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, ao,
a = dv/dt = ao = constant
By integrating, the velocity can be determined as a function of the
acceleration and time, giving
Next, the velocity can be integrated to express the position as a function of the acceleration and time, giving,
x(t) = xo + vo (t
- to) + ao (t - to)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.
It should be stressed that above equations are only valid when the acceleration is expressed as a constant.