The potential energy of a particle of mass 5 kg moving in the x-y plane is given by U=(-7 x+24

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The potential energy of a particle of mass 5 kg moving in...

The potential energy of a particle of mass $5 \mathrm{~kg}$ moving in the $x-y$ plane is given by $U=(-7 x+24 y)$ J. $x$ and $y$ being in meter. If the particle starts from rest from origin then speed of particle at $t=2 \mathrm{sec}$ is: (a) $5 \mathrm{~m} / \mathrm{s}$ (b) $14 \mathrm{~m} / \mathrm{s}$ (c) $17.5 \mathrm{~m} / \mathrm{s}$ (d) $10 \mathrm{~m} / \mathrm{s}$

The potential energy of a particle of mass $5 \mathrm{~kg}$ moving in the $x-y$ plane is given by $U=(-7 x+24 y)$ J. $x$ and $y$ being in meter. If the particle starts from rest from origin then speed of particle at $t=2 \mathrm{sec}$ is:
(a) $5 \mathrm{~m} / \mathrm{s}$
(b) $14 \mathrm{~m} / \mathrm{s}$
(c) $17.5 \mathrm{~m} / \mathrm{s}$
(d) $10 \mathrm{~m} / \mathrm{s}$

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Key Concepts

Potential Energy Function

In mechanics, the potential energy function represents the stored energy of a system as a function of configuration. Its gradient relates directly to the forces in the system, and when the energy is conservative, the work done by these forces around any closed path is zero.

Conservative Forces and the Gradient Relationship

A conservative force can be derived from the potential energy function using the gradient operator, where the force is equal to the negative of the gradient of the potential energy. This relationship simplifies the analysis of dynamics by linking energy concepts with forces.

Newton's Second Law

Newton's second law states that the net force acting on an object is equal to the product of its mass and acceleration. This principle is fundamental in translating force information, obtained either directly or through the potential energy gradient, into the motion (acceleration) of the object.

Kinematics Under Constant Acceleration

When acceleration is constant, the kinematic equations provide a means to compute the velocity and displacement of a particle over time. These equations relate the initial conditions, acceleration, and elapsed time to determine the final state of motion, making them crucial for solving problems involving uniform acceleration.

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