Rearranging Formulas for Orbital Velocity and Radius

[dukg08]

Homework Statement

(i)
Combine Equations A and B to give a single equation for v and substitute this in place of v in Equation C.
(ii)
Rearrange your answer to part (i) to make r the subject of the equation.

Homework Equations

Equation A
z = Δλ/λ0

Equation B
v = zc

For a planet such as the one orbiting Gliese, the speed v is linked to the mass of the star M and the planet’s orbital radius, by Equation C:

Equation C
v² = GM/r

G is Newton’s universal constant of gravitation.

The Attempt at a Solution

(i) v = (Δλ/λ0)c -> (Δλ/λ0)c² = GM/r
(ii) this is where I'm not sure, is the above correct? I always get roots confused.

 

[Ouabache]

Are you sure about your result for v^2 in equ. (i)?
It looks like you only squared part of your expression for v.

 

[dukg08]

((Δλ/λ0)c)² = GM/r

 

[Ouabache]

That looks good to me, now they ask you to rearrange your expression
to make r the subject of the equation. Shall we interpret this to mean solving for r ?

 

[dukg08]

((Δλ/λ0)c)² = GM/r

So do I multiply the r to both sides now or remove the root?

(Δλ/λ0)c = √(GM/r) or ((Δλ/λ0)c)²/r = GM

 

[Ouabache]

what are you solving for? that will help you decide which direction to take.

 

[dukg08]

I'm trying to make r the subject.

((Δλ/λ0)c)²/r = GM

r = ((Δλ/λ0)c)²xGM

 

[Ouabache]

Your algebra doesn't look quite right.
To this expression: ((Δλ/λ0)c)2 = GM/r

are you sure you multiplied both sides by r ??

 

[dukg08]

((Δλ/λ0)c)2 = GM/r

((Δλ/λ0)c)² x r = GM

totally not sure

 

[Ouabache]

Looks good, you're almost there now.

 

[dukg08]

so then:

r = GM/((Δλ/λ0)c)²

 

[Ouabache]

well done