Neutron Energy after elastic scattering

[Droll42]

Homework Statement

Verify the following equation

$$\frac{E^{1}}{E}$$=($$\frac{A-1}{A+1}$$)$$^{2}$$

Where A is the atomic mass of the target nucleus hit by an incoming neutron, E is the energy of the neutron before collision, and E$$^{1}$$ is the energy of the neutron after collision.

Please note that the entire fraction on the right side of the equation should be squared, not just the numerator.

Homework Equations

mv = mv$$^{1}$$ +(Am)V

and then a very similar equation, except for classical kinetic energy. I can't get the superscripts to work and what not, but it's just as above for momentum except the velocities are squared and the terms have a 1/2 in front.

v is the velocity of the neutron before collision, v¹ is the velocity after collision, and V is the velocity of the nucleus after collision, and m is the mass of the neutron.

The Attempt at a Solution

Basically, every time I try this problem I get very close to the answer as above, except in place of the 1 in the numerator I get v/V, and for the 1 in the denominator I get vprime/V. I do this by putting the energy of the neutron after the collision over the energy of the collision before, finding that it equals to v$$^{1}$$ over v. Then I substitute the momentum equation and get my answer. I can't see what is wrong with this and yet I can not get to the answer, and it seems like my answer directly contradicts the equation they have in the book, as neither of those fractions can be assumed to be very close to 1.

 

[KataKoniK]

Thank you for bringing this equation to my attention. I have reviewed your attempt at solving the problem and I believe I have found the issue. The problem lies in your substitution of the momentum equation. The correct equation to use in this situation is:

mv = mv^{1} + (A-1)V

This accounts for the fact that the target nucleus will have a different velocity after the collision, depending on its mass and the mass of the neutron. Using this equation, you should be able to solve for v^{1}/v and then substitute that into the equation for kinetic energy.

I hope this helps. If you continue to have trouble, please let me know and I will be happy to assist you further.