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

A block of mass $\mathrm{m}=1 \mathrm{kg}$ slides with velocity $\mathrm{v}=6 \mathrm{m} {\mathrm{s}}^{-1}$ on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown. The rod is pivoted about $\mathrm{O}$ and swings as a result of the collision making angle $\theta$ before momentarily coming to rest. if the rod has mass $M=2 \mathrm{kg},$ and length $\ell =1 \mathrm{m},$ the value of $\theta$ is approximately$\left.(\mathrm{take} \mathrm{g}=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

A ball of mass $10 \mathrm{kg}$ moving with a velocity $10\sqrt{3}{ \mathrm{m} \mathrm{s}}^{-1}$ along $X$-axis, hits another ball of mass $20 \mathrm{kg}$ which is at rest. After the collision, the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along $Y$-axis at a speed of $10 \mathrm{m} {\mathrm{s}}^{-1}$. The second piece