrangian:L kg m² s⁻²,J	MLT⁻²		torque:τ,moment:M kg m rad s⁻²,N m
MT⁻³		yank:Y kg m s⁻³, N s⁻¹	power:P kg m² s⁻³, W	MLT⁻³		rotatum:P kg m rad s⁻³, N m s⁻¹