Dimensional Analysis and the mathematical steps throughout a process.

[Olly_price]

One would assume that:

$$t \propto h^\alpha m^\beta g^\gamma$$

Where t = time taken for object to fall, h = height dropped from, m = mass, g = acceleration due to gravity.

By doing some dimensional analysis one can find that:

$$t \propto h^\frac{1}{2} g^\frac{-1}{2}$$ and that t is independant of the objects mass.

From this, one can derive that:

$$t = C \surd\frac{h}{g}$$

Where C is some unknown constant of proportionality.

MY QUESTION:

How does one get from $$t \propto h^\frac{1}{2} g^\frac{-1}{2}$$ to $$t = C \surd\frac{h}{g}$$. I need to know all the mathematical processes and each step in detail.

 

[Integral]

x^-1/2=$\sqrt{ \frac 1 x}$

How can you not know this?

 

[Olly_price]

x^-1/2=$\sqrt{ \frac 1 x}$

How can you not know this?

Probably cos I'm 14 and haven't been taught it...

 

[Olly_price]

x^-1/2=$\sqrt{ \frac 1 x}$

Besides, that's not even an answer to my question!

I asked for each mathematical step and all you give me is $$x^-1/2=√\frac{1}{x}$$

So come on, what's each mathematical step?

 

[pmacias]

To get from $$t \propto h^\frac{1}{2} g^\frac{-1}{2}$$ to $$t = C \surd\frac{h}{g}$$, we first need to understand what the symbol $\propto$ means. This symbol indicates that there is a proportional relationship between two quantities, in this case, between time (t) and the other variables (h, m, and g). In other words, as one variable changes, the other variable changes in proportion to it.

Next, we can use dimensional analysis to find the relationship between these variables. Dimensional analysis is a mathematical method used to check the validity of equations and to convert between different units of measurement. It is based on the principle that physical quantities can be expressed in terms of fundamental dimensions such as length, mass, and time.

In this case, we have four variables (t, h, m, and g) and three fundamental dimensions (length, mass, and time). This means that we can express the equation as a product of these three dimensions raised to some exponents. We can represent this as:

$$t \propto L^a M^b T^c$$

Where L represents length, M represents mass, and T represents time. The exponents a, b, and c are unknown and we need to find them.

To find the exponents, we will use the given information that t is proportional to h and g. This means that the ratio of t to h and the ratio of t to g must be constant. We can express this as:

$$\frac{t}{h} = K$$

$$\frac{t}{g} = K$$

Where K is the constant of proportionality.

Next, we can rearrange these equations to isolate t:

$$t = Kh$$

$$t = Kg$$

Now, we can substitute these values for t in our original equation:

$$Kh \propto h^\alpha m^\beta g^\gamma$$

$$Kg \propto h^\alpha m^\beta g^\gamma$$

Since the constant K is the same in both equations, we can equate the exponents of h, m, and g on both sides:

$$1 = \alpha + \beta$$

$$1 = \gamma$$

Solving for the exponents, we get:

$$\alpha = 1 - \beta$$

$$\gamma =