Physics - Error Question with Solution | TestHub

Given initial velocity = v₀

final velocity = 0

$a=-{\text{x}}^2$

now we know that$a=\frac{\text{dv}}{\text{dt}}$and$\text{v}=\frac{\text{dx}}{\text{dt}}$

thus$\text{a}=\frac{\text{dv}}{\text{dt}}=\frac{\text{dv}}{\text{dt}}·\frac{\text{dx}}{\text{dx}}=\frac{\text{vdv}}{\text{dx}}$

$\frac{\text{vdv}}{\text{dx}}=-{\text{x}}^2$

$\text{vdv}=-{\text{x}}^2\text{dx}$

integrating we get

$\underset{{\text{v}}_0}{\overset{0}{\int }}\text{vdv}=\underset{0}{\overset{\text{x}}{\int }}-{\text{x}}^2\text{dx}$

${\left(\frac{{\text{v}}^2}{2}\right)}_{{\text{v}}_0}^0=-{\left[\frac{{\text{x}}^3}{3}\right]}_0^{\text{x}}$

$-\frac{{\text{v}}_0^2}{2}=-\frac{{\text{x}}^3}{3} \Rightarrow \text{x}={\left(\frac{3{\text{v}}_0^2}{2}\right)}^{\frac{1}{3}}$