Working with the Laws of Newtonian Motion
Issue: 4.1 (September/October 2005)
Author: JC Cruz
Article Description: No description available.
Article Length (in bytes): 20,120
Starting Page Number: 17
RBD Number: 4110
4110.sit Updated: Monday, September 19, 2005 at 12:19 PM
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Excerpt of article text...
In the previous article, I have covered the Euler Method, a basic mathematical algorithm used for solving ODE problems. Today, I shall use the Euler Method in its first practical application: simulating an object in free-fall.
I will introduce the basic Newtonian laws governing an object in motion. I will use the SpriteSurface control and Sprite class to animate the simulation results. Finally, I will show how to incorporate other factors such as air resistance, wind speed, and ground collision in order to make the simulation more realistic.
The Physics of Falling Objects
We have always observed that an object of mass m released at a height h will fall and eventually hit the ground. What we do not easily observe (as illustrated in Figure 1) is that the speed of a falling object steadily increases at each point in time. Consequently, the height of the object as it falls does not decrease in equal incrementsover similar time intervals.
This non-linear behavior is due to the fact that a falling object is actually accelerating. Mathematically, the position and speed of any moving object can be described by the pair of ordinary differential equations (ODEs) shown in Figure 2. For a falling object, the rate of acceleration a(t) is replaced by the constant g, also known as the acceleration due to gravity. Since it is caused by gravity, the value of g is different for every planetary body. On Earth, g has been measured as
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