Algebra problem involving work using specific heat ratio

[gmaverick2k]

NO TEMPLATE BECAUSE SUBMITTED TO NON-HOMEWORK FORUM

I'm stuck on substituting the following (where gamma is the specific heat ratio):
W=(P1v1 - P2v2) / (gamma-1)
P1v1^gamma = P2v2^gamma
substituting for v2... <= this is where I get stuck...
W = [(P1v1) / (gamma-1)] * [(P2 / P1)^((gamma-1)/gamma) -1]

How did it get from the bold to the final equation?

 

[Chestermiller]

Are you saying that you haven't been able to figure out how to do the algebra?

 

[gmaverick2k]

Yes

 

[Chestermiller]

Yes

OK. I'm going to move this to the Precalculus Math forum. I'm assuming that this is a homework problem. If so, in the future, please submit homework problems to one of the Homework forums, along with using the required template.

 

[Mark44]

NO TEMPLATE BECAUSE SUBMITTED TO NON-HOMEWORK FORUM

I'm stuck on substituting the following (where gamma is the specific heat ratio):
W=(P1v1 - P2v2) / (gamma-1)
P1v1^gamma = P2v2^gamma
substituting for v2... <= this is where I get stuck...
W = [(P1v1) / (gamma-1)] * [(P2 / P1)^((gamma-1)/gamma) -1]

How did it get from the bold to the final equation?

In your second equation in bold, is it $P_1v_1^{\gamma} = P_2v_2^{\gamma}$ (which is what you wrote) or did you mean $(P_1v_1)^{\gamma} = (P_2v_2)^{\gamma}$? I'm not familiar enough with your equations to be sure that what you wrote is what you meant.

 

[Buffu]

Yes

Can you please tell what is to be proved and what is given ? I can't figure out from our post.

 

[Chestermiller]

In your second equation in bold, is it $P_1v_1^{\gamma} = P_2v_2^{\gamma}$ (which is what you wrote) or did you mean $(P_1v_1)^{\gamma} = (P_2v_2)^{\gamma}$? I'm not familiar enough with your equations to be sure that what you wrote is what you meant.

It is correct as he has it written.

 

[Chestermiller]

The first step is to factor out $P_1V_1$ from the term in parenthesis in the numerator.

 

[gmaverick2k]

Can you please tell what is to be proved and what is given ? I can't figure out from our post.

It's from Coulson & Richardson Volume 4 problem 2.1... I'm confused with the algebra:
W = (P1*v1 - P2*v2) / (gamma-1)
P1*v1^gamma = P2*v2^gamma

 

[haruspex]

NO TEMPLATE BECAUSE SUBMITTED TO NON-HOMEWORK FORUM

I'm stuck on substituting the following (where gamma is the specific heat ratio):
W=(P1v1 - P2v2) / (gamma-1)
P1v1^gamma = P2v2^gamma
substituting for v2... <= this is where I get stuck...
W = [(P1v1) / (gamma-1)] * [(P2 / P1)^((gamma-1)/gamma) -1]

How did it get from the bold to the final equation?

$\left(\frac{v_2}{v_1}\right)^{\gamma}=\frac{P_1}{P_2}$
$\frac{v_2}{v_1}=\left(\frac{P_2}{P_1}\right)^{-\frac 1{\gamma}}$

 

[gmaverick2k]

$\left(\frac{v_2}{v_1}\right)^{\gamma}=\frac{P_1}{P_2}$
$\frac{v_2}{v_1}=\left(\frac{P_2}{P_1}\right)^{-\frac 1{\gamma}}$

Thanks