Kinematic Equation/Relation For Uniformly Accelerated motion - Graphical method

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

\[-x_{0} = Area \;OABD\]

\[= \frac{1}{2}(OA+BD)\times AC\]

\[= \frac{1}{2}(v_{0}+v) \times t\]

\[v=v_{0}+at\]


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