Another energy and momentum SR problem

[zimo]

Homework Statement

A 100kev photon scatters from a free electron initially at res. Find the recoil velocity of the electron if the photon scattering angle is 180 degrees.

Homework Equations

Conservation of Energy, Conservation of Momentum
(It is not allowed to use Compton's effect equation!)

The Attempt at a Solution

I've written:
$$\eqalign{ & {{hc} \over \lambda } + m_e c^2 = {{hc} \over {\lambda '}} + {{m_e c^2 } \over {\sqrt {1 - {{\bar v^2 } \over {c^2 }}} }} \cr$$
$$& {h \over \lambda } = {{m_e \bar v} \over {\sqrt {1 - {{\bar v^2 } \over {c^2 }}} }} - {h \over {\lambda '}} \cr}$$

but as I develop the equations on trying to solve the equations, somewhat it doesn't work out...

 

[tiny-tim]

I've written:
$$\eqalign{ & {{hc} \over \lambda } + m_e c^2 = {{hc} \over {\lambda '}} + {{m_e c^2 } \over {\sqrt {1 - {{\bar v^2 } \over {c^2 }}} }} \cr$$
$$& {h \over \lambda } = {{m_e \bar v} \over {\sqrt {1 - {{\bar v^2 } \over {c^2 }}} }} - {h \over {\lambda '}} \cr}$$

but as I develop the equations on trying to solve the equations, somewhat it doesn't work out...

Hi zimo!

Well, you want v, but you don't want $$\lambda'$$ …

so just multiply the second equation by c, and add it to the first equation.

 

[zimo]

OK.

Did I write the equations correctly?

 

[tiny-tim]

OK.

Did I write the equations correctly?

Looks ok to me.

Why are you asking?

 

[zimo]

because I get v=0.483c, in contrast to the final answer in the book, which is 0.319c.

 

[tiny-tim]

because I get v=0.483c, in contrast to the final answer in the book, which is 0.319c.

Unless you show your full calculations, we can't see where you've gone wrong.

 

[zimo]

I attached my solution. (starting with the lines indicating I) and II) )

 

[zimo]

can someone read my uploaded solution, please, and reveal my mistake?

 

[Hootenanny]

can someone read my uploaded solution, please, and reveal my mistake?

Your attachment needs to be approved by a moderator before we can view it. Be patient, attachments are usually approved fairly quickly. If it hasn't been approved by this evening I'll see if I can get someone's attention.

 

[tiny-tim]

because I get v=0.483c, in contrast to the final answer in the book, which is 0.319c.

Hi zimo!

√(1 - u²)0.711 MeV = 0.511(u + 1) MeV

(1 - u) = 1.936(1 + u)
…
u = .483

That's with (.711/.511)² = 1.936.

The book seems to have used .711/.511 = 1.391, giving the answer of .319

The book seems to be wrong. ​

 

[zimo]

No.

I made an error during the strike-out of (u+1) on both sides, I needed to leave one (u+1) at the right side of the eqation.

Finally, I got 0.319 as requested.