A particle moves in a straight line with retardation proportional to its displacement. Its loss in kinetic energy for any displacement x is proportional to

Options

A. (a)X
B. (b)`e^(x)`
C. (c)`x^(2)`
D. (d)`log_(e) x`

Answer

C

Explanation

According to problem,
Retardation displacement
or `v(dv)/(dx) = -kx " or " int_(v_1)^(v_2) vdv = -k int_(0)^(x) xdx " or " (v_2^2)/(2) - (v_1^2)/(2) = -(kx^2)/(2)`
or `1/m ((mv_1^2)/(2) - (mv_2^2)/(2)) = (kx^2)/(2) " or "(k_1 - k_2) = (mk)/2 x^2`
Thus, loss of kinetic energy is proportional to `x^2`.