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

step 1 In this question we have given a potential function of a particle and we have given its total me

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 [2006] (A) $\frac{3}{\sqrt{2}}$ (B) $\sqrt{2}$ (C) $\frac{1}{\sqrt{2}}$ (D) 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 [2006]
(A) $\frac{3}{\sqrt{2}}$
(B) $\sqrt{2}$
(C) $\frac{1}{\sqrt{2}}$
(D) 2

Key Concepts

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces (like friction), the total mechanical energy of a system (the sum of its kinetic and potential energy) remains constant over time. It allows us to relate potential energy and kinetic energy at different positions to solve for quantities like speed.

Potential Energy

Potential energy is the energy stored in a system due to its position or configuration relative to a force field. In conservative force fields, such as gravitational or elastic forces, the potential energy function describes how energy varies with position, which is key to determining the motion of a particle.

Kinetic Energy

Kinetic energy is the energy associated with the motion of a particle, typically given by one-half the mass times the velocity squared. It represents the energy that can be converted from the potential energy in conservative systems, and understanding this conversion is critical when analyzing speed and motion.

Turning Points

Turning points in a potential energy diagram are the positions where the kinetic energy momentarily becomes zero (all energy is potential), and beyond which the particle cannot move without additional energy. Identifying these points helps in determining where the particle reaches maximum speed, since the maximum speed occurs where the potential energy is minimized and kinetic energy is maximized.

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