Question

Derive the equation for position time relation(S=ut+1/2at2) through

graphical representation?

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Answer 1

To derive:

The 2nd equation of kinematics through graphical method :

$\boxed{ \bf{s = ut + \dfrac{1}{2} a {t}^{2} }}$

Derivation:

We know that area under the velocity time graph will give us displacement :

Let displacement be denoted as "s" :

$\rm{ \therefore \: s = area \: under \: v - t \: graph}$

$\rm{ = > \: s = area \: of \: trapezium}$

$\rm{ = > \: s = \dfrac{1}{2}(sum \: of \: parallel \: sides)(distance) }$

$\rm{ = > \: s = \dfrac{1}{2}(v + u)(t) }$

Now , expressing "v" in terms of "u" and acceleration:

$\rm{ = > \: s = \dfrac{1}{2} \{(u + at) + u \}(t) }$

$\rm{ = > \: s = \dfrac{1}{2} \{2u + at\}(t) }$

$\rm{ = > \: s = \dfrac{1}{2} \{2u t+ a {t}^{2} \} }$

$\rm{ = > \: s = ut+ \dfrac{1}{2} a {t}^{2} }$

[Hence derived].

HOPE IT HELPS.