We can begin by being rather complete. We have the ball, its properties (R, m, color, deformability,...), and its kinematic descriptive parameters (position = (x, y, z), velocity = (vx, vy, vz), acceleration = ...). We have the systems external to the ball acting on it (the earth's gravitational field = g, the density of the air = ρ,...).
Clearly, the three dimensions are going to be a problem. (We can improve things by using the vector character and writing vector-correct equations.) For now, let's simplify by assuming we are only working with one dimension -- up and down (typically, y). This then reduces us to 4 dimensioned parameters, y, vy, m, and g. It is clear that the initial position is going to matter, but it should come in easily. The height it goes after we release it should just be measured from our starting point. Changing our starting point will just change the result in a simple linear way. So let's take the initial value of y = 0 at our release point.
We are now left with three dimensioned variables.
From these, how can we construct a distance?
Since [m] M, [v] = L/T and [g] = L/T2, this combination has the dimensions:
If we want this to be a length ([h] = L) we must satisfy the equations:
These are easily solved to find:
This gives the resultL
We use the "~" instead of "=" since there may be a dimensionless constant in front. In fact, there is: a "1/2" that we cannot derive using dimensional analysis. But we get pretty close without actually solving for any motion.
In general, if we want to combine dimensioned quantities to get a quantity of a particular dimension we can use a procedure like this. Notice, however, how much physics we put into the argument. Using dimensional analysis to create a solution is an art, relying heavily on one's understanding of the physics. It's not an algorithm.
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Last revision 11. September, 2005.