A particle moves in a straight line with retardation...

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for and displacement $x$ is proportional to
(a) $x^{2}$
(b) $e^{z}$
(c) $x$
(d) $\log \mathrm{e}^{x}$

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Key Concepts

Variable Force Integration

When a force varies with displacement—such as a force proportional to the displacement—the work done must be calculated by integrating the force over the distance traveled. This integration often produces a quadratic dependence (for example, the integral of x with respect to x gives a term proportional to x²), explaining why energy changes can exhibit a squared relationship with displacement.

Proportional Relationships in Physics

Proportionality is a fundamental concept indicating that one quantity changes at a constant rate relative to another. In problems involving forces where parameters like retardation depend linearly on displacement, the mathematical operations (such as integration) can transform these linear relations into quadratic or other forms, thereby determining how quantities such as kinetic energy vary with displacement.

Work-Energy Principle

This principle states that the work done by forces on an object results in a change in its kinetic energy. When a force is applied, the net work performed is equal to the change in kinetic energy of the particle, providing a direct connection between the force acting over a displacement and the energy lost or gained by the particle.

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