Help me derive the Doppler effect ratio

[Comscistudent]

Homework Statement

I'm watching this lecture and wanted to attempt to derive the expression for ΔTo/ΔTs

Homework Equations

ΔTo = ΔTs + VΔTo / U

The Attempt at a Solution

I have worked out that ΔTs = 1-V / U, however I'm stuck trying to get ΔTs / ΔTo because of the ΔTo on both sides of the original equation, it just turns into a recursive thing which I know must be wrong.

I'd appreciate any pointers that would help me get there myself rather than a solution. I'm only refreshing myself on algebra currently so I'm doubtless missing something simple

 

[Mark44]

Homework Statement

I'm watching this lecture and wanted to attempt to derive the expression for ΔTo/ΔTs

Homework Equations

ΔTo = ΔTs + VΔTo / U

The Attempt at a Solution

I have worked out that ΔTs = 1-V / U

That's not what I get, starting from your relevant equation. What happened to ΔTo?
Please show us what you did.

, however I'm stuck trying to get ΔTs / ΔTo because of the ΔTo on both sides of the original equation, it just turns into a recursive thing which I know must be wrong.

I'd appreciate any pointers that would help me get there myself rather than a solution. I'm only refreshing myself on algebra currently so I'm doubtless missing something simple

 

[Comscistudent]

Sorry that was a mistake I forgot to divide the ΔTs by ΔTo. Now I've gotten this far but still end up with the variables on both sides

ΔTo = ΔTs + VΔTo / U
ΔTo = ΔTs / ΔTo + V / U
ΔTo / ΔTs = 1 / ΔTo + V/U / ΔTs
ΔTo / ΔTs = 1 / ΔTo + V / UΔTs

 

[Mark44]

Sorry that was a mistake I forgot to divide the ΔTs by ΔTo. Now I've gotten this far but still end up with the variables on both sides

ΔTo = ΔTs + VΔTo / U
ΔTo = ΔTs / ΔTo + V / U

The equation above is incorrect. You divided the right side by ΔTo, but not the left side. Any arithmetic operation you apply to one side of an equation, you must apply the same operation to the other side.
Instead, bring the terms with ΔTo to one side, and then factor ΔTo out.
Here's a start.
ΔTo - VΔTo / U = ΔTs
Now factor ΔTo from the two terms on the left side.

ΔTo / ΔTs = 1 / ΔTo + V/U / ΔTs
ΔTo / ΔTs = 1 / ΔTo + V / UΔTs

 

[Comscistudent]

Wow I'm stupid, thanks for putting up with me. I finally got there I think -

ΔTo - VΔTo / U = ΔTs
ΔTo( 1 - V / U ) = ΔTs
ΔTo / ΔTs ( 1 - V / U ) = 1
ΔTo / ΔTs = 1 / ( 1 - V / U )
ΔTo / ΔTs = (1 - V / U )^-1

 

[Mark44]

Wow I'm stupid, thanks for putting up with me. I finally got there I think -

ΔTo - VΔTo / U = ΔTs
ΔTo( 1 - V / U ) = ΔTs
ΔTo / ΔTs ( 1 - V / U ) = 1
ΔTo / ΔTs = 1 / ( 1 - V / U )
ΔTo / ΔTs = (1 - V / U )^-1

Since $1 - \frac V U = \frac{U - V}{U}$ you could go from the next to last line above, to this:
ΔTo / ΔTs = $\frac U {U - V}$

 

[Comscistudent]

That's even nicer, I wonder why they use the (1 - V / U)^-1 in the lecture? Probably works out easier to apply or something, I'm sure I'll find out when I watch the rest of it.

Thanks again!