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

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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: $$ V(x)=\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right) \mathrm{J} $$ The total mechanical energy of the particle is $2 \mathrm{~J}$ Then, the maximum speed (in $\mathrm{m} / \mathrm{s}$ ) is: (a) 2 (b) $\frac{3}{\sqrt{2}}$ (d) $\frac{1}{\sqrt{2}}$ (c) $\sqrt{2}$

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

Key Concepts

Identifying Extremal Points in a Potential Function

Finding the extrema of a potential energy function through differentiation helps determine positions where the particle has maximum or minimum potential energy. These extremal points are important for analyzing the motion, as the maximum speed occurs when the energy difference (total energy minus the minimum potential energy) is at its largest.

Relationship between Kinetic Energy and Potential Energy

This principle involves using the energy conservation equation (total energy = kinetic energy + potential energy) to calculate the particle's kinetic energy at a given position. In scenarios where the potential energy reaches its minimum value, the kinetic energy is maximized, leading to the maximum speed of the particle.

Conservation of Mechanical Energy

This concept states that in a system with no non-conservative forces, the total mechanical energy (sum of kinetic and potential energies) remains constant during the motion. It is critical in problems where potential energy is converted to kinetic energy and vice versa, allowing the determination of quantities like speed from energy differences.

Potential Energy Function

A potential energy function describes how the potential energy of a particle depends on its position. Understanding the shape and critical points of this function (such as minima and maxima) is essential because these points indicate where the particle may have either maximal kinetic energy (when potential energy is minimum) or turning points in its motion.

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