Note that this velocity term is a step behind the position term, since this is for the velocity at time t , not t + Δ Δ t , meaning that v n = x n + 1 − − x n − − 1 2 Δ Δ t is a second-order approximation to v ( t n ) . With the same argument, but halving the time step, v n + 1 2 = x n + 1 − − x n Δ Δ t is a second-order approximation to v ( t n + 1 2 ) , with t n + 1 2 = t n + 1 2 Δ Δ t .