Making something 1% larger than something

[RJLiberator]

Homework Statement

In the Michelson-Morley experiment
a) What would v/c have to be to make tACA 1% larger than tABA?

Homework Equations

$$t_{ACA} =\frac{2l}{c}\frac{1}{1 - v^2/c^2}$$

$$t_{ABA} =\frac{2l}{c}\frac{1}{\sqrt{1 - v^2/c^2}}$$

The Attempt at a Solution

The way I set up my equation is as follows.

I let x = v/c

$$t_{ACA} - \frac{1}{100}t_{ACA} = t_{ABA}$$

I plug in the equations, omitting 2l/c as it cancels, into wolfram alpha and it finds a value of x = sqrt(199)/100.

I then plug it into the calculations to check and find

$$t_{ACA} = 1.01989, t_{ABA} = 1.009899, \frac{1}{100}t_{ACA} = 0.010198$$

That is rather close to what I am looking for.

Is this the correct way to solve this type of mathematical problem?

 

[TomHart]

Let's say I have 2 barrels of apples - one labeled T_ABA and one labeled T_ACA.
If T_ACA has 100 apples and TABA has 99 apples, is that the same as if T_ACA has 101 apples and T_ABA has 100 apples?

 

[RJLiberator]

No, if Taca has 100 apples and Taba has 99 apples then 99.99 would be 1% larger than Taba.

If Taca has 101 apples and Taba has 100 apples then 101 would be 1% larger than 100 and indeed Taca is correct.

 

[TomHart]

I guess I would write it like this:
If T_ACA = 100 and T_ABA = 99, then TACA/T_ABA = 1.0101 = 101.01%
If T_ACA = 101 and T_ABA = 100, then T_ACA/T_ABA = 1.0100 = 101.00%, making T_ACA 1% larger than T_ABA

 

[Ray Vickson]

Homework Statement

In the Michelson-Morley experiment
a) What would v/c have to be to make tACA 1% larger than tABA?

Homework Equations

$$t_{ACA} =\frac{2l}{c}\frac{1}{1 - v^2/c^2}$$

$$t_{ABA} =\frac{2l}{c}\frac{1}{\sqrt{1 - v^2/c^2}}$$

The Attempt at a Solution

The way I set up my equation is as follows.

I let x = v/c

$$t_{ACA} - \frac{1}{100}t_{ACA} = t_{ABA}$$

I plug in the equations, omitting 2l/c as it cancels, into wolfram alpha and it finds a value of x = sqrt(199)/100.

I then plug it into the calculations to check and find

$$t_{ACA} = 1.01989, t_{ABA} = 1.009899, \frac{1}{100}t_{ACA} = 0.010198$$

That is rather close to what I am looking for.

Is this the correct way to solve this type of mathematical problem?

Yes, but an even easier way would be to take $\sqrt{1-v^2/c^2} = y$, so your equation becomes $1/y^2 = 1.01/y$ after cancelling the $2l/c$ as you did. Thus , $y = 100/101$.

You really should AVOID using tools like Wolfram Alpha at this stage of your learning experience; doing much of the work by hand and then using computer assistance at the end (primarily for doing arithmetic) is the very best way to learn the material.

 

[RJLiberator]

Ah, that makes sense.

I should have made an easier substitution instead of trying to do the calculation in one big wolfram alpha enhanced step.