Question

$\LARGE{\boxed {\mathscr {Question:-}}}$

Prove that

$\sf s=ut+\dfrac {1}{2}at^2$​

Answer 1

Question:-

Prove that :-

$\sf s=ut+\dfrac {1}{2}at^2$

Required Answer;-

Let us assume that an article has a initial velocity 'u' and a uniform acceleration 'a' for time 't' so that its final velocity becomes 'v'.

And let the distance travelled by the article be 's' . The distance travelled by a moving body in time 't' can be found out by considering its average velocity.

Since the initial velocity of the body is 'u' and its final velocity is 'v', the average velocity is given by :-

$\\ \sf{\blue{\bold{\bar{v} = \frac{u + a}{2} }}} - - - - - - - - (1)$

From the first equation of motion, we know that:-

$\\\sf{\bold{v}} = u + at$

Putting the value of 'v' in equation 1 ,

$\\ \sf{\bold{s = ( \frac{u + u + at}{t} )t}}$

$\\\sf{\implies{\sf{\bold{s} = ( \frac{2u + at}{2})t}}}$

$\\\sf{\implies{\bold{s} = \frac{2ut + a {t}^{2} }{2} }}$

$\\\sf{\implies{\bold{s} = \frac{\cancel{2}ut}{\cancel{2}} + \frac{a {t}^{2} }{2}}}$

$\\\sf{\implies{\bold{s} = ut + \frac{at {}^{2} }{2}}}$

$\\\sf{\therefore{\bold{\red{s = ut + \frac{1}{2} a {t}^{2} }}}}$

where,

• s= distance travelled by the article in time t
• u= initial velocity of the article
• a= acceleration

$\\\\\rm{\bold{\green{Hence, \: proved!!}}}$

Answer 2

$\huge\textit{Question :-}$

➦ acceleration is defined as change in velocity per unit time.

★$\underline{\bf{=> (v - u) / (t - 0) = a => v = u + at}}$★

➦ Distance traveled is = average velocity * time duration

$\begin{gathered} S = \frac{(initial + final\: velocity)}{2} * duration \\ \\ S = \frac{u + v}{2} * t \\ \\ S = \frac{1}{2} (u + u + a t ) * t \\ \\ S = \frac{2ut + a t^2}{2} \\ \\ S = u t + \frac{1}{2} a t^2 \\ \\ \end{gathered}$

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Thankyou :)