The velocity  v  of a moving body is the time rate of change of its position in a specified direction,

          .                         (1)

The equality includes two time-dependent functions,  s(t)  and  v(t).
If  v = v(t)  is a given function of  t  then equality (1) is a directly integrable differential equation for the dependent variable  s = s(t), the general solution of which has the form

.

Let us consider a few particular cases.

1. Assume that the velocity of a body is a constant value  v. Then a displacement of the body is a linear function of  t:
   s(t) = v t + C.
   The value of the constant  C  is determined by an initial condition, e.g. by the position of the body at time  t = 0:  C = s(0) = s0.
   Thus,
   s(t) = v t + s0.

   Graphic Illustrations Using MATLAB.

2. In case of a uniformly accelerated motion of a body, the velocity is a linear function,
   v(t) = a t + v0,
   where  a  is the acceleration, and  v0  is an initial velocity (at  t = 0).
   Then
   .
   The value of the constant  C  is determined by the position of the body at time point  t = 0:
   C = s(0) = s0.
   Therefore,
   .

   Graphic Illustrations Using MATLAB.

Indefinite Integrals Definite Integrals Differential Equations