Using calcus method prove that v²-u²=2as

1.	Using calcus method prove that v²-u²=2as
Answer» We know that, acceleration, $\displaystyle\longrightarrow\sf{a=\dfrac{dv}{dt}}$ $\displaystyle\longrightarrow\sf{a=\dfrac{dv}{dx}\cdot\dfrac{dx}{dt}}$ $\displaystyle\longrightarrow\sf{a=\dfrac{dv}{dx}\cdot v}$ $\displaystyle\longrightarrow\sf{a\ dx=v\ dv}$ Now, before integration, integration is done for displacement from x = 0 to x = s. integration is done for velocity from u to v. Then, $\displaystyle\longrightarrow\sf{\int\limits_0^sa\ dx=\int\limits_u^vv\ dv}$ SINCE acceleration is independent of displacement (MOTION is not SIMPLE HARMONIC), $\displaystyle\longrightarrow\sf{a\big[x\big]_0^s=\left[\dfrac{v^2}{2}\right]_u^v}$ $\displaystyle\longrightarrow\sf{a(s-0)=\dfrac{v^2-u^2}{2}}$ $\displaystyle\longrightarrow\sf{as=\dfrac{v^2-u^2}{2}}$ $\displaystyle\longrightarrow\sf{\underline{\underline{v^2-u^2=2as}}}$ HENCE the Proof!

Answer»

We know that, acceleration,

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