A particle performs simple harmonic motion according to the equation x = A ( t + 3 ) . Let t₁ be the first time after t=0 when the particle's kinetic energy becomes equal to its potential energy, and t₂ be the first time after t=0 when the particle comes to rest. The ratio t₁ t₂ is

Step-by-step solution

The equation of motion is x = A ( t + 3 ) . The particle comes to rest at the extreme position, where the phase is 2 . For the first time this occurs at t₂ : t₂ + 3 = 2 t₂ = 2 - 3 = 6 The kinetic energy equals the potential energy when the displacement is x = A 2 . This corresponds to phases 4 , 3 4 , 5 4 , etc. Given the initial phase is 3 (which is greater than 4 ), the first time the particle reaches this state is at phase 3 4 . For the first time this occurs at t₁ : t₁ + 3 = 3 4 t₁ = 3 4 - 3 = 5 12 The ratio is