Question

Derive the first equation of motion ​

Answer 1

v =u + at .

We know, acceleration(a) = Change in velocity / Time taken

= Final velocity (v) - Initial velocity (u) / Time taken (t)

So, a = (v-u)/t

=> at = v-u

=> v = u + at

Thus, the first equation of motion is derived

Answer 2

$\huge{\underline{\sf{AnswEr-}}}$

First equation of motion using differentiation and integration -

Diagram -

t = 0⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀t = t

_______⠀⠀⠀S⠀⠀_________

________________________

u⠀⠀⠀⠀a = constant⠀⠀⠀⠀⠀⠀v

Derivation -

$\sf{a = \dfrac{dv}{dt}}$

Integrating it,

$\implies$ $\sf{\displaystyle\int\limits_{u}^{v} dv = \displaystyle\int\limits_{0}^{t} a.dt}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\implies$ $\sf{[ \: v \: ]_u^v = a [ \: t \: ]_0^t}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\implies$ $\sf{(v - u) = a [t - 0]}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\implies$ $\sf{(v - u) = at}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\implies$ $\sf{\blue{v = u + at}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Hence, we get the required expression.