Question

A block of mass $2kg$ is kept at origin at $t=0$ and is having velocity $4\sqrt{5}m/s$ in positive $x-$ direction. The only force on it is a conservative and its potential energy is defined as $U=-x^3+6x^2+15$ ( $SI$ units). Its velocity when the force acting on it is minimum (after the time $t=0$ ) is

• A. $8m/s$
• B. $4m/s$
• C. $10\sqrt{24}m/s$
• D. none of these

Answer 1

Answer: (A)

At $x=0$
$K=\frac{1}{2}\times 2\times 80=80J$ and $U=15J$
$\therefore$ Total energy is,
$E=K+U=95J$
Force, $F=\frac{-dU}{dx}$
$\Rightarrow F=3x^2-12x$
For $F$ to be minimum, $\frac{dF}{dx}=0$
$\Rightarrow x=2m$
At $x=2m$
$E=K+U$
$\Rightarrow 95=\frac{1}{2}\times 2\times v^2+\left(-8+24+15\right)$
$\Rightarrow v=8m/s$