A particle executes a linear S.H.M. In two of its positions the velocities are V₁, V₂ and accelerations are a₁ and a₂ respectively (0 < a₁ < a₂ ) . The distance between the positions is

Step-by-step solution

The displacement in linear simple harmonic motion is governed by x = A ( t + ) , with acceleration related to displacement by a = - ^2 x and velocity satisfying V^2 = ^2(A^2 - x^2) . For positions with accelerations a₁ and a₂ where 0 The velocity expressions yield V₁^2 - V₂^2 = ^2(x₂^2 - x₁^2) = (a₂^2 - a₁^2)/ ^2 , which simplifies to 1/ ^2 = (V₁^2 - V₂^2)/(a₂^2 - a₁^2) . Substituting gives d = (a₂ - a₁) V₁^2 - V₂^2 a₂^2 - a₁^2 = V₁^2 - V₂^2 a₁ + a₂ . The expression corresponds to option C .