Question

The potential energy of a particle varies with $x$ (position) according to relation $U=27x^3-64x$. We get maximum speed at $x=$

• A. $\frac{8}{9}$ units
• B. $\frac{3}{4}$ units
• C. $\frac{12}{7}$ units
• D. $\frac{5}{9}$ units

Answer 1

The correct option is A $\frac{8}{9}$ units

Kinetic energy and speed are maximum when potential energy is minimum
$\frac{dU}{dx}=81x^2-64=0$
$x^2=\frac{64}{81}$
$x=\pm \frac{8}{9}$
for $x=\frac{8}{9}$, $\frac{d^{2}u}{d{x}^2}=162x-0=162\left(\frac{8}{9}\right)=+ve$
at $x=\frac{8}{9}$, $\frac{du}{dx}=0$ & $\frac{d^{2}u}{d{x}^2}=+ve$
$\therefore$ U is min at $x=\frac{8}{9}$