(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

An electron, massm, collides head-on with massM, initially at rest. As a result of the collision, a characteristic amount of energyEis stored internally within the atom. What is the minimum initial speedv_{0}that the electron must have? (Hint: Conservation principles lead to a quadratic equation for the final electron speedvand a quadratic equation for the final atom speedV. The minimum valuev_{0}follows from the requirement that the radical in the solutions forvandVbe real.)

2. Relevant equations

[tex] P=mv [/tex]

[tex] KE = mv^2/2 [/tex]

3. The attempt at a solution

At first glance, I can tell its a momentum problem since its a collision and I'm given variables for mass and velocity. Since characteristic energy E is stored as a result of the collision, I know that this is an inelastic collision because kinetic energy is not conserved.

Since I know that the conservation of momentum still applies, I can say:

(1) mv_{0}= mv + MV

And also, since I know that total energy is conserved, I can also say:

(2) (1/2)mv_{0}^{2}= (1/2)mv^{2}+ (1/2)MV^{2}+ E

I've worked through this problem several times, and I know that somewhere I will get a quadratic equation for v or V, depending on which of the two I eliminate, then end up using the quadratic formula to get a solution.

If I eliminate the variable V (via substitution or whatever technique), I'm "supposed" to end up with the equation:

(3) (1 + m/M)v^{2}- (2m/M)v_{0}v + 2E - (1 - m/M)v_{0}^{2}= 0.

I understand the concepts behind how the result is gotten, the problem is that when I do the substitution, I get the same result, except I end up with a 2E/m instead of a 2E. I'm not sure if I'm right or not, because if I attach units to equation (3) that my book and the TA gave me, it doesn't make sense to add Joules (kg m^2/s^2) to an equation where the rest of the units are m^2/s^2.

Moving along, solving for v yields:

(4) v = (m/M)v_{0}(+/-) [v_{0}^{2}- 2E(m+M)/mM]^(1/2)

from there you can just say that the minimum value of v_{0}is the square root of 2E(m+M)/mM.

As I stated before, I know the basic concepts, but I'm not sure how they got (3) from (1) and (2). Did the book and the TAs make a mistake?

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# Electron-Atom 1-D Collision

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