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