Linear motion

Classical mechanics
${\vec {F}}=m{\vec {a}}$ Second law of motion
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Formulations Newton's laws of motion Analytical mechanics Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton–Jacobi equation Appell's equation of motion Udwadia–Kalaba equation Koopman–von Neumann mechanics
Core topics Damping (ratio) Displacement Equations of motion Euler's laws of motion Fictitious force Friction Harmonic oscillator Inertial / Non-inertial reference frame Mechanics of planar particle motion Motion (linear) Newton's law of universal gravitation Newton's laws of motion Relative velocity Rigid body dynamics Euler's equations Simple harmonic motion Vibration
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Scientists Galileo Newton Kepler Horrocks Halley Euler d'Alembert Clairaut Lagrange Laplace Hamilton Poisson Daniel Bernoulli Johann Bernoulli Cauchy

Linear motion (also called rectilinear motion^[1]) is a one dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; non uniform linear motion with variable velocity or non-zero acceleration. The motion of a particle (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 100m along a straight track.^[2]

Linear motion is the most basic of all motion. According to Newton's first law of motion, objects that do not experience any net 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 as gravity and friction 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 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.^[2]

Neglecting the rotation and other motions of the Earth, an example of linear motion is the ball thrown straight up and falling back straight down.

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 traveled by an object in particular direction is the same as displacement.^[4] The SI unit of displacement is the metre.^[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 in rotational motion is the angular displacement $\theta$ measured in radian. 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 = displacement / time. Velocity is defined as the rate of change of displacement with respect to time.^[7] The SI unit of velocity is $ms^{{-1}}$ or metre per second.^[6]

Average velocity

The average velocity is the ratio of total displacement $\Delta x$ taken over time interval $\Delta t$ . Mathematically, it is given by:^[8]^[9]

${\mathbf {v_{{av}}}}={\frac {\Delta x}{\Delta t}}={\frac {x_{2}-x_{1}}{t_{2}-t_{1}}}$

where:
$t_{1}$ is the time at which the object was at position $x_{1}$
$t_2$ is the time at which the object was at position $x_{2}$

Instantaneous velocity

The instantaneous velocity can be found by differentiating the displacement with respect to time.

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

Speed

Speed is the absolute value of velocity i.e. speed is always positive. The unit of speed is metre per second.^[10] If $v$ is the speed then,

$v=\left|{\mathbf {v}}\right|=\left|{{\frac {dx}{dt}}}\right|$

The magnitude of the instantaneous velocity is 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.^[11] The SI unit of acceleration is $ms^{{-2}}$ or metre per second squared.^[6]

If ${\mathbf {a_{{av}}}}$ is the average acceleration and $\Delta {\mathbf {v}}={\mathbf {v_{2}}}-{\mathbf {v_{1}}}$ is the average velocity over the time interval $\Delta t$ then mathematically,

${\mathbf {a_{{av}}}}={\frac {\Delta {\mathbf {v}}}{\Delta t}}={\frac {{\mathbf {v_{2}}}-{\mathbf {v_{1}}}}{t_{2}-t_{1}}}$

The instantaneous acceleration is the limit of the ratio $\Delta {\mathbf {v}}$ and $\Delta t$ as $\Delta t$ approaches zero i.e.,

${\mathbf {a}}=\lim _{{\Delta t\to 0}}{\Delta {\mathbf {v}} \over \Delta t}$ $={\frac {d{\mathbf {v}}}{dt}}={\frac {d^{2}x}{dt^{2}}}$

Jerk

The rate of change of acceleration, the third derivative of displacement is known as jerk.^[12] The SI unit of jerk is $ms^{{-3}}$ . 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.^[12] The SI unit of jounce is $ms^{{-4}}$ which can be pronounced as metres per quartic second.

Equations of kinematics

In case of constant acceleration, the four physical quantities acceleration, velocity, time and displacement can be related by using the Equations of motion^[13]^[14]^[15]

\mathbf {V_{f}} =\mathbf {V_{i}} +\mathbf {a} \mathbf {t} \;\!

