The potential energy φin joule of a particle of mass 1 kg, moving in the x-y palne, obeys the l

step 1 The potential energy of the particle is given by the equation 3x plus 4y. Therefore, the force a

The potential energy φin joule of a particle of mass 1 kg,...

The potential energy $\varphi$ in joule of a particle of mass $1 \mathrm{~kg}$, moving in the $x-y$ palne, obeys the law, $\varphi=3 x+4 y$, where $(x, y)$ are the coordinates of the particle in metre. If the particle is at rest at $(6,4)$ at time $t=0$, then: (a) the particle has constant acceleration. (b) the work done by external forces from the position of rest of the particle and the instant of the particle crossing $x$ -axis is $25 \mathrm{~J}$. (c) the speed of the particle when it crosses the $y$ -axis is $10 \mathrm{~ms}^{-1}$. (d) the coordinates of the particle at time $t=4 \mathrm{~s}$ are $(-18,-28)$.

The potential energy $\varphi$ in joule of a particle of mass $1 \mathrm{~kg}$, moving in the $x-y$ palne, obeys the law, $\varphi=3 x+4 y$, where $(x, y)$ are the coordinates of the particle in metre. If the particle is at rest at $(6,4)$ at time $t=0$, then:
(a) the particle has constant acceleration.
(b) the work done by external forces from the position of rest of the particle and the instant of the particle crossing $x$ -axis is $25 \mathrm{~J}$.
(c) the speed of the particle when it crosses the $y$ -axis is $10 \mathrm{~ms}^{-1}$.
(d) the coordinates of the particle at time $t=4 \mathrm{~s}$ are $(-18,-28)$.

Key Concepts

Work-Energy Theorem

The work-energy theorem relates the work done by all forces acting on a particle to the change in its kinetic energy. This theorem simplifies the analysis of motion by providing a link between the net work performed, including work done by conservative forces, and the dynamics of the particle’s velocity. It is particularly useful when the forces can be expressed as gradients of potential energy.

Potential Energy

Potential energy is a scalar energy associated with the configuration of a system within a force field. It often depends on position and is used to determine the work done by conservative forces in the system. In mechanics, potential energy allows one to analyze the energy exchange between kinetic energy and stored energy in the system.

Force as the Gradient of Potential Energy

In conservative force fields, force is related to potential energy by the gradient. The negative gradient of the potential energy function gives the force acting on a particle. This relationship is fundamental in understanding how a system's configuration influences its dynamics, making it easier to derive equations of motion from energy considerations.

Newton's Second Law and Constant Acceleration

Newton's second law states that the net force acting on a particle equals the mass of the particle multiplied by its acceleration. When a particle is subject to a constant force, the resulting acceleration is also constant. This concept is crucial for analyzing motion in many physical situations and forms the basis for deriving the kinematic equations of motion.

Kinematics Equations

Kinematics involves the study of motion without regard to the causes of motion. The kinematics equations describe relationships between displacement, velocity, acceleration, and time. They are extensively used to predict the position and velocity of a particle moving with constant acceleration, making them essential for solving many dynamics problems.

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