4.  How and why the simulations work.

The kinematics equations that describe an object's motion are simple and straightforward, but they can only be applied if the acceleration is constant. Since acceleration depends on the net force and the mass of the object, it will not be constant if either varies.

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Varying net forces are found in situations such as an object moving over a circular track in a vertical plane, any situation involving a spring, the presence of air resistance (drag), a swinging pendulum, gravitational forces in cases where the object is not close to the Earth’s surface and is moving over large distances, Coulombic forces between charged particles, etc. Although energy methods or differential equations are typically applied to analyze some of these situations theoretically, they cannot be used to simulate an object’s motion. 

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Objects with varying mass are most commonly found in rocket propulsion: as the fuel is burned to produce thrust, the mass of the rocket decreases. Even if the thrust is constant the acceleration will vary with the varying mass. 

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What allows a computer simulation to work in the presence of varying acceleration is that objects can be moved over small distances by using small and often incremental time intervals. Though the acceleration still varies, it doesn't vary much, and is approximately constant. In the limit as the time interval is reduced the error approaches zero. (If you remember your calculus, this is precisely the limit process that underlies integration.)

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The object's trajectory is thus broken down into a series of movements over suitably small time intervals. The process is known as incremental iteration. The length of a time interval is typically in milliseconds, or less.  But in the case of an object orbiting the Earth, or travelling to the Moon, the time intervals can be hundreds of seconds since the gravitational force will not have changed much for the velocities involved and distances moved in each interval.   

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The final position and velocity at the end of each small movement, or iteration, are used as the initial conditions for the next iteration (cycle), over and over again. 

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The process is programmed using a loop, which is a code structure that repeats itself until some condition is satisfied -- say the total allowable time is reached, or the user presses a pause or stop button.

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The code used to simulate the motion of a rocket car moving in a straight line along a road (ignoring air resistance and the varying mass for the time being) can be viewed by clicking here.