Joseph Rao writes in to ask about AP Physics B question 6 on the 2011 exam. (Can't post it, but follow the link.) He notes:
The question said that an electron with KE of 4.8 eV can excite an electron in an atom upon collision even if the energy does not match the transitional energy from n=1 to n-=2. The energy difference between states was for 3 eV.
Huh... this confuzzled me momentarily, too. I would've gotten it wrong on first attempt.
What makes this problem unusual is that it's an electron -- a massive particle -- causing the transition.
It's a PHOTON that can either be absorbed entirely or not. There's no such thing as absorbing just some of a photon's energy.
However, an electron is a massive, classical particle, with speed that can vary. It had 4.8 eV to begin with due to its 1.3 x 106 m/s speed. Nothing wrong with the electron slowing down, giving some of its kinetic energy to the bound electron in the atom. The bound electron jumps states by gaining 3.0 eV. (The bound electron must absorb only the amount of energy necessary to jump states.) The free electron goes on its merry way, now with (4.8 eV - 3.0 eV) = 1.8 eV of kinetic energy. You can work out its speed to be smaller now.