Dimensional Analysis: Finding Relation Between v, p and p

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Homework Statement

The velocity,v of the sound in a gas depends on the density, p and the pressure, p of the gas. By using dimensional analysis, find a possible relation between v, p and p.

Homework Equations

[v] = LT^-1
[p] = ML^-3
[p] = ML^-1T^-2

The Attempt at a Solution

well i have no idea how to do it, so far i can only express the variables in dimension ways

 

[RoyalCat]

Here, have a ρ to express the density.

The whole concept of dimensional analysis is based on the fact that you can only equate two quantities if they have the same dimensions and units.

You are told that the velocity of sound in a gas, $$v$$ depends on two factors, that is to say, that it is a function of these two factors.

$$v=f(\rho , p)$$

Naively, we say that it is a product of these two quantities, raised to some powers $$\alpha, \beta$$:

$$v=\rho^{\alpha}\cdot p^{\beta}$$

Now use dimensional analysis to find $$\alpha, \beta$$

Do this by constraining the system so that the dimensions on the RHS are the same as the dimensions on the LHS.