Question

$\bf Namaste \: Everyone!$
$\sf State \: the \: Law \: related \: to \: \\ \sf following \: Formula/Equation$
$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \sf s=ut + \frac{1}{2} {at}^{2} }$
$\sf \rightarrow \: \: Derive \: the \: given \: equation$
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Answer 1

$\huge\underline{\sf Answers:}$

${\sf The \: Formula/Equation \: is \: related \: to \: the \: Second \: Law \: of \: Motion \: which \: defines \: that \: the \: rate \: of \: change \: of \: momentum \: of \: an \: object \: is \: proportional \: to \: the \: applied \: unbalanced \: force \: in \: the \: direction \: of \: the \: force.}$

$\large{\sf Derivation \: of \: the \: Equation}$

${\sf We \: know, \: acceleration \: a = \frac{change in velocity}{time taken}}$

${ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{v - u}{t} }$

${ \sf \: or \: at \: = v - u}$

${ \sf \: or \: v \: = \: u + at}$

${\sf An \: object \: is \: moving \: with \: initial \: velocity \: u \: and \: uniform \: acceleration \: a \: for \: time \: t. \: As \: the \: object \: is \: moving \: with \: uniform \: acceleration, \: its \: velocity \: is \: non-uniform. \: After \: time \: t, \: the \: velocity \: of \: the \: object \: will \: be \: v = u + at}$

${ \sf \: average \: velocity \: of \: the \: body \: = \frac{v + u}{2} }$

${ \sf \: So \: the \: path \: travelled \: s \: = average \: velocity \times time \: t}$

${ \sf \: \: \: \: \: \: \: \: = \frac{v + u}{2} \times t}$

${ \sf \: \: \: \: \: \: \: \: = \frac{u + at + u}{2} \times t}$

${ \sf \: \: \: \: \: \: \: \: = \frac{2u + at}{2} \times t}$

${ \sf \: \: \: \: \: \: \: \: = ut + \frac{1}{2} {at}^{2} }$

${ \sf \: Hence, \: proved}.$