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

Here, a=−kx, where a is acceleration, x is   displacement and k is proportionality constant.  ∴ vdvdx=−kx Vdv=−kxdx Integrating within proper limits, we get ∫v0v vdv=−k∫0x xdx v22v0v=−kx220x v2−v022=−kx22 multiplying m on both sides, we get m v2−v022=−mkx22∞x2  Δk=−mkx22∝x2