Linear motion

For the class of linkages, see straight line mechanism.

Linear motion is 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). Linear motion is also called as rectilinear motion.

Linear motion is the most basic of all the motions. According to Newton's first law of motion, objects that not subjected to forces will continue to move uniformly in a straight line indefinitely. Under every-day circumstances, external forces such as gravity and friction will cause objects to deviate from linear motion and can cause them to rest at a point.. For linear motion embedded in a higher-dimensional space, the velocity and acceleration should be described as vectors, made up of two parts: magnitude and direction. The direction part of these vectors is the same and is constant for linear motion.

An example of linear motion is that of a ball thrown straight up and falling back straight down.

Displacement

Main article: displacement

Since Linear motion is a motion in a single dimension, distance traveled by an object is same as displacement. The SI unit of displacement is metre. If $\,x_{1}$ is the Initial position of an object and $\,x_{2}$ is the Final position, then mathematically displacement is given by putang ina mo,

$\Delta x=x_{2}-x_{1}$

The equivalent of displacement in rotational motion is the angular displacement $\theta$ measured in radian.

Velocity

Main article: velocity

Velocity is defined as the rate of change of displacement with respect to the time. The SI unit of velocity is $ms^{-1}$ or metre per second.

Average Velocity

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

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

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 position with respect to time.

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

Speed

Main article: speed

Speed is the absolute value of velocity i.e., whatever may be the sign of velocity speed is always positive. This unit of speed is same as that of velocity. If $v$ is the speed then,

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

The magnitude of instantaneous velocity is instantaneous speed.

Acceleration

Main article: acceleration

Acceleration is defined as the rate of change of velocity with respect to time. Acceleration is the second derivative of the position i.e., acceleration can be found be differentiating position with respect to time twice or differentiating velocity with respect to time only once. This SI unit of acceleration is $ms^{-2}$ or metre per second squared.

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.,

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

Equations of kinematics

Main article: Equations of motion

The four physical quantities acceleration, velocity, time and displacement can be related by using the Equations of motion

$\mathbf {v} =\mathbf {u} +\mathbf {a} \mathbf {t} \;\!$

$\mathbf {s} =\mathbf {u} \mathbf {t} +{\begin{matrix}{\frac {1}{2}}\end{matrix}}\mathbf {a} \mathbf {t} ^{2}$

${\mathbf {v} }^{2}={\mathbf {u} }^{2}+2{\mathbf {a} }\mathbf {s}$

$\mathbf {a} ={\tfrac {1}{2}}\left(\mathbf {v} +\mathbf {u} \right)\mathbf {t}$

Here,
$\mathbf {u}$ is the initial velocity
$\mathbf {v}$ is the final velocity
$\mathbf {a}$ is the acceleration
$\mathbf {s}$ is the displacement
$\mathbf {t}$ is the time

These relationships can be demonstrated graphically. The gradient of a line on the 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 velocity.

Analogy between Linear Motion and Rotational motion

Analogy between Linear Motion and Rotational motion

Linear motion	Rotational motion
Displacement = $\mathbf {s}$	Angular displacement = $\theta$
Velocity = $\mathbf {v}$	Angular velocity = $\omega$
Acceleration = $\mathbf {a}$	Angular acceleration = $\alpha$
Mass = $\mathbf {m}$	Moment of Inertia = $\mathbf {I}$
Force = $\mathbf {F} =\mathbf {m} \mathbf {a}$	Torque = $\mathrm {T} =\mathbf {I} \alpha$

See also

• Motion graphs and derivatives
• Jounce
• Jerk (physics)
• Uniformly accelerated linear motion
• Linear motor

Further Reading

• 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.