Conservation of energy regarding electrons

[theowne]

Homework Statement

An atom of mass M is initially at rest, in its ground state. A moving (nonrelativistic) electron of mass m_e collides with the atom. The atom+electron system can exist in an excited state in which the electron is absorbed into the atom. The excited state has an extra, "internal," energy E relative to the atom's ground state.

To be honest, I am very confused reading this. The problem system I am going through has a section where you can ask for hints and for the most part that's mostly how I've been trying to complete this, but a more clear explanation of what this is saying would be helpful.

Homework Equations

Kinetic energy = 1/2 mv^2
Momentum = mv

The Attempt at a Solution

Well using the hints, first I was asked to find the final velocity of the atom, supposing the initial velocity was v_0. So using conservation of momentum I found it.

Here's the work I did so far:

So I eliminated it and got K final = (Kinitial * m_e) / (m_e + M)

But where to go from here? I tried for a long time but I don't understand this problem at all. The answer given is this:

K_e = ([M + m_e] / M) * E

Can someone please tell me how they got this answer?

As you can see I don't need the answer, I just want to understand this problem~! The most confusing aspect to me is what this E is supposed to represent, where it comes in, I don't get it..

 

[Dick]

The collision is inelastic. Energy and momentum are conserved, but not kinetic energy. The bound state energy E should be equal to K_initial-K_final. K_initial is K_e, the energy of the incoming electron.

 

[theowne]

How do you know it is inelastic from the question? What is "bound state energy"? And how is that used to get the answer that the software gives me?

 

[Dick]

It's inelastic because K_initial is not equal to K_final. The difference is the "bound state energy", the "extra internal energy E" that the problem talks about. And if had read my answer you would know how to find it. I told you: E=K_initial-K_final.

 

[theowne]

I'm sorry, I do not understand the initial question's wording about "bound state energy". I have never learned such a thing and I don't quite understand the question...I know how to plug things into formulas, I was hoping for a conceptual explanation of what this question describes...but you seem annoyed so I will ask elsewhere, thanks anyways.

 

[Dick]

Apologies for being annoyed. That wasn't very patient of me. In a general inelastic collision where the two colliding objects stick together some energy is lost. You can compute the amount just by conservation of momentum. That's what the question is asking you to do. Whatever is lost in kinetic energy has to appear some place else. That's the E. If it's two pieces of putty, then you say the lost energy goes into heat and acoustic vibration. The question is encouraging you to think of it as going into pushing the target atom into an excited state, for example an atom with a extra bound electron. But it doesn't really matter where you think of it as going. You can still compute the amount by the difference between the kinetic energy of the incoming state minus the kinetic energy of the outgoing state. Sorry again to be annoyed.