(14) surprising result !!! As time goes on the displacement gets bigger and bigger. In this simple model, the objects just keeps moving towards the right. Numerical Approach We can also find the velocity and displacement of the object by solving Newton’s Second Law of motion a finite difference method. We start with (4) In the finite difference method we calculate the velocity and displacement at v x discrete times tk at fixed time intervals N t , = 0 = (k-1) = 1, 2, 3, , t1, t2, tk, tN t =t2 - t1 t1 tk t k N … …, … The acceleration is approximated by the difference formula a Therefore, the velocity at time is tk+2 Hence, to calculate the velocity we need to know the velocity at the two previous time steps and . We know = 0 and = = t1 v(t1) v(0) v0. We estimate the velocity at the second time step t2 (8) we have assumed a constant acceleration in the first time step. We can improve our estimate of by using an average value of the acceleration in the first time step We can now calculate the velocity at all times from = to = v(t) t t1 t tN. The acceleration at each time step is (9)
(14) surprising result !!! As time goes on the displacement gets bigger and bigger. In this simple model, the objects just keeps moving towards the right. Numerical Approach We can also find the velocity and displacement of the object by solving Newton’s Second Law of motion a finite difference method. We start with (4) In the finite difference method we calculate the velocity v and displacement x at N discrete times tk at fixed time intervals t t1, t2, … , tk, …, tN t =t2 - t1 t1 = 0 tk = (k-1) t k = 1, 2, 3, … , N The acceleration a is approximated by the difference formula Therefore, the velocity at time tk+2 is Hence, to calculate the velocity we need to know the velocity at the two previous time steps and . We know t1 = 0 and v(t1) = v(0) = v0. We estimate the velocity at the second time step t2 (8) we have assumed a constant acceleration in the first time step. We can improve our estimate of by using an average value of the acceleration in the first time step We can now calculate the velocity v(t) at all times from t = t1 to t = tN. The acceleration at each time step is (9)