Mirror Equations: Derivation of Newtonian Form

[Zeros]

Hallo,

First time I've experienced a real problem rearding equations in physics, so:

As conclusions to the lab on focal points and and center of curvatures (which will be explained at a later date), we were given the following relationships:

So = Do - f
Si = Di - f

And, we were ably to verify the Gaussian? form of the equation through a series of experiments:

1/Do + 1/Di = 1/f

And yadayada, now the final conclusion for the lab simply asks for a relationship between f, So, and Si that can be expressed analytically, graphically, and verbally.

After wrecking my brain on this question, I eventually checked the following site: http://www.sasked.gov.sk.ca/docs/physics/u3b32phy.html to find this relationship:

SoSi = f^2

However, I obviously cannot go that far, lol. Here is, as best as I can transcribe, how far I got:

So = do -f ----> do = So + f
Si = di -f ----> di = Si + f

then, I substituted those into the equation, getting this stuff:

1/(So +f) - 1/(Si + f) = 1/f

I know for sure (I think, well w/e) that that is the first correct step. After that, I tried a couple of different things, the most prominent being the LCM/LCD approach:

(Si + f) - (So + f)
all over or divided by
(So + f)(Si + f)

all equals = 1/f

then...some simplification:

Si - So
all over or divided by
(So + f)(Si + f)

all equals = 1/f

from there, a smart guy at school (whose done some college level physics...I thinK) told me I could flip the numerator/denominator in both sides of the equation...giving...

(Si + f)(So + f)
all over or divided by
Si - So

all equals = f

(I just put (si + f) in the front b/c it looks a bit better I think)
However, after this we were both rather clueless as to how further simplify the expression...I tried expanding the numerator:

SiSo + Sif + Sof + f^2
all over or divided by
Si - So

all equals = f

but, like, then what? Si, So, f...nothing is common in the numerator anyhow. Eventually I got kinda berserk and tried taking out a Si and an f from the numerator...not pretty:

(Si)(So + f) + (f)(So + f)
all over or divided by
Si - SO

all equals = f

yawn..now what?...mm then this:

(So + f)(Si + f)
all over or divided by
Si - So

all equals = f

and that last part, I'm not even sure about my algebra lol. so yeah, I'm sort of hopeless, and I can't find much besides the aformentioned site for the derviation of this formula, but I think I'm on the right track so if anyone has any ideas...go ahead.

--Zeros

 

[Galileo]

Substituting the equations for the D's gives:
$$\frac{1}{S_0+f} + \frac{1}{S_i+f}=\frac{1}{f}$$

You seem to have a minus sign wrong, which is probably the source of all your troubles. Try it now.

Since you want to get rid of the denominators, multiply both sides of the equation by $f(S_0+f)(S_i+f)$.

 

[Doc Al]

However, I obviously cannot go that far, lol. Here is, as best as I can transcribe, how far I got:

So = do -f ----> do = So + f
Si = di -f ----> di = Si + f

then, I substituted those into the equation, getting this stuff:

1/(So +f) - 1/(Si + f) = 1/f

First thing, correct that minus sign.

1/(So +f) + 1/(Si + f) = 1/f

Just multiply both sides by (So + f)(Si + f)f. Simplify. (Just get rid of the denominators.)

[Galileo beat me to it!]

 

[Zeros]

Eheh ^^;...that would explain a lot...

*seven minutes later*

O_O;;; Yes, where were we...:

1/(so + f) + 1/(si + f) = 1/f

now I'll take LCD/LCM:

(Si + f) + (So + f)
all over
(Si + f)(So + f)

all equals = 1/f

ok then I'll do what you guys said (needed to do LCD step, right? or else it gets...weird) (oh I'm not going to expand to see if stuff cancels, that's how you do it, right?)

f((si + f) + (so + f))

all equals

(So + f)(Si +f)

...er, now what? simplification/expansion gives the following:

f(si + so + 2f)

all equals

(sosi + sof + sif + f^2)

...yikes. erm, did I sort of screw up somewhere again? But I see, I think it's right, I just don't know what to do from here. But thanks for the help so far!

--Zeros

 

[Galileo]

1/(so + f) + 1/(si + f) = 1/f

now I'll take LCD/LCM:

(Si + f) + (So + f)
all over
(Si + f)(So + f)

all equals = 1/f

That's overly complicated. Just multiply both sides by f(si + f)(so + f) right away. The reason is that you have an equation of the form:
1/a + 1/b = 1/c.
If you multiply both sides by abc, you get bc+ac=ab. You've immediately rid yourself of fractions.

ok then I'll do what you guys said (needed to do LCD step, right? or else it gets...weird) (oh I'm not going to expand to see if stuff cancels, that's how you do it, right?)

f((si + f) + (so + f))

all equals

(So + f)(Si +f)

...er, now what?

You should expand to see if stuff cancels. How else are you going to simplify it?