Question

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. $e^x$
• C. $x$
• D. ${\log }_{e}x$

Answer 1

The correct option is A $x^2$

The motion of particle will be described by SHM.

Let amplitude be $A$, and angular frequency be $\omega$.

so for displacement $x$,

change in kinetic energy is, $\frac{1}{2}m(A\omega {)}^2-\frac{1}{2}m(\omega \sqrt{A^2-x^2}{)}^2=\frac{1}{2}m{x}^2{\omega }^2\propto x^2$