A particle is executing SHM along a straight line. Its velocities at distances x1 and x2 from the mean position are V1 and V2, respectively. Its time period is:

Using the expressions for velocity for SHM,
v₁ = ω√(A² - x₁²)
v₂ = ω√(A² - x₂²)
Squaring both,
v₁² = ω²(A² - x₁²) (3)
v₂² = ω²(A² - x₂²) (4)
Subtracting (4) from (3),
v₁² - v₂² = ω²(x₂² - x₁²) ⇒ 4π²(x₂² - x₁²) = T²(v₁² - v₂²)
⇒T = 2π√(x₂² - x₁²)/(v₁² - v₂²)
where we have used ω = 2π/T