\mathbf {d} =\mathbf {V_{i}} \mathbf {t} +{\begin{matrix}{\frac {1}{2}}\end{matrix}}\mathbf {a} \mathbf {t} ^{2}

{\mathbf {V_{f}} }^{2}={\mathbf {V_{i}} }^{2}+2{\mathbf {a} }\mathbf {d}

\mathbf {d} ={\tfrac {1}{2}}\left(\mathbf {V_{f}} +\mathbf {V_{i}} \right)\mathbf {t}

here,
$\mathbf {V_{i}}$ is the initial velocity
$\mathbf {V_{f}}$ is the final velocity
$\mathbf{a}$ is the acceleration
$\mathbf {d}$ is the displacement
${\mathbf {t}}$ is the time

These relationships can be demonstrated graphically. The gradient 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 an acceleration time graph gives the change in velocity.

Analogy between linear and rotational motion

The following table refers to rotation of a rigid body about a fixed axis: $\mathbf {s}$ is arclength, $\mathbf {r}$ is the distance from the axis to any point, and ${\mathbf {a}}_{{\mathbf {t}}}$ is the tangential acceleration, which is the component of the acceleration that is parallel to the motion. In contrast, the centripetal acceleration, ${\mathbf {a}}_{{\mathbf {c}}}=v^{2}/r=\omega ^{2}r$ , 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 ${\mathbf {F}}_{\perp }$ . The sum is over ${\mathbf j}\ =1\ {\mathbf {to}}\ N$ particles and/or points of application.

Analogy between Linear Motion and Rotational motion^[16]
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 = ${\mathbf {I}}$	${\mathbf {I}}=\sum {\mathbf {m_{j}}}{\mathbf {r_{j}}}^{2}$
Force = ${\mathbf {F}}={\mathbf {m}}{\mathbf {a}}$	Torque = $\tau ={\mathbf {I}}\alpha$	$\tau =\sum {\mathbf {r_{j}}}{\mathbf {F}}_{\perp }{\mathbf {_{j}}}$
Momentum= ${\mathbf {p}}={\mathbf {m}}{\mathbf {v}}$	Angular momentum= ${\mathbf L}={\mathbf {I}}\omega$	${\mathbf L}=\sum {\mathbf {r_{j}}}{\mathbf {p}}{\mathbf {_{j}}}$
Kinetic energy = ${\frac 12}{\mathbf {m}}{\mathbf {v}}^{2}$	Kinetic energy = ${\frac 12}{\mathbf {I}}\omega ^{2}$	${\frac 12}\sum {\mathbf {m_{j}}}{\mathbf {v}}^{2}={\frac 12}\sum {\mathbf {m_{j}}}{\mathbf {r_{j}}}^{2}\omega ^{2}$

The following table shows the analogy in derived SI units:

Classical mechanics SI units

Dimensions	1	L	L²	Dimensions	1	1	1
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: $f$ s⁻¹, Hz	angular speed: $ω$ , angular velocity: $ω$ rad s⁻¹
T⁻²		acceleration: $a$ m s⁻²		T⁻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			ML²	moment of inertia: $I$ kg m²
MT⁻¹		momentum: $p$ , impulse: $J$ kg m s⁻¹, N s	action: $𝒮$ , actergy: $ℵ$ kg m² s⁻¹, J s	ML²T⁻¹		angular momentum: $L$ , angular impulse: $Δ L$ kg m² s⁻¹	action: $𝒮$ , actergy: $ℵ$ kg m² s⁻¹, J s
MT⁻²		force: $F$ , weight: $F g$ kg m s⁻², N	energy: $E$ , work: $W$ kg m² s⁻², J	ML²T⁻²		torque: $τ$ , moment: $M$ kg m² s⁻², N m	energy: $E$ , work: $W$ kg m² s⁻², J
MT⁻³		yank: $Y$ kg m s⁻³, N s⁻¹	power: $P$ kg m² s⁻³, W	ML²T⁻³		rotatum: $P$ kg m² s⁻³, N m s⁻¹	power: $P$ kg m²s⁻³, W

References

↑ Resnick, Robert and Halliday, David (1966), Physics, Section 3-4
1 2 "Basic principles for understanding sport mechanics".
↑ "Motion Control Resource Info Center". Retrieved 19 January 2011.
↑ "Distance and Displacement".
↑ "SI Units".
1 2 3 "SI Units".
↑ "Speed & Velocity".
↑ "Average speed and average velocity".
↑ "Average Velocity, Straight Line".
↑ "Speed and velocity".
↑ "Acceleration".
1 2 "What is the term used for the third derivative of position?".
↑ "Equations of motion" (PDF).
↑ "Description of Motion in One Dimension".
↑ "What is derivatives of displacement?".
↑ "Linear Motion vs Rotational motion" (PDF).

External links

Media related to Linear movement at Wikimedia Commons

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