An electron possessing a kinetic energy of 5 eV strikes a hydrogen...

The energy of a hydrogen atom in its ground state ($n=1$) is $-13.6 \text{ eV}$.

The energy of the first excited state ($n=2$) is $-3.4 \text{ eV}$.

The minimum energy required to excite the hydrogen atom from the ground state to the first excited state is $\Delta E = -3.4 \text{ eV} - (-13.6 \text{ eV}) = 10.2 \text{ eV}$.

The incident electron has a kinetic energy of $5 \text{ eV}$, which is less than the minimum excitation energy of $10.2 \text{ eV}$.

Since the energy levels are quantized, the hydrogen atom cannot absorb any fraction of this energy. Thus, the internal energy of the hydrogen atom remains unchanged.

Because there is no loss of kinetic energy to internal excitation, the collision must be perfectly elastic.