Linear motion
Encyclopedia
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
, which varies with 
(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.
traveled by an object in particular direction is the same as displacement
. The SI
unit of displacement is the metre
. If
is the initial position of an object and 
is the final position, then mathematically the displacement is given by:
The equivalent of displacement in rotational motion is the angular displacement
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.
unit of velocity is
or metres per second
.
taken over time interval 
. Mathematically, it is given by:
where,
is the time at which the object was at position 
is the time at which the object was at position 
If
is the speed then,
The magnitude of the instantaneous velocity is the instantaneous speed.
unit of acceleration is
or metre per second squared
.
If
is the average acceleration and 
is the average velocity over the time interval 
, then mathematically,
The instantaneous acceleration is the limit of the ratio
and 
as 
approaches zero i.e.,
Here,
is the initial velocity
is the final velocity
is the acceleration
is the displacement
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.
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...
along a straight line
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
, and can therefore be described mathematically using only one spatial dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
. 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
, which varies with (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
Since linear motion is a motion in a single dimension, the distanceDistance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
traveled by an object in particular direction is the same as displacement
Displacement (vector)
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...
. The SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
unit of displacement is the metre
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...
. If
is the initial position of an object and is the final position, then mathematically the displacement is given by:The equivalent of displacement in rotational motion is the angular displacement
measured in radianRadian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
.
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
Velocity is defined as the rate of change of displacement with respect to time. The SISi
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
unit of velocity is
or metres per secondMetre per second
Metre per second is an SI derived unit of both speed and velocity , defined by distance in metres divided by time in seconds....
.
Average Velocity
The average velocity is the ratio of total displacement taken over time interval . Mathematically, it is given by:where,
is the time at which the object was at position is the time at which the object was at position Instantaneous Velocity
The Instantaneous Velocity can be found by differentiating the displacement with respect to time. 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
is the speed then,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 be differentiating position with respect to time twice or differentiating velocity with respect to time once. This SISi
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
unit of acceleration is
or metre per second squaredMetre per second squared
The metre per second squared is the unit of acceleration in the International System of Units . As a derived unit it is composed from the SI base units of length, the metre, and the standard unit of time, the second...
.
If
is the average acceleration and is the average velocity over the time interval , then mathematically,The instantaneous acceleration is the limit of the ratio
and as approaches zero i.e., Equations of kinematics
The four physical quantities acceleration, velocity, time and displacement can be related by using the Equations of motionHere,
is the initial velocity is the final velocity is the acceleration is the displacement is the timeThese relationships can be demonstrated graphically. The gradient
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
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
| Linear motion | Rotational motion |
|---|---|
Displacement =  |
Angular displacement =  |
Velocity =  |
Angular velocity =  |
Acceleration =  |
Angular acceleration =  |
Mass =  |
Moment of Inertia =  |
Force =  |
Torque =  |
See also
- Motion graphs and derivativesMotion graphs and derivativesIn mechanics, the derivative of the position vs. time graph of an object is equal to the velocity of the object. In the International System of Units, the position of the moving object is measured in meters relative to the origin, while the time is measured in seconds...
- Jounce
- Jerk (physics)
- Uniformly accelerated linear motion
- Linear motorLinear motorA linear motor is an electric motor that has had its stator and rotor "unrolled" so that instead of producing a torque it produces a linear force along its length...
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.