The potential energy of a 1 kg particle free to move along the x -axis is given by V=((x^4)/(4)

step 1 We have given the potential energy of a 1 kilogram particle which is moving in x direction and T

The potential energy of a 1 kg particle free to move along...

The potential energy of a $1 \mathrm{~kg}$ particle free to move along the $x$ -axis is given by $V=\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right) \mathrm{J}$. The total mechanical energy of the particle is $2 \mathrm{~J}$. Then the maximum speed is (a) $\frac{3}{\sqrt{2}} \mathrm{~m} / \mathrm{s}$ (b) $\sqrt{2} \mathrm{~m} / \mathrm{s}$ (c) $\frac{1}{\sqrt{2}} \mathrm{~m} / \mathrm{s}$ (d) $2 \mathrm{~m} / \mathrm{s}$

The potential energy of a $1 \mathrm{~kg}$ particle free to move along the $x$ -axis is given by $V=\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right) \mathrm{J}$. The total mechanical energy of the particle is $2 \mathrm{~J}$. Then the maximum speed is
(a) $\frac{3}{\sqrt{2}} \mathrm{~m} / \mathrm{s}$
(b) $\sqrt{2} \mathrm{~m} / \mathrm{s}$
(c) $\frac{1}{\sqrt{2}} \mathrm{~m} / \mathrm{s}$
(d) $2 \mathrm{~m} / \mathrm{s}$

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Key Concepts

Energy Transformation

Energy transformation in mechanical systems refers to the process where potential energy is converted into kinetic energy and vice versa. This exchange is central to determining variables like maximum speed, as it occurs in systems where the total mechanical energy remains constant.

Conservation of Mechanical Energy

This principle states that the total energy in a closed system—typically the sum of kinetic energy and potential energy—remains constant when only conservative forces are acting. It is fundamental in dynamics, as it allows one to relate the motion of an object to its energy changes without having to solve complex differential equations.

Potential Energy

Potential energy represents the energy stored in an object due to its position or configuration within a force field. In mechanics, changes in potential energy, when balanced with kinetic energy, describe how an object's speed varies as it moves through different regions of space.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. It is calculated as one half of the mass times the square of its velocity. This concept is essential in understanding how energy is distributed in a moving system, especially when potential energy is converted into kinetic energy.

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