Limitations of Dimensional Analysis in Predicting Proportional Relationships

[physics user1]

Our professor introduced us to dimensional analysis and told us that we can use it to predict how some variables are proportional to others, for example:

I have a ball at a certain height and i want to know the time it requires to touch the grond, i can make a guess that it will depend on the height with dimension [L] on g.[L]/[T]^2and on the mass m [M]...
Making calculations: T~ [M]^a [L]^b [L]^c [T]^-2c and i find a=0 b= 1/2 and c= -1/2 that leads to t~h^1/2 * g^(-1/2)

But what if i said in the assumption that the time depends also on the friction force? Or the initial velocity?
Why can't i use dimensional analysis to find a relation between time and these others quantities?

 

[hilbert2]

If you add too much quantities in the problem, the system of linear equations from which the exponents are calculated becomes under-determined, which means that you can't find a unique solution.

 

[physics user1]

If you add too much quantities in the problem, the system of linear equations from which the exponents are calculated becomes under-determined, which means that you can't find a unique solution.

So... How do i solve this problem? If i can't set a system?

 

[hilbert2]

So... How do i solve this problem? If i can't set a system?

Then it can't be solved by simple dimensional analysis. If you have too many quantities $a,b,c,\dots$ that the thing to be calculated depends on, then there are many different products $a^\alpha b^\beta c^\gamma\dots$ that have the correct dimensions, and you can't tell which one of them is correct.

 

[physics user1]

Then it can't be solved by simple dimensional analysis. If you have too many quantities $a,b,c,\dots$ that the thing to be calculated depends on, then there are many different products $a^\alpha b^\beta c^\gamma\dots$ that have the correct dimensions, and you can't tell which one of them is correct.

So there's a limit where i can go with dimensional analysis? In this case is 3 because 3 are the fundqmentals dimensions in mechanics? L, M and T?

Thanks

 

[hilbert2]

So there's a limit where i can go with dimensional analysis? In this case is 3 because 3 are the fundqmentals dimensions in mechanics? L, M and T?

Thanks

If you know some kind of a physical reason why the result should depend on a particular power of a given quantity, then you can remove one unknown from the linear system and it may become possible to find a solution by dimensional analysis. The dimensional analysis alone works only for a very limited set of problems.

 

[Ibix]

The dimensional analysis alone works only for a very limited set of problems.

For instance, the OP's example has a constant factor of $\sqrt {2}$ that cannot be detected by dimensional analysis.

I tend to regard dimensional analysis as a sanity check. If your dimensions don't match, your maths is wrong. If they do match it might be right.