Neutron colliding with an atom

[zorro]

Homework Statement

A neutron moving with a speed v makes a head-on collision with a hydrogen atom in ground state kept at rest. Find the minimum kinetic energy of the neutron, for which inelastic collision may take place. Mass of neutron = Mass of Hydrogen = 1.67 x 10^-27kg

The Attempt at a Solution

Let the final velocities of the nuetron and hydrogen atom be v1 and v2.
I used conservation of momentum and energy to get the following equations-
v = v₁ + v₂
v² = v₁² + v₂² + 2E/m

where E is the energy released due to electronic transition.
How do I find out the min. kinetic energy?

 

[gneill]

Let the final velocities of the nuetron and hydrogen atom be v1 and v2.
I used conservation of momentum and energy to get the following equations-
v = v₁ + v₂
v² = v₁² + v₂² + 2E/m

where E is the energy released due to electronic transition.
How do I find out the min. kinetic energy?

If you square your first equation and compare the result to the second equation, you may spot an interesting relationship between the E and the product of the velocities.

Then consider that for maximum energy availability for the excitation you want the difference between the final velocities to head towards zero.

 

[zorro]

Beautiful!
I was thinking all time about using derivatives.
Thank You.

Btw we have to consider the minimum energy availability for the excitation