I am trying to figure out how undefined formulas in mathematics relates to physics. Take the following formula for terminal velocity.
$$V_\text{terminal} = \sqrt{ mg \over{c \rho A}} $$
Say we have a an air density of 0; $ \rho = 0 $ (a vacuum)
Logic tells me the particle would continue to accelerate and never reach terminal velocity, but in mathematics this formula would be undefined.
Obviously this is one of many examples of what can happen in physics problems, but what does undefined actually mean in terms of physics? I hope I am explaining myself clearly.
Best Answer
Yes, the particle would continue to accelerate and would never reach a terminal velocity. But that is not what this equation tells you. This equation tells you what the terminal velocity is, given the parameters of the function. When in a vacuum, there is no terminal velocity. It is not zero, it is not infinity. A terminal velocity literally does not exist and that is exactly what the equation tells you.