How to Make v the Subject of a Formula: Step-by-Step Guide

[anonymouse123]

Hi.

I am struggling with this question:

How do I make v the subject of this formula?

e=mc^2/sqrt(1-v^2/c^2)

Please reply with a step by step guide.

I have ended up with all kind of derivations to make v the subject so would appreciate your help.

Thanks.

 

[berkeman]

Hi.

I am struggling with this question:

How do I make v the subject of this formula?

e=mc^2/sqrt(1-v^2/c^2)

Please reply with a step by step guide.

I have ended up with all kind of derivations to make v the subject so would appreciate your help.

Thanks.

Welcome to the PF.

What is the context of the question? What is your math background so far?

To get you started, bring the sqrt() up to the LHS of the equation, and move the e to the denominator of the RHS. Then square both sides, and see what you end up with...

 

[epenguin]

Hi.

I am struggling with this question:

How do I make v the subject of this formula?

e=mc^2/sqrt(1-v^2/c^2)

Please reply with a step by step guide.

I have ended up with all kind of derivations to make v the subject so would appreciate your help.

Thanks.

Oh dear to make something the 'subject' is constantly needed and you will find in science textbooks that it is considered so evident, or the student sufficiently used to it that they will often jump from a formula like that to v= something without any step-by-step - so quickly get used.

The principle used is (as you will probably recall hearing said?) that if you do the same thing to both sides of an equation it remains true. You try to to those things which bring it to the form v = something. Those things are usually fairly obvious if v occurs only once in the equation.

For instance here if you multiply both sides by sqrt(1-v^2/c^2), divide both sides by e, square both sides, subtract 1 from both sides, multiply both sides by -1, and take the square roots of both sides, in that order, I think you'll get it.

Something to realize and get beyond rather quickly. Textbooks of physics etc. assume you can see it or do it for yourself.

 

[anonymouse123]

Thanks for both your replies so far.

The questions is exactly that, to make v the subject. I think the question is aimed at getting people to rearrange complex physics equations to prepare for further physics work,

So far my steps have been E*sqrt(1-v^2/c^2) = mc^2

Then (1-v^2/c^2) = m^2c^4 / E^2 ( I am unsure whether this is correct)

Then 1 - m^2c^4 / E^2 = v^2/c^2

c^2 ( 1 - m^2c^4 / E^2 ) = v ^ 2 ( Is this stage correct... i.e. should I not expand?)

To then get v on its own I know that I need to square root everything thus getting:

sqrt(c^2 ( 1 - m^2c^4 / E^2 ) ) = v

If the above last step is correct, how do I go about simplifying further. Would my answer be c sqrt( 1 - m^2c^4 / E^2 ) = v

Thanks for the help

 

[epenguin]

Yes your final answer looks right to me, hope for a second opinion.

It cannot be further simplified. The factor inside the square root is a difference of two squares so it is
√[(1 - mc²⁾/E)(1 + mc²/E)] - you cannot call that a simplification but it might be useful for some purposes.

Although you have done the operations I suggested I hope you see why they would come to mind: first the sought for v is in a denominator and you want it on top hence the first step, then it is under a square root which is nasty so you get it out by squaring both sides, etc...
By laws of algebra (arithmetic really) you can change the order in which you do some of these operations but not all.

 

[anonymouse123]

Thank you epenguin. I think all the powers is what had me stumped, I knew the method to do it.

I hope some one does give me a second opinion on the answer.