Derive the second equation of motion by graphical method.

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Solution

Step 1: To prove:
$S = u t + \frac{1}{2} a t^{2}$
Step 2: Derivation of the equation of motion by the graphical method:
From the graph above, we can say that
$D i s \tan c e t r a v e l l e d (s) = A r e a o f f i g u r e O A B C = A r e a o f r e c \tan g l e O A D C + A r e a o f t r i a n g l e A B D$
$T h e r e f o r e S = (O A \times O C) + \frac{1}{2} (A D \times B D)$
Since, $B D = E A$ , the above equation becomes
$S = (O A \times O C) + \frac{1}{2} (A D \times E A) o r S = u t + \frac{1}{2} t (E O - A O) (∵ E A = E O - A O) o r S = u t + \frac{1}{2} t (v - u) o r S = u t + \frac{1}{2} t (a t) (∵ v = u + a t o r v - u = a t) o r S = u t + \frac{1}{2} a t^{2}$
Hence the second equation of motion $S = u t + \frac{1}{2} a t^{2}$ has been derived graphically.

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Derive the following equation of motion by the graphical method :

$v^{2} = u^{2} + 2 a$

where the symbols have their usual meanings.

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