ElectronAtom 1D Collision
1. The problem statement, all variables and given/known data
An electron, mass m, collides headon with mass M, initially at rest. As a result of the collision, a characteristic amount of energy E is stored internally within the atom. What is the minimum initial speed v_{0} that the electron must have? (Hint: Conservation principles lead to a quadratic equation for the final electron speed v and a quadratic equation for the final atom speed V. The minimum value v_{0} follows from the requirement that the radical in the solutions for v and V be 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? 
Re: ElectronAtom 1D Collision
Welcome to Physics Forums.
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If these terms are to be added to and/or subtracted from each other, then the units must all agree. Do they? Finally, what happens if you have 2E/m instead of 2E? 
Re: ElectronAtom 1D Collision
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Re: ElectronAtom 1D Collision
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However, note that your final answer has units of Energy^{1/2}, which is not a velocity, whereas the book's answer does result in velocity units. Whenever I get a result with the wrong units, I backtrack through my work to find the step where the units first became inconsistent. That's almost certainly where the error happened. 
Re: ElectronAtom 1D Collision
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In short, the book answer was right, but equation (3) provided by the book and my TA was wrong, as I reached the answer using 2E/m in place of 2E. Thanks for everything Redbelly! 
Re: ElectronAtom 1D Collision
You're welcome. :smile: Hooray for units!

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