A body is said to experience rectilinear motion if any two particles of the body travel the same distance along two parallel straight lines. The figures below illustrate rectilinear motion for a particle and body.

Rectilinear motion for a particle:

Rectilinear motion for a body:

In the above figures,

Given the position of the particles,

A common assumption, which applies to numerous problems involving rectilinear motion, is that acceleration is constant. With acceleration as constant we can derive equations for the position, displacement, and velocity of a particle, or body experiencing rectilinear motion.

The easiest way to derive these equations is by using Calculus.

The acceleration is given by

where

Integrate the above equation with respect to time, to obtain velocity. This gives us

where

Integrate the above equation with respect to time, to obtain position. This gives us

where

The constants

At time

At time

Substituting these two initial conditions into the above two equations we get

Therefore

This gives us

For convenience, set

Displacement is defined as Δ

If we wish to find an equation that doesn’t involve time

Equations (1), (2), (3), and (4) fully describe the motion of particles, or bodies experiencing rectilinear (straight-line) motion, where acceleration

For the cases where acceleration is not constant, new expressions have to be derived for the position, displacement, and velocity of a particle. If the acceleration is known as a function of time, we can use Calculus to find the position, displacement, and velocity, in the same manner as before.

Alternatively, if we are given the position

For example, let's say the position

Thus, the velocity

The acceleration

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