Special Relativity derivations ....

In summary: Under the following conditions:-The masses of the particles are the same-The velocities of the particles are the same-The distances between the particles are the sameThus, in summary, under these conditions, the energy "equals" the mass times the speed of light squared.
  • #1
VoteSaxon
25
1
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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  • #2
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)?
 
  • #3
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? ...
 
  • #4
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?
 
Last edited:

What is the concept of Special Relativity?

Special Relativity is a theory developed by Albert Einstein in 1905, which explains the relationship between space and time. It states that the laws of physics are the same for all observers in uniform motion and that the speed of light is constant regardless of the observer's frame of reference.

What are the key equations used in Special Relativity derivations?

The two key equations used in Special Relativity derivations are the Lorentz transformation equations and the time dilation equation. The Lorentz transformation equations describe how measurements of space and time change between two observers in relative motion, while the time dilation equation shows how time is perceived differently by observers moving at different velocities.

What is the significance of the speed of light in Special Relativity?

The speed of light, denoted by the symbol c, plays a crucial role in Special Relativity. According to the theory, the speed of light is constant and is the maximum speed at which all objects in the universe can travel. This means that nothing can move faster than the speed of light, and it remains the same regardless of the observer's frame of reference.

How does Special Relativity impact our understanding of time and space?

Special Relativity has revolutionized our understanding of time and space. It introduced the concept of spacetime, which combines the three dimensions of space and the dimension of time into a four-dimensional continuum. It also explains the phenomena of time dilation, length contraction, and the relativity of simultaneity, challenging our traditional Newtonian view of absolute time and space.

What are some real-world applications of Special Relativity?

Special Relativity has numerous practical applications, such as in the fields of GPS technology, nuclear energy, and particle accelerators. The theory's principles are also essential in understanding the behavior of objects traveling at high speeds, such as spacecraft and satellites, and have been used to make accurate predictions in astrophysics and cosmology.

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