Gareth56 Posted November 6, 2009 Share Posted November 6, 2009 In a book I have the following step isn't clear to me:- r = m x sqrt(kT/m) ------> r = sqrt(mkT) I just can't see the steps that go from the first equation to the second. Could anyone show me the missing step(s)? Link to comment Share on other sites More sharing options...
timo Posted November 6, 2009 Share Posted November 6, 2009 Here's a possible explanation. Feel free to skip any step if you find it unnecessary and also free to ask if one of the steps is not clear to you: [math] m \sqrt{ kT/m } = \sqrt{m^2} \sqrt{ kT/m } = \sqrt{ m^2kT/m } = \sqrt{mkT} [/math]. Edit: I was assuming that m is a mass, k the Boltzman constant and T the temperature and that hence all of these three variables are positive. You might have to be careful if some of them can be negative, e.g. m=sqrt(m^2) does not hold when m<0. Link to comment Share on other sites More sharing options...
Gareth56 Posted November 6, 2009 Author Share Posted November 6, 2009 (edited) Hi can I just ask how and why [math]m[/math] can become [math]\sqrt{m^2}[/math] ? I've just realised that [math]m[/math] is the same as [math]\sqrt{m^2}[/math] So was it really necessary to simplify [math]m \sqrt{ kT/m }[/math] to [math]\sqrt{mkT}[/math] or could you just have left it as it was? Edited November 6, 2009 by Gareth56 Penny dropped!!! Link to comment Share on other sites More sharing options...
timo Posted November 6, 2009 Share Posted November 6, 2009 It's not absolutely needed and depending on the context either version could be preferable. But usually, the shorter an expression and the less frequent symbols appear, the better. OLD POST, REDUNDANT: Sure, I am just not sure how to explain it. Seems pretty obvious that the square root of a number squared gives the number. The square root is something like the anti-operation (the technical term is "inverse") of squaring a number. Would you agree that [math]\sqrt{x}^2 = x[/math]? In that case you can rearrange it as [math] \sqrt{m^2} = \sqrt{m\cdot m} = \sqrt m \sqrt m = \sqrt{m}^2 = m[/math]. Perhaps you should also just do some reading on what the square root is and how you rearrange expressions involving it. Or a somewhat more intuitive explanation: The square root of X tells me which value Y=sqrt(X) I'd have to square to get X. So which value do I have to square to get m-squared? m. Hence m = sqrt(m*m). 1 Link to comment Share on other sites More sharing options...
Gareth56 Posted November 6, 2009 Author Share Posted November 6, 2009 Many thanks. 'O' level maths was so long ago Link to comment Share on other sites More sharing options...
Bignose Posted November 7, 2009 Share Posted November 7, 2009 m is almost certainly the symbol for mass, which will always be a positive number. In that case, the square root of the square is always positive as well - there is no ambiguity in taking the square root. I think that [math]\sqrt{mkT}[/math] looks better than [math]m\sqrt{\frac{kT}{m}}[/math], but in the end it is just aesthetics. Link to comment Share on other sites More sharing options...
the tree Posted November 7, 2009 Share Posted November 7, 2009 So was it really necessary to simplify [math]m \sqrt{ kT/m }[/math] to [math]\sqrt{mkT}[/math] or could you just have left it as it was?When you're trying to find a value for this, two multiplications and a square root is preferable to two multiplications, a square root and a division. Partially because any operation is going to take up computation time and cause a loss of accuracy and also because division in particular is very sensitive to small inaccuracies. Link to comment Share on other sites More sharing options...
magicalprimate Posted December 9, 2009 Share Posted December 9, 2009 The first equation was m^1 x (kT)^1/2/(m)^1/2, which is (kT)^1/2(m^1)/[m^1/2]. m^1 divided by m^1/2 is m^1/2, and (kT)^1/2 is multiplied by that in the above to equal (kT^1/2)(m^1/2), and since the square root of a a dimensionless quantity multiplied by the square root of another dimensionless quantity is the same as the square root of both of those things multiplied together, the previous equation is (kT x m)^1/2 which is of course the same as sqrt(mkT) Link to comment Share on other sites More sharing options...
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