Until now, the motion of an object was determined using the simple procedure,

• Use Newton's Second Law to determine the acceleration, then...
• Integrate to find analytical (closed-form) solutions for velocity and acceleration.

In many situations, the functions describing the force (and thus the acceleration) are too complicated to integrate and find closed-form solutions.

Approximate solutions can be obtained using numerical integration.

The finite-difference method is a method of numerical integration that determines changes in dependent variables over finite intervals of time.

Consider an object of mass m in straight-line motion along the x-axis. The force acting on the object may depend on time, position, and velocity.

The corresponding acceleration will also be a function of time, position, and velocity, and thus cannot easily be integrated to find velocity and position.

     a = F/m = 1/m ΣF_x (t,x,v_x)                            (1)

Suppose that at a particular time, t_o, the position x(t_o) and velocity v_x(t_o) is known. By definition, the acceleration of the object at t_o is

            = 1/m ΣF_x(t_o, x(t_o), v_x(t_o))                    (2)

The time derivative of v_x at t_o can be approximated as

Substituting this into Eq. 2 gives an approximate expression for the velocity at t_o + Δt as

     v_x(t_o + Δt) =  v_x(t_o)
                       + 1/m ΣF_x(t_o, x(t_o), v_x(t_o))Δt        (3)

The relationship between the velocity and position can be approximated as

From this an approximate expression for the position at time t_o + Δt can be determined as

     x(t_o + Δt) = x(t_o) + v_x(t_o) Δt                          (4)

Thus, if the position and velocity are known at
time t_o, then their values can be approximated at t_o + Δt by using Eqs. 3 and 4. The procedure can then be repeated using x(t_o + Δt) and v(t_o + Δt) as the initial conditions to determine the approximate position and velocity at t_o + 2 Δt. Continuing in this fashion, an approximate solution for the position and velocity can be obtained in terms of time.

Because of the number of calculations required, this process is greatly simplified with the use of a computer.

The position and velocity of an object in curvilinear motion can also be determined using the same approach.

If an object moves in the x-y plane, the components of force may depend on time, position, and velocity,

     ΣF_x = ΣF_x(t,x,y,v_x,v_y)

     ΣF_y = ΣF_y(t,x,y,v_x,v_y)

If the position and time are known at time t_o, a derivation similar to that for Eqs. 3 and 4 can be used to find the position and velocity at t + Δt.

     x(t_o + Δt) = x(t_o) + v_x(t_o) Δt                          (5a)

     y(t_o + Δt) = y(t_o) + v_y(t_o) Δt                          (5b)

                                      v_x(t_o + Δt) =  v_x(t_o) + (1/m) ΣF_x(t_o, x(t_o), y(t_o), v_x(t_o), v_y(t_o))Δt         (5c)

                                      v_y(t_o + Δt) =  v_y(t_o) + (1/m) ΣF_y(t_o, x(t_o), y(t_o), v_x(t_o), v_y(t_o))Δt         (5d)