Question

derive the second equation of motion

Answer 1

Average velocity = Initial velocity + final velocity \ 2

Displacement = A.V × Time


$s \: = \frac{u + v}{2} \times t \\ s = \frac{u + u + at}{2} \times t \: \: \: \: \: v = u + at \\ s = \frac{ut + ut + {at}^{2} }{2} \\ s = \frac{2ut + {at}^{2} }{2} \\ s = \frac{2ut}{2} + \frac{ {at}^{2} }{2} \\ s = ut + \frac{1}{2} {at}^{2}$

Answer 2

Second Equation of Motion

Consider a body has a initial velocity 'u' , final velocity 'v' and acceleration 'a' and for time 't'.

Let the distance traveled by the body in time be 's'

$Average\:velocity=\dfrac{Initial+Final\:velocity}{2}$

$A.V=\dfrac{u+v}{2}$

Distance Travelled = Average Velocity ×Time

$S=\bigg(\dfrac{u+v}{2}\bigg)\times t$

From the first equation of motion we have

$v=u+at\\\\\\S=\bigg(\dfrac{u+u+at}{2}\bigg)\times t\\\\\\S=\bigg(\dfrac{2u+at}{2}\bigg)\times t\\\\\\S=\dfrac{2ut+at^2}{2}\\\\\\S=\dfrac{2ut}{2}+\dfrac{at^2}{2}\\\\\\S=ut+\dfrac{1}{2}\:at^2 ...........(2) Eq.\:of\:motion$