Question

derive s=ut+1/2at 2
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Answer 1

⇒To prove s=ut + 1/2at² first we have to prove v = u +at

Derivation

⇒ For one dimensional motion with a = constant

We can write as               dv = a dt

Integrating both sides , we have

$\rm \implies \int dv = a \int dt$

At t = 0 , velocity is u and at t = t velocity is v . hence

$\rm\implies\int\limits^v_u {} \, dv=a\int\limits^t_0 {} \, dt$

$\rm\implies [v]^v_u= a[t]^t_o$

$\implies\rm v-u=at$

$\boxed{\rm v=u +at}$

we can write as           ds = v dt

= (u +at)dt

At time t=0 suppose s = 0 and at t = t , displacement is s , the

$\rm \implies \int\limits^s_0 {} \, ds=\int\limits^t_0 {} \,(u + at) dt$

$\rm\implies [s]^s_0=\bigg[ut +\dfrac{1}{2} at^{2} \bigg]^t_0$

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

Hence proved