**Second Equation of Dimensional Motion**

In one dimensional motion a body moves along a straight line. So quantities associated with motion, for example displacement, velocity, acceleration etc., have only one component (moving along X-axis will have X-component and Y and Z components will be zero). In deriving equations of linear motion we will consider that the body is moving along X-axis. In that case subscripts associated with different quantifies of motion may be omitted. Normally, v_{x} will be represented by v and a_{x} by a.

**Second Equation:**

**Equation of motion relating final velocity, acceleration and time (v = v _{0} + at or v_{x} = v_{x0} + a_{x}t) **

Derivation: Let an object’s velocity at time, t = 0 be represented by v_{0} and at a later time t, its velocity is v.

Now the change of velocity is v – v_{0} during a time t – 0; its average acceleration during that time is

a = (v – v_{0})/( t – 0)

Since the object is moving with uniform acceleration, so “a” is equal to the instantaneous acceleration “a”.

so, a = (v – v_{0})/ t

or, (v – v_{0}) = at

or, v = v_{0} + at … … … (1)

If initial velocity of the object is zero; i.e., the object starts from rest, then v_{0} = 0. So equation (1) become

v = at

For one dimensional motion, say along X-axis, let, v_{0} = v_{x0}; v = v_{x} and a = a_{x}; then we get from equation (1):

**v _{x} = v_{x0} + a_{x}t**

Similarly, for motion along Y and Z-axis equation (1) reduces to,

**v _{y} = v_{y0} + a_{y}t**

**v _{z} = v_{z0} + a_{z}t**

Instead of uniform acceleration if the objects moves with uniform retardation, then “-a” should be inserted in the above equation.