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A heavy nucleus at rest breaks into two fragments which fly off with velocities in the ratio $3: 1$. The ratio of radii of the fragments is
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$1: 3^{1 / 3}$
From law of conservation of momentum
$\begin{aligned} m v=\text { constant } & \\ V \rho v & =k \\ \frac{4}{3} \pi R^3 \rho v & =k\end{aligned}$
$\left(\frac{R_1}{R_2}\right)^3=\frac{v_2}{v_1} \Rightarrow \frac{R_1}{R_2}=\left(\frac{v_2}{v_1}\right)^{1 / 3}=\left(\frac{1}{3}\right)^{1 / 3}=1: 3^{1 / 3}$
$\begin{aligned} m v=\text { constant } & \\ V \rho v & =k \\ \frac{4}{3} \pi R^3 \rho v & =k\end{aligned}$
$\left(\frac{R_1}{R_2}\right)^3=\frac{v_2}{v_1} \Rightarrow \frac{R_1}{R_2}=\left(\frac{v_2}{v_1}\right)^{1 / 3}=\left(\frac{1}{3}\right)^{1 / 3}=1: 3^{1 / 3}$
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