Linear motion, also called rectilinear motion,[1] is 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 (zero acceleration); and non-uniform linear motion, with variable velocity (non-zero acceleration). The motion of a particle (a point-like object) along a line can be described by its position
x
t
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]
See 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, 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
x1
x2
\theta
See main article: Velocity and Speed. 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
m ⋅ s-1,
The average 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]
where:
t1
x1
t2
x2
\left|vavg\right|
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
\Deltat
The magnitude of the instantaneous velocity
|v|
See 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
m ⋅ s-2 |
If
aavg
\Deltav=v2-v1
\Deltat
The instantaneous acceleration is the limit, as
\Deltat
\Deltav
\Deltat
See 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
m ⋅ s-3 |
The rate of change of jerk, the fourth derivative of displacement is known as jounce.[11] The SI unit of jounce is
m ⋅ s-4 |
See main article: Equations of motion. In case of constant acceleration, the four physical quantities acceleration, velocity, time and displacement can be related by using the equations of motion.[12] [13] [14]
vf=vi+at
d=vit+
1 | |
2 |
at2
2 | |
v | |
f |
=
2 | |
v | |
i |
+2ad
d=
t | |
2 |
\left(vf+vi\right)
Here,
vi
vf
a
d
t
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 a graph of acceleration versus time is equal to the change in velocity.
The following table refers to rotation of a rigid body about a fixed axis:
s
r
at
ac=v2/r=\omega2r
F\perp
j
1
N
Linear motion | Rotational motion | Defining equation | |
---|---|---|---|
Displacement = x | Angular displacement = \theta | \theta=s/r | |
Velocity = v | Angular velocity = \omega | \omega=v/r | |
Acceleration = a | Angular acceleration = \alpha | \alpha=
| |
Mass = m | Moment of Inertia = I | ||
Force = F=ma | Torque = \tau=I\alpha | ||
Momentum= p=mv | Angular momentum= L=I\omega | ||
Kinetic energy = | Kinetic energy = | ||