Potential energy (in joule) of a particle of mass 1 kg moving in x-y plane is U=3 x+4 y, here x

step 1 In this question we have given the potential energy of a particle which is a function of x and y

Potential energy (in joule) of a particle of mass 1 kg...

Potential energy (in joule) of a particle of mass $1 \mathrm{~kg}$ moving in $x-y$ plane is $U=3 x+4 y$, here $x$ and $y$ are in meter. If at time $t=0$, particle is at rest at point $P(6 \mathrm{~m}$, $4 \mathrm{~m}$ ). Then (A) acceleration of particle is $(3 \hat{i}+4 \hat{j}) \mathrm{m} / \mathrm{s}$. (B) time when it crosses $y$-axis is $t=1 \mathrm{~s}$. (C) speed of particle when it crosses $y$-axis is $10 \mathrm{~m} / \mathrm{s}$. (D) it crosses $y$-axis at $y=-8 \mathrm{~m}$.

Potential energy (in joule) of a particle of mass $1 \mathrm{~kg}$ moving in $x-y$ plane is $U=3 x+4 y$, here $x$ and $y$ are in meter. If at time $t=0$, particle is at rest at point $P(6 \mathrm{~m}$, $4 \mathrm{~m}$ ). Then
(A) acceleration of particle is $(3 \hat{i}+4 \hat{j}) \mathrm{m} / \mathrm{s}$.
(B) time when it crosses $y$-axis is $t=1 \mathrm{~s}$.
(C) speed of particle when it crosses $y$-axis is $10 \mathrm{~m} / \mathrm{s}$.
(D) it crosses $y$-axis at $y=-8 \mathrm{~m}$.

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Key Concepts

Newton's Second Law

Newton's second law of motion establishes the relationship between force, mass, and acceleration, stated as F = ma. This principle is used to determine the acceleration of an object when the net force acting on it is known, linking the concepts of force derived from potential energy to motion.

Kinematics in Two Dimensions under Constant Acceleration

When an object is subject to constant acceleration, its motion can be described using kinematic equations in each coordinate direction independently. This approach allows the determination of displacement, velocity, and time in problems involving motion in the plane, integrating the initial conditions with the computed acceleration.

Potential Energy

Potential energy is the energy stored in an object due to its position in a force field. In mechanics, it is often defined as a function of position and represents the work that can be done by conservative forces when the object moves without energy loss.

Force from Potential Energy (Gradient)

In conservative force fields, the force acting on an object can be derived from the potential energy function using the gradient operation. Specifically, the force is equal to the negative gradient of the potential energy, which indicates both the magnitude and direction in which the potential energy decreases most rapidly.

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