A Simple Numerical Approximation Method
Simplest method:
0. Remember to work in rectangular components (x,y,z).
1. Using the initial conditions xo and vo, calculate the initial amount and direction of the forces, and find the rectangular components of these forces.
2. Knowing the initial forces, calculate the initial acceleration, ao, (in rectangular components) from Newton's Second Law: SF = ma.
3. Knowing the initial acceleration, ASSUME that this acceleration stays constant during a small time interval, Dt. Then use the equation for constant acceleration:
v1(t=to+Dt) = vo + ao(Dt).
Remember to do this for each rectangular component.
4. Then ASSUME that the velocity during this small time interval is a constant equal to the average of the initial and final velocities to calculate the new position:
x1(t=to+Dt) = xo + 1/2(vo+v1)Dt
again remembering to do this for each rectangular component.
5. Now repeat the process (go to step 1) with vo being replaced with v1, xo being replaced with x1 and to being replaced with to+Dt.
As you decrease Dt, the approximation should get better but the number of calculations will increase.
Better method (sometimes):
You can improve your approximation by delaying step 5 (repeat the process) and inserting the following steps after step 4:
4-1. Recalculate the forces using v1 and x1.
4-2. Recalculate a new acceleration, a1.
4-3. Now average the initial acceleration from step 2, ao, and the acceleration from step 4-2, a1 to get a better acceleration. Then proceed with steps 3, 4 and 5 as above.
This improved approximation takes more computing steps per iteration, and so may not be more efficient than simply reducing Dt. It depends on the type of forces you have.
Advantage of numerical method:
You don't have to solve a differential equation to see what will happen.
Disadvantage of numerical method:
If you wish to change a parameter, you will have to re-do the process to see what will happen. In the analytical method, you can look at the symbolic result to see what will happen.