1. Physical condition for inelastic collision

A stationary hydrogen atom is initially in the ground state. For an inelastic collision to occur, the neutron must transfer at least the excitation energy required to take hydrogen from ground state to first excited state.

For hydrogen,

So the minimum excitation energy is

However, the neutron cannot transfer all of its kinetic energy because of conservation of momentum. So we must find the maximum possible energy transfer in a head-on collision.

2. Use center-of-mass / two-body collision result

The neutron collides head-on with a stationary hydrogen atom. Since the hydrogen atom mass is approximately equal to neutron mass,

Let the initial neutron speed be $v$. Initial kinetic energy of neutron:

In a collision of equal masses, the maximum kinetic energy that can be transferred from the moving particle to the target is the entire kinetic energy associated with relative motion in the center-of-mass frame. For equal masses, at threshold for inelasticity, the final two bodies move together with center-of-mass speed.

Center-of-mass speed is

At threshold, after collision both move with speed $V_{cm}=v/2$, so total final translational kinetic energy is

Initial kinetic energy is

Hence maximum energy available for excitation is

But since

we get

So for equal masses, at most half of the neutron's initial kinetic energy can go into excitation.

3. Threshold condition

For minimum neutron kinetic energy, this maximum transferable energy must just equal $10.2\,\text{eV}$:

Therefore,

4. Check options

• A: $10.2\,\text{eV}$ — not enough, because all energy cannot be transferred.
• B: $16.8\,\text{eV}$ — half is $8.4\,\text{eV}$, still insufficient.
• C: $12.1\,\text{eV}$ — half is $6.05\,\text{eV}$, insufficient.
• D: $20.4\,\text{eV}$ — half is $10.2\,\text{eV}$, exactly the threshold.

Therefore the correct option is