Inelastic collision energy relationship

[gills]

Homework Statement

An atom of mass M is initially at rest, in its ground state. A moving (nonrelativistic) electron of mass me 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.

Find the kinetic energy that the electron must have in order to excite the atom.
Express your answer in terms of E, me, and M.

Homework Equations

Inelastic collision:
m1*v1 + m2*v2 = (m1+m2)Vf

and possibly:

$$\Delta$$K = Ki - Kf

The Attempt at a Solution

What I've got so far:
m1*v1 + m2*v2 = (m1+m2)Vf -->

me*ve + (0) = (me + M)*Vf -->

Vf = (me*ve)/(me + M)$$\Delta$$K = Ki - Kf -->

Kf = (1/2)(me + M) * [(meve)/(me + M)]^2 --> simplified -->

(me*ve)^2/2(me+M)

Ki = just the Kinetic energy of the electron = (1/2)(me*ve^2 --> so..

(me*ve)^2/2(me+M) - (1/2)(me*ve^2) = E(?.. an assumption) --->

so K_e = (m_e * v_e)^2/2(m_e + M) - E --->

there's the problem, i have v_e in my solution. But first i need to know I'm on the right track which i think i am. Any advice?

 

[Doc Al]

(me*ve)^2/2(me+M) - (1/2)(me*ve^2) = E(?.. an assumption) --->

You want K_i - K_f = E, since kinetic energy is lost to internal energy.

so K_e = (m_e * v_e)^2/2(m_e + M) - E --->

there's the problem, i have v_e in my solution. But first i need to know I'm on the right track which i think i am.

You are exactly on track, except for the error that I pointed out. Correct that, of course.

Then: Eliminate the v_e in your solution by expressing it in terms of K_e, and then solving for K_e in terms of E and the masses.

 

[gills]

You want K_i - K_f = E, since kinetic energy is lost to internal energy.You are exactly on track, except for the error that I pointed out. Correct that, of course.

Then: Eliminate the v_e in your solution by expressing it in terms of K_e, and then solving for K_e in terms of E and the masses.

I'm having trouble isolating K_e when i substitute V_e = (2k_e)/m_e

Here's my new eq. as K_i - K_f and V_e in terms of K_e:

K_e =[(m_e *(2K_e/m_e)]^2/[2(m_e + M)] - E ---> I'm having trouble isolating K_e. Is this setup correct? --->

4(K_e)^2/[2(m_e + M)] - E = K_e

 

[Doc Al]

I'm having trouble isolating K_e when i substitute V_e = (2k_e)/m_e

That's not quite right. K_e = 1/2 m_e v_e^2 (Don't forget that v_e is squared!)

 

[gills]

That's not quite right. K_e = 1/2 m_e v_e^2 (Don't forget that v_e is squared!)

wow, i keep making dumb mistakes.

I've ended up with this:

K_e = E(m_e + M)/2m_e + M --->

 

[gills]

nevermind ^ that.

I got K_e = E(m_e + M)/-M

I don't think the negative sign is correct. I got to check my algebra. What do you think?

 

[Doc Al]

The negative sign is not correct. Did you correct the sign error I pointed out in post #2? (Other than that, it's good.)