Center of Mass, Momentum & Collision

JEE Main level, numerical answer, medium

10 of 130

Question 10 of 130

A nucleus disintegrates into two nuclear parts, in such a way that ratio of their nuclear sizes is 1:21/31:{2}^{1/3}. Their respective speed have a ratio of n:1n:1. The value of nn is_________

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Answer: 2

Using the law of Momentum conservation
m1v1+m2(−v2)=0{m}_{1}{v}_{1}+{m}_{2}(-{v}_{2})=0
⇒m1v1=m2v2\Rightarrow {m}_{1}{v}_{1}={m}_{2}{v}_{2}
⇒(m1m2)=(v2v1)⇒4πρr3134πρr323=(v2v1)\Rightarrow \left(\dfrac{{m}_{1}}{{m}_{2}}\right)=\left(\dfrac{{v}_{2}}{{v}_{1}}\right) \Rightarrow \dfrac{\dfrac{4\pi \rho {{r}^{3}}_{1}}{3}}{\dfrac{4\pi \rho {{r}^{3}}_{2}}{3}}=\left(\dfrac{{v}_{2}}{{v}_{1}}\right)
It is given
r1r2=1213\dfrac{{r}_{1}}{{r}_{2}}=\dfrac{1}{{2}^{\dfrac{1}{3}}}
and(r1r2)3=(v2v1)Therefore,(12)=(v2v1)\text{and}{\left(\dfrac{{r}_{1}}{{r}_{2}}\right)}^{3}=\left(\dfrac{{v}_{2}}{{v}_{1}}\right) \text{Therefore}, \left(\dfrac{1}{2}\right)=\left(\dfrac{{v}_{2}}{{v}_{1}}\right)
⇒v1v2=21\Rightarrow \dfrac{{v}_{1}}{{v}_{2}}=\dfrac{2}{1}

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