Question

Prove that S=ut+1/2at 2

Answer 1

acceleration is defined as change in velocity per unit time.

           =>   (v - u) / (t - 0)  = a        =>  v = u + at

Distance traveled is = average velocity * time duration

Answer 2

To Prove: ${ S }={ ut }+\frac { 1 }{ 2 } { at }^{ 2 }$

We know that, $Average\quad Velocity\quad =\quad \frac { total\quad distance\quad travelled }{ total\quad time\quad taken }$

$Average\quad Velocity\quad =\quad \frac { S }{ t }$

Average velocity is also written as ${ u }+{ \frac { v }{ 2 } }$

By equating both the above equations

$\frac { S }{ t } =u+\frac { v }{ 2 }$

We know that , "first equation of motion" v = u+at

By substituting the value of 'v' in equation 1, we get

$\frac { S }{ t } ={ u }+\frac { { u }+{ at } }{ 2 }$

By simplifying, $S=\left( \frac { 2u+at }{ 2 } \right) t$

$S=\frac { 2ut+a{ t }^{ 2 } }{ 2 }$

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

The above equation is the second equation of motion.