0Linear motion is motion along a straight line, and can therefore be described mathematically using only one spatial dimension. It can be uniform, that is, with constant velocity (zero acceleration), or non-uniform, that is, with a variable velocity (non-zero acceleration). The motion of a particle (a point-like object) along the line can be described by its position x, which varies with t (time).
An example of linear motion is that of a ball thrown straight up and falling back straight down.
The average velocity v during a finite time span of a particle undergoing linear motion is equal to v = Px/Pt, where Px is the total displacement and Pt denotes the time needed.
The instantaneous velocity of a particle in linear motion may be found by differentiating the position x with respect to the time variable t: v = dx/dt. The acceleration may be found by differentiating the velocity: a = dv/dt. By the fundamental theorem of calculus the converse is also true: to find the velocity when given the acceleration, simply integrate the acceleration with respect to time; to find displacement, simply integrate the velocity with respect to time.
This 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.
