A particle of mass 5 kg moving in the X-Y plane has its potential energy given by U=(-7 x+24 y)

step 1 The potential energy u is given as minus 7x plus 24y. Now we can find the forces by using this p

A particle of mass 5 kg moving in the X-Y plane has its...

A particle of mass $5 \mathrm{~kg}$ moving in the $X-Y$ plane has its potential energy given by $U=(-7 x+24 y)$ Joule. The particle is initially at origin and has velocity $\vec{u}=(14.4 \hat{i}+4.2 \hat{j}) \mathrm{m} / \mathrm{s}$ (A) The particle has speed $25 \mathrm{~m} / \mathrm{s}$ at $t=4 \mathrm{~s}$. (B) The particle has an acceleration $5 \mathrm{~m} / \mathrm{s}^{2}$. (C) The acceleration of particle is normal at its initial velocity. (D) None of these.

A particle of mass $5 \mathrm{~kg}$ moving in the $X-Y$ plane has its potential energy given by $U=(-7 x+24 y)$ Joule. The particle is initially at origin and has velocity $\vec{u}=(14.4 \hat{i}+4.2 \hat{j}) \mathrm{m} / \mathrm{s}$
(A) The particle has speed $25 \mathrm{~m} / \mathrm{s}$ at $t=4 \mathrm{~s}$.
(B) The particle has an acceleration $5 \mathrm{~m} / \mathrm{s}^{2}$.
(C) The acceleration of particle is normal at its initial velocity.
(D) None of these.

Key Concepts

Newton's Second Law

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass, usually expressed as F = ma. This key principle is used to relate the computed force from the potential energy gradient to the acceleration of the particle, helping to determine its motion characteristics.

Tangential and Normal Acceleration

In the analysis of particle motion, acceleration can be decomposed into tangential and normal components. The tangential component is responsible for changes in the speed (magnitude) of the velocity, while the normal component is responsible for changes in the direction of the velocity. Recognizing this decomposition is essential when assessing whether an acceleration is acting along the direction of motion or perpendicular to it.

Conservative Forces

Conservative forces are those forces that can be derived from a scalar potential energy function. The work done by conservative forces when moving an object between two points is path independent, relying only on the start and end positions. This concept allows the use of potential energy to determine the force acting on an object through the negative gradient of the potential energy function.

Force from Potential Energy Gradient

The force acting on a particle in a potential field is found by taking the negative gradient of its potential energy function. This relationship, F = -?U, connects the spatial variation of potential energy directly to the force experienced by the particle, which in turn influences its acceleration and subsequent motion.

Kinematics with Constant Acceleration

Kinematics under constant acceleration involves using equations of motion to predict the future state of a moving object, including its velocity and displacement over time. When acceleration is constant, the kinematic equations allow for the determination of speed at any moment by integrating the acceleration over time, considering initial conditions.

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