When a particle moves in a straight line with constant acceleration, then the position , time , velocity and acceleration of the particle are represented by equation knowns as kinematic Equations of motion.
(i) Velocity attained after time t :
The velocity-time graph for positive constant acceleration of a particle is shown in fiqure .
Let u be the initial velocity of the particle at t=0 and v is final velocity of the particle after time t .
Consider two point A and B on the curve corresponding to t=0 and t =t respectively. Draw BD perpendicular on time axis . Also draw AC perpendicular on BD.
OA = CD = u ; BC = ( v-u) and OD = t
Now Slope of v - t graph = acceleration (a)
\[a = Slope \;of \;v-t \;graph\]\[= tan \Theta = \frac{BC}{AC} = \frac{BC}{OD}\]
\[a = \frac{v-u}{t}\] \[or \;v-u=at\]
\[v=u+at\]
Which is the expression for the velocity atttained by uniformly accelerated particle in time t.
(ii) Distance travelled in time interval t :
Let xo= Position of the particle at t=0 from the origin.
x = position of the particle at t=t from the origin.
Therefore distance travelled by a particle in the given time interval = area under velocity time graph
\[-x_{0} = Area \;OABD\]
\[= \frac{1}{2}(OA+BD)\times AC\]
\[= \frac{1}{2}(v_{0}+v) \times t\]
\[v=v_{0}+at\]