A particle of mass m moves on a straight line with its...

A particle of mass $m$ moves on a straight line with its velocity varying with the distance travelled according to the equation $v=a \sqrt{x}$, where $a$ is a constant. Find the total work done by all the forces during a displacement from $x=0$ to $x=d$.

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Key Concepts

Work-Energy Theorem

This principle states that the net work done on a particle is equal to the change in its kinetic energy. It is a central concept in mechanics that allows the determination of work without explicitly calculating the forces along the path, simply by comparing the initial and final kinetic energies.

Kinetic Energy

Kinetic energy is the energy a body possesses due to its motion and is defined as one-half of the mass multiplied by the square of the velocity. In problems where the velocity of a particle changes with position, it is used to determine the change in energy as the particle moves from one point to another.

Integration of Motion Equations

When velocity or acceleration is given as a function of displacement or time, integration becomes a key mathematical tool. It allows one to derive relationships between the variables of motion, such as computing the net work done by integrating forces over a displacement or converting between differential forms of kinetic energy and displacement.

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