Special Relativity derivations ....

AI Thread Summary
The discussion focuses on deriving linear relations between momentum, energy, and mass using special relativity formulas. The initial attempt suggests that momentum can be expressed as p = γmv = γmβc, but uncertainty remains about its classification as a linear relation. For energy, there is confusion regarding the expectation to derive E = mc², while the relationship between energy and momentum is also under scrutiny, particularly for massless particles. Clarification is sought on the conditions under which E = mc² holds true, emphasizing that E is not equal to mc² in general. The conversation highlights the need for a deeper understanding of the relationships among these quantities in the context of special relativity.
VoteSaxon
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Homework Statement
Using the special relativity formulae
p = mv / [1 - (v/c)2]
E2 = p2c2 + m2c4
derive linear relations between:
(i) momentum and mass;
(ii) energy and mass;
(iii) energy and momentum,
which involve only c, c2, β = v/c, and γ (= 1/sqrt(1 - β2))

The attempt at a solution
I am pretty sure the answer to (i) is p = γmv = γmβc, although I am unsure if this counts as a linear relation.
I suppose for (ii) I should be aiming for E = mc2, and for (iii) maybe I should be trying to get to E = pc (although I think this only applies to massless particles), but I haven't had much luck thus far.
I know these aren't really that hard, but for some reason my brain is just drawing blanks with these.

Many thanks for help and patience.
 
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I think that here "linear" means that the two variables being related are each of first-power (no squares or square roots).
So, (i) is fine.
It seems that you have three quantities: p,m,E (mutually related by the above)... and you wish to find the relations between pairs chosen from those three.
So, what would you get for (ii)?
 
robphy said:
I think that here "linear" means that the two variables being related are each of first-power (no squares or square roots).
So, (i) is fine.
It seems that you have three quantities: p,m,E (mutually related by the above)... and you wish to find the relations between pairs chosen from those three.
So, what would you get for (ii)?

Sorry, a bit confused. I am pretty sure for (ii) I am meant to get E = mc2. My main problem is getting there from the formulae the question provided. Does that make sense? ...
 
VoteSaxon said:
Sorry, a bit confused. I am pretty sure for (ii) I am meant to get E = mc2. My main problem is getting there from the formulae the question provided. Does that make sense? ...

From your given relations,
E is the relativistic energy, p is the relativistic momentum, and m is the [invariant] rest mass.
With these symbols, ##E\neq mc^2## in general.
You can see this immediately by plugging in "what you think your E should be" into "E2 = p2c2 + m2c4".
When (that is, Under what conditions) will E="what you think your E should be" be true?
 
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