If the path of the motion of a particle is a straight line, it is called ‘Rectilinear motion

Fig. 1.2 shows the movement of a particle r moving along a straight line. O’ is the initial position of Pat t = 0 and S =S₀. P is the position at time t and P’ is the position at time t+ ∆t

Displacement in time                         ∆t =∆S

Displacement m unit time =∆S /∆t

∆S /∆t =v _av [Average Velocity]. In the limiting case as ∆t à 0, we have

v = Lt .∆S /∆t= dS /dt , where v is called instantaneous velocity.

v = dS/dt =S

v = v _{o ­}at t =0 (i.e., at initial point 0,)

The average acceleration during ∆t is defined as

a _{av =}∆v / ∆t. In  the limiting case

where a is called the instantaneous acceleration of the particle.

a = dv /dt =v

-a is called instantaneous deceleration or retardation

Thus ,              a d²S / dt = S

From (1.1)                               dt = dS /v and from (1.2) we have

dt = dv / a. thus ,dt = dS /v = dv /a

v dv =a dS