Derive p^2/2m from relativistic equations

[Mancuso]

Given the relativistic equation for energy E² = (pc)² + (mc²)²
I want to find the non-relativistic approximation for kinetic energy in non-relativistic terms,
K_nr = p²/2m

I start off with subtracting the rest energy
E₀=mc₂
from the above equation.

So K = E - E₀

and assume that c is very large.
I've messed around for hours on the algebra and I need help.
I want to show that K ≈ K_nr

I am doing this using a linear approximation. I've written the energy as E = E₀ √1+x

And using the function f(x)=√1+x about x = 0

I've derived the linearization as L(x) = 1 + x/2

But I am struggling with relating to the equations above to show that K ≈ K_nr

 

[TSny]

I am doing this using a linear approximation. I've written the energy as E = E₀ √1+x

And using the function f(x)=√1+x about x = 0

I've derived the linearization as L(x) = 1 + x/2

You have the right approach. Can you tell us how x is related to p when you write E = E₀ √(1+x) ?

 

[Mancuso]

That's where I get stuck, relating the two equations

1. E = E₀√(1+x) and
2. E = √( (pc)² + (mc²)²)

I've arranged the relativistic equations as such
3. K = E - E₀
3. K = √( (pc)² + (mc²)²) - mc²

I don't know what x represents in equation 1 and so I don't know where I go next.

I tried a lot of algebraic manipulation of equation 2. including substituting mv for p
In an effort to find a structural relationship, but I'm lost. I would appreciate help relating x to p.
This question comes from a numerical methods section of a calculus course. I completely understand the numerical methods, but I'm struggling with the physics and how to apply a linear approximation to relate the two equations. I don't understand which part of equation 3. I am substituting with Eq. 1 and if any variables should be eliminated or rearranged.

 

[TSny]

1. E = E₀√(1+x) and
2. E = √( (pc)² + (mc²)²)

Can you rewrite equation 2 using E₀ instead of mc²? That might help you see how to identify x in equation 1.

 

[Mancuso]

Yes,

mc² = √(E² - (pc)²)

sub into eq 1?

E = √(E² - (pc)²)⋅√(1 + x)

I still don't see it. Please help

 

[TSny]

Your equation 2 can be written as $E = \sqrt{(pc)^2 + E_0^2}$. Is there any way to "pull" the $E_0$ outside the square root so that it looks more like equation 1?

 

[Mancuso]

Yes!

E = E ₀ √((pc)²/E₀² +1)

So x represents (pc)²/E₀²

This problem has taken up my entire night and it's only worth 0.2% of my final mark haha. The problem was so interesting that I could not put it down. I will tackle the rest of the problem in the morning, might bug you again for a hint if I get stuck. Thanks so much for your help, you're awesome !

 

[TSny]

E = E ₀ √((pc)²/E₀² +1)

So x represents (pc)²/E₀²

Yes, good.