Many times, the motion of an object is specified by an acceleration that is constant with time. For example, an object that falls for a short distance in the earth's (or any other planet's) atmosphere experiences a constant acceleration. That constant is written as, a_{o},
a = dv/dt = a_{o} = constant
By integrating, the velocity can be determined as a function of the
acceleration and time, giving

v(t) = v_{o} + a_{o} (t
 t_{o})


Next, the velocity can be integrated to express the position as a function of the acceleration and time, giving,

x(t) = x_{o} + v_{o} (t
 t_{o}) + a_{o} (t  t_{o})^{2}
/ 2


Using the chain rule, the acceleration can be expressed as
This relationship can be integrated to express the velocity as a function of the acceleration and position.

v^{2} = v_{o}^{2} +
2 a_{o} (x  x_{o})


It should be stressed that above equations are only valid when the acceleration is expressed as a constant.
