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In what follows we will
describe the motion of a body moving under constant aceleration a_{0}.
That might f.x. be a body making a free fall. 
If we look at the (t,v)graph for such a motion we end up with a straight line, because the increase in velocity per unit time is a constant. The acceleration a_{0} is determined as the slope of the line, as we know that v'(t) = a.

From the fact that we are dealing with a linear function of velocity we can write down the velocity equation
v(t)= v_{0} + a_{0}^{.}t ,
where v_{0} is the initial velocity.
We wish to determine the function of position s(t) of the motion.
From what we have seen in the preceeding section we know that the covered distance Ds at the time t is equal to the area lying under the (t,v)graph.
Fortunately the graph of velocity is a straight line, making it possible for us to determine the area with the help of some simple geometrical considerations.

Start!
2. Slice the area in two. The total area is the sum of the rectangular area and the triangular area. 3. Put on notation. Now, we know that v(t) concists of two terms, and as the height of the rectangle is v_{0} , then the height of the triangle must be given by the second term  that is a_{0}^{.}t. 
In other words the covered distance Ds can be calculated as
Ds = ½^{.}a_{0}^{.}t^{2} + v_{0}^{.}t
The covered distance Ds only expresses the change in the bodys position.
The final positionen is s(t), and if we denotes the initial positionen s_{0}, we get
Ds = s(t)  s_{0} = ½^{.}a_{0}^{.}t^{2} + v_{0}^{.}t
or
s(t) = ½^{.}a_{0}^{.}t^{2} + v_{0}^{.}t + s_{0} .
The graph of position for the motion under constant acceleration is a parabola.
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 Sorry, no translations.
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