A particle move in a straight line with retardation proportional to its displacement its loss of kinectic energy for any displacement `x` is proportional to

Options

A. x
B. `e^(x)`
C. `x^(2)`
D. `log_(e) x`

Answer

C

Explanation

According to problem
Retardation `prop` displacement
or `(dv)/(dt) = kx`
or `v(dv)/(dt) = kx`
or `vdv = kxdx`
or `underset(v_(1)) overset(v_(2)) int vdv = underset(0) overset(x) int kdx`
or `(v_(2)^(2))/(2) -(v_(1)^(2))/(2) =(kx^(2))/(2)`
or `(1)/(m) ((mv_(2)^(2))/(2) -(mv_(1)^(2))/(2)) = (kx^(2))/(2)`
or `(k_(2)-k_(1)) = (mk)/(2) x^(2)`.
Thus, loss of kinetic enery is proportional to `x^(2)`.