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Question

# A nucleus moving with a velocity $\stackrel{\to }{v}$ emits an α-particle. Let the velocities of the α-particle and the remaining nucleus be v1 and v2 and their masses be m1 and m2. (a) $\stackrel{\to }{v},{\stackrel{\to }{v}}_{1}\mathrm{and}{\stackrel{\to }{v}}_{2}$ must be parallel to each other. (b) None of the two of $\stackrel{\to }{v},{\stackrel{\to }{v}}_{1}\mathrm{and}{\stackrel{\to }{v}}_{2}$ should be parallel to each other. (c) $\stackrel{\to }{{v}_{1}}+\stackrel{\to }{{v}_{2}}$ must be parallel to $\stackrel{\to }{v}$ (d) ${m}_{1}\stackrel{\to }{{v}_{1}}+{m}_{2}\stackrel{\to }{{v}_{2}}$ must be parallel to $\stackrel{\to }{v}$

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Solution

## (d) ${m}_{1}\stackrel{\to }{{v}_{1}}+{m}_{2}\stackrel{\to }{{v}_{2}}$ must be parallel to $\stackrel{\to }{v}$ By the law of conservation of linear momentum, we can write: $\mathrm{Initial}\mathrm{momentum}=\mathrm{Final}\mathrm{momentum}\phantom{\rule{0ex}{0ex}}⇒m\stackrel{\to }{v}={m}_{1}{\stackrel{\to }{v}}_{1}+{m}_{2}{\stackrel{\to }{v}}_{2}\phantom{\rule{0ex}{0ex}}⇒\left({m}_{1}{\stackrel{\to }{v}}_{1}+{m}_{2}{\stackrel{\to }{v}}_{2}\right)\mathrm{must}\mathrm{be}\mathrm{parallel}\mathrm{to}\stackrel{\to }{v}$

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