The potential energy of 2 kg particle, free to move along x axis is given by U(X)=[{x^4 / 4}-{x

step 1 Here we have given the potential energy of 2 kg particle. The mass of particle is 2kG, 3 to move

The potential energy of 2 kg particle, free to move along x...

The potential energy of $2 \mathrm{~kg}$ particle, free to move along $\mathrm{x}$ axis is given by $\mathrm{U}(\mathrm{X})=\left[\left\{\mathrm{x}^{4} / 4\right\}-\left\{\mathrm{x}^{2} / 2\right\}\right] \mathrm{J}$. If its mechanical energy is $2 \mathrm{~J}$, its maximum speed is $\ldots \mathrm{m} / \mathrm{s}$ (A) $(3 / 2)$ (B) $\sqrt{2}$ (C) $(1 / \sqrt{2})$ (D) 2

The potential energy of $2 \mathrm{~kg}$ particle, free to move along $\mathrm{x}$ axis is given by $\mathrm{U}(\mathrm{X})=\left[\left\{\mathrm{x}^{4} / 4\right\}-\left\{\mathrm{x}^{2} / 2\right\}\right] \mathrm{J}$. If its mechanical
energy is $2 \mathrm{~J}$, its maximum speed is $\ldots \mathrm{m} / \mathrm{s}$
(A) $(3 / 2)$
(B) $\sqrt{2}$
(C) $(1 / \sqrt{2})$
(D) 2

Key Concepts

Kinetic Energy

Kinetic energy is the energy associated with an object's motion, calculated as half of the mass multiplied by the square of its velocity. Understanding kinetic energy is essential because any change in the potential energy of a system directly affects the kinetic energy, allowing one to derive velocity or speed from energy conservation principles.

Energy Partitioning and Maximum Speed

When analyzing a particle's motion under conservative forces, the speed is maximized when the potential energy is at a minimum, as the mechanical energy then is entirely converted into kinetic energy. This concept of energy partitioning is critical in determining the maximum speed, where the sum of the kinetic and potential energies equals the constant mechanical energy of the system.

Conservation of Mechanical Energy

This principle states that in an isolated system with only conservative forces acting, the total energy—comprising both kinetic and potential components—remains constant over time. It allows one to equate the sum of the kinetic and potential energies at different positions, facilitating the determination of quantities like speed when potential energy is known.

Potential Energy

Potential energy represents the energy stored in a system because of its position or configuration relative to a force field. The functional form of the potential energy informs us how the energy landscape varies with position, highlighting regions where the system might be in stable or unstable equilibrium.

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