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 any displacement $x$ is proportional to:
(a) $x^{2}$
(b) $\mathrm{e}^{\mathrm{x}}$
(c) $x$
(d) $\log _{e} x$

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Key Concepts

Work-Energy Principle

The work-energy principle states that the work done by the net force acting on a particle is equal to the change in its kinetic energy. When a retarding force acts on a particle, integrating this force over the displacement shows how its kinetic energy decreases.

Differential Relationship Between Force and Energy Change

In mechanics, the rate of change of kinetic energy with respect to displacement is directly related to the force acting on the particle. When the force is proportional to displacement, this differential relation can be integrated, leading to a quadratic dependence of the total energy loss on displacement.

Retardation Proportional to Displacement

This concept refers to a force (or resulting acceleration) whose magnitude increases linearly with displacement. In dynamics, such a relationship implies that the further the particle moves from a reference point, the greater the opposing force acting on it, resembling systems governed by restoring forces in oscillatory motion.

Kinetic Energy

Kinetic energy is the energy of motion, defined as one-half the mass of the particle times the square of its velocity. Changes in kinetic energy occur as the result of work being done on or by the particle, which is central to understanding energy transformations during deceleration.

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