In this section, we are going to derive relations among displacement, velocity, acceleration and time.
We know that,
dv = adt
On integrating both sides, we get
òdv = òadt
v = at + c1 where c1 = constant of integration
If v0 be the initial velocity of the particle then at t = 0, v = v0. This implies that c1 = v0
v = v0 + at … (1)
Also we know that
or dx = (v0 + at)dt
On integrating, we get
òdx = òv0dt + òatdt
x = v0t + at2 + c2 where c2 = constant
If x0 be the initial position of the particle, then at t = 0; x = x0. This implies that c2 = x0
x = x0 + v0t + at2 … (2)
Eliminating t from equations (1) and (2), we get
v2 = … (3)
The equation (2) may be rewritten as
From equation (1), we have
thus … (4)
The equations (1) to (4) are called the equations of kinematics in the direction along the x-axis.
The equations of kinematics are summarized as
v = v0 + at (1)
x = x0 + v0t + (2)
v2 = (3)
x = x0 + (v0 + v)t (4)
where x0 = initial position coordinate; x = Final position coordinate
v = final velocity; a = acceleration (constant)
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