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

ò*dv* = ò*adt*

*v *= *at *+ *c*1 where *c*1 = constant of integration

If *v*0 be the initial velocity of the particle then at *t *= 0, *v* = *v*0. This implies that *c*1 = *v*0

*v* = *v*0 + *at* … (1)

Also we know that

or *dx *= (*v*0 + *at*)*dt*

On integrating, we get

ò*dx *= ò*v*0*dt* + ò*atdt*

*x *= *v0t *+ *at*2 + *c*2 where *c*2 = constant

If *x*0 be the initial position of the particle, then at *t *= 0; *x *= *x0*. This implies that *c*2 = *x*0

*x *= *x*0 + *v*0*t* + *at*2 … (2)

Eliminating *t *from equations (1) and (2), we get

*v*2 = … (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 *= *v*0 + *at* (1)

*x *= *x0 *+ *v0t *+ (2)

*v*2 = (3)

*x *= *x*0 + (*v*0 + *v*)*t* (4)

where *x0 *= initial position coordinate; *x *= Final position coordinate

*v *= final velocity; *a *= acceleration (constant)

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