Relativity question involving proofs of beta and gamma

[AegisFLCL]

Homework Statement

(a) The ratio v/c is very often denoted by the single symbol β. Show that if β<<1, the following are valid through terms of order β^2

E = mc^2 + (mv^2)/2 = (mc^2)[1+(β^2)/2]
K = (mv^2)/2 = (mc^2)[(β^2)/2]
pc = mvc = m(c^2)β
γ = 1+(β^2)/2

(b) Show that if γ = ε^-1 >> 1, the following are valid through terms of order ε^2

β = 1-(ε^2)/2
E = (ε^-1)*m*c^2
K/E = 1-ε
pc/E = 1-(ε^2)/2
K/pc = 1-ε+(ε^2)/2

Homework Equations

Either none are necessary or I am missing something...

The Attempt at a Solution

First I'm not entirely clear on what the question is asking (especially part b). Am I supposed to substitute in some arbitrary value that goes along with the statement or are they simply asking for algebraic solutions? If someone could clarify and provide hints it would be much appreciated. I did try the first part of part a however.

E = mc^2 + (mv^2)/2 = (mc^2)[1+(β^2)/2]

mc^2 + (mv^2)/2 = (mc^2)[1+(β^2)/2]

1+(β^2)/2 = 1+[(mv^2)/2]/(mc^2)

(β^2)/2 = [(mv^2)/2]*(mc^2)^-1

β^2 = (mv^2)/(mc^2)

β = v/c

 

[cepheid]

Welcome to PF AegisFLCL,

I think you're kind of on the wrong track here. For your answer to part a, you've started by assuming what you were supposed to prove! What you need to do in the case of the first one (energy) is to start with the full, proper definition of relativistic energy: E = γmc² where γ = (1 - β)^-1/2. You need to show that the expression they gave you is equivalent to this, to second order in beta. In other words, if you take a Taylor series expansion (around β = 0) of the expression for the relativistic energy, and you keep all of the terms in this series approximation up to second order, you should end up with the expression given in the problem. Since β is small, this series approximation should be reasonably accurate, even though you discarded all the higher order terms (above second order).

The point of this question (the first equation) is to show that if β is small, you can just think of the total energy of a particle as being the sum of its rest energy and its non-relativistic (Newtonian) kinetic energy, and this is true to second order.

 

[AegisFLCL]

Welcome to PF AegisFLCL,

I think you're kind of on the wrong track here. For your answer to part a, you've started by assuming what you were supposed to prove! What you need to do in the case of the first one (energy) is to start with the full, proper definition of relativistic energy: E = γmc² where γ = (1 - β)^-1/2. You need to show that the expression they gave you is equivalent to this, to second order in beta. In other words, if you take a Taylor series expansion (around β = 0) of the expression for the relativistic energy, and you keep all of the terms in this series approximation up to second order, you should end up with the expression given in the problem. Since β is small, this series approximation should be reasonably accurate, even though you discarded all the higher order terms (above second order).

The point of this question (the first equation) is to show that if β is small, you can just think of the total energy of a particle as being the sum of its rest energy and its non-relativistic (Newtonian) kinetic energy, and this is true to second order.

Thanks for the welcome and speedy reply. I'm still having trouble understanding the problem. Does each line (separate eq) require a separate solution? After taking calc 1, 2, DE and multivariable calc I don't remember learning/using a Taylor series expansion. The chapter this question pertains to makes no mention of gamma, beta or any relative equations aside from the one you've given me. In general the clarity of the book is just very poor (for me).

 

[cepheid]

Thanks for the welcome and speedy reply. I'm still having trouble understanding the problem. Does each line (separate eq) require a separate solution?

Yes, each one is separate approximation that you have to show holds to second order. EDIT: Although, once you do the first one, the second one just follows from it without any additional work.

After taking calc 1, 2, DE and multivariable calc I don't remember learning/using a Taylor series expansion.

This is a pretty important topic and is usually covered in calc 1, since it is an application of derivatives. I'd say you'll have to look it up and refresh your memory. But basically, it's a method for expressing a function in terms of an infinite power series in some variable. So, truncating the Taylor series after n+1 terms amounts to finding an nth order polynomial that is an approximation to the function. It should be exactly equal to the function at the point around which you're taking the expansion, but deviate from it as you move away from that point, with an error that decreases as you make n larger (i.e. as you include more and more terms from the infinite series). In your case, you only have to find a second order approximation.

The chapter this question pertains to makes no mention of gamma, beta or any relative equations aside from the one you've given me. In general the clarity of the book is just very poor (for me).

The book certainly must provide definitions for gamma, for the relativistic energy, for the relativistic kinetic energy, and for the relativistic momentum. (If not, look them up on Wikipedia ) None of these are given in your problem. The equations given in the problem are approximations to these that hold true in the classical (Newtonian) limit. You have to show that they are valid approximations.

 

[paranormal]

This is the definition of β, so it is already valid through terms of order β^2. I am not sure what else the question is asking for. It seems like it may be asking for algebraic manipulations to show that these equations are valid through terms of order β^2, but it is not entirely clear. In any case, I would suggest rephrasing the question to make the desired answer more clear.