Use the following data to find the velocity and acceleration

Use the following data to find the velocity and acceleration at t = 10 seconds: Use second-order correct (a) centered finite-difference, (b) forward finite-difference, and (c) backward finite-difference methods.

Solution

a) Centered finite-difference

Velocity v=dx/dt=x\'(t_i)= [x(t_i+1)-x(t_i-1)]/2h

                               = (7.3-5.1)/2x2

                               = 2.2/4=0.55m/s

Acceleration a=d²x/dt²=x\"(t_i)=[x(t_i+1)-2x(t_i)+x(ti-1)]/h²

                                          =[7.3-2(6.3)+5.1]/2²

                                         = -0.05m/s²

b) Forward finite-difference

Velocity v=dx/dt=x\'(t_i)= [x(t_i+1)-x(ti)]/h

                                = (7.3-6.3)/2=0.5m/s

Acceleration a=d²x/dt²=x\"(t_i)= [x(t_i+2)-2x(t_i+1)+x(t_i)]/h²

                                                        = [8-2(7.3)+6.3]/4

                                          = -0.075m/s²

c) Backward finite-difference

Velocity v=dx/dt=x\'(t_i)= [x(t_i)-x(t_i-1)]/h

                                = (6.3-5.1)/2=0.6m/s

Acceleration a=d²x/dt²=x\"(t_i)= [x(t_i)-2x(t_i-1)+x(t_i-2)]/h²

                                                        = [6.3-2(5.1)+3.4]/4

                                          = -0.275m/s²

time	0	2	4	6	8	10	12	14	16
distance	0	0.7	1.8	3.4	5.1	6.3	7.3	8.0	8.4
				xi-2	xi-1	xi	xi+1	xi+2