Linear motion

For the class of linkages, see straight line mechanism.

Linear motion (also called rectilinear 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 the most basic of all motion. According to Newton's first law of motion, objects that experience no net force will continue to move in a straight line with a constant velocity until they are subject 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 it's 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: 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.

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, the distance traveled by an object in particular direction is the same as displacement. The SI unit of displacement is the metre. If $\, x_{1}$ is the initial position of an object and $\, x_{2}$ is the final position, then mathematically the displacement is given by:

Δx = x₂ − x₁

The equivalent of displacement in rotational motion is the angular displacement θ measured in radian. The displacement of an object cannot be greater than the distance distance. To understand this, consider the journey you take to work and back everyday. Overall your displacement is zero, since you end up back where you started, but the distance you travel is clearly non zero.

Velocity

Main article: velocity

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

Average Velocity

The average velocity is the ratio of total displacement Δx taken over time interval Δt. Mathematically, it is given by:

$\mathbf{v_{av}} = \frac {\Delta x}{\Delta t} = \frac {x_2 - x_1}{t_2 - t_1}$

where,
t₁ is the time at which the object was at position x₁
t₂ is the time at which the object was at position x₂

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

Main article: speed

Speed is the absolute value of velocity i.e. 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 the instantaneous velocity is the 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 displacement i.e. acceleration can be found be differentiating position with respect to time twice or differentiating velocity with respect to time 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 Δt, 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 Δt as Δt approaches zero i.e.,

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

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

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

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

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

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

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 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 velocity.

Analogy between Linear Motion and Rotational motion

Analogy between Linear Motion and Rotational motion

Linear motion	Rotational motion
Displacement = $\mathbf{s}$	Angular displacement = θ
Velocity = $\mathbf{v}$	Angular velocity = ω
Acceleration = $\mathbf{a}$	Angular acceleration = α
Mass = $\mathbf{m}$	Moment of Inertia = $\mathbf{I}$
Force = $\mathbf{F} = \mathbf{m} \mathbf{a}$	Torque = $\Tau = \mathbf{I} \alpha$

See also

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

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.