A particle of mass 200 , MeV/c^2 collides wit | vedclass

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(D) Let $m = 200 \, MeV/c^2$ be the mass of the particle and $M = 1000 \, MeV/c^2$ be the mass of the hydrogen atom.Let $v$ be the initial velocity of

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$51$

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(D) Let $m = 200 \, MeV/c^2$ be the mass of the particle and $M = 1000 \, MeV/c^2$ be the mass of the hydrogen atom.Let $v$ be the initial velocity of the particle and $V$ be the recoil velocity of the atom.By conservation of linear momentum: $mv = MV = p$,where $p$ is the momentum.The energy required to excite the hydrogen atom to its first excited state is $\Delta E = 13.6 \, eV 	imes (1 - 1/4) = 10.2 \, eV$.By conservation of energy: $K_{initial} = K_{final} + \Delta E$.$K_{initial} = \frac{p^2}{2m}$ and $K_{final} = \frac{p^2}{2M}$.Thus,$\frac{p^2}{2m} - \frac{p^2}{2M} = 10.2 \, eV$.$\frac{p^2}{2m} (1 - \frac{m}{M}) = 10.2 \, eV$.Given $m/M = 200/1000 = 0.2$,we have $K_{initial} (1 - 0.2) = 10.2 \, eV$.$K_{initial} (0.8) = 10.2 \, eV$.$K_{initial} = \frac{10.2}{0.8} = 12.75 \, eV$.We are given $K_{initial} = \frac{N}{4}$,so $\frac{N}{4} = 12.75$.$N = 12.75 	imes 4 = 51$.

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