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  4. A block of mass 2 kg is kept at origin at t=0 and is having velocity 4√5 m/s in positive x- direction. The only force on it is a conservative and its potential energy is defined as U=-x3+6x2+15 ( SI units). Its velocity when the force acting on it is minimum (after the time t=0 ) is

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Q. A block of mass $2 \, kg$ 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

NTA AbhyasNTA Abhyas 2020Work, Energy and Power

Solution:

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