Question

Using calcus method prove that v²-u²=2as

Answer 1

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!