⟨ ⟨ T ⟩ ⟩ = 1 t 2 − − t 1 ∫ ∫ t 1 t 2 ∑ ∑ k = 1 N 1 2 m k | v k ( t ) | 2 d t = 1 t 2 − − t 1 ∫ ∫ t 1 t 2 ( 1 2 m | v 1 ( t ) | 2 + 1 2 m | v 2 ( t ) | 2 ) d t = m v 2 . Taking center of mass as the origin, the particles have positions r 1 (t) and r 2 (t) = − r 1 (t) with fixed magnitude r. The attractive forces act in opposite directions as positions, so F 1 (t) ⋅ r 1 (t) = F 2 (t) ⋅ r 2 (t) = − Fr. Applying the centripetal force formula F = mv / r results in: − − 1 2 ∑ ∑ k = 1 N ⟨ F k ⋅ ⋅ r k ⟩ = − − 1 2 ( − − F r − − F r ) = F r = m v 2 r ⋅ ⋅ r = m v 2 = ⟨ ⟨ T ⟩ ⟩ , as required. Note: If the origin is displaced then we'd obtain the same result. This is because the dot product of the displacement with equal and opposite forces F 1 (t), F 2 (t) results in net cancellation.