A neutron collides head-on with a stationary | vedclass

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(D) Let the mass of the neutron and the hydrogen atom be $m$. Let the initial velocity of the neutron be $v$ and the hydrogen atom be $0$. After the c

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$(A)$ and $(C)$ both.

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(D) Let the mass of the neutron and the hydrogen atom be $m$. Let the initial velocity of the neutron be $v$ and the hydrogen atom be $0$. After the collision,let their velocities be $v_1$ and $v_2$. By conservation of momentum: $mv = mv_1 + mv_2 \implies v = v_1 + v_2$. The kinetic energy available in the center of mass frame is $K_{cm} = \frac{1}{2} \mu v_{rel}^2$,where $\mu = \frac{m \cdot m}{m+m} = \frac{m}{2}$ and $v_{rel} = v$. Thus,$K_{cm} = \frac{1}{2} (\frac{m}{2}) v^2 = \frac{1}{4} mv^2 = \frac{K_{initial}}{2}$. For the hydrogen atom to be excited from the ground state $(n=1)$ to the first excited state $(n=2)$,the energy required is $\Delta E = E_2 - E_1 = -3.4 \, eV - (-13.6 \, eV) = 10.2 \, eV$. Since only half of the initial kinetic energy is available for excitation,we need $\frac{K_{initial}}{2} \ge 10.2 \, eV$,which implies $K_{initial} \ge 20.4 \, eV$. If $K_{initial} < 20.4 \, eV$,the energy is insufficient to excite the atom,so the collision must be elastic. Thus,statements $(A)$ and $(C)$ are correct.

$A$ neutron collides head-on with a stationary hydrogen atom in its ground state. Which of the following statements are correct? (Assume that the hydrogen atom and neutron have the same mass):

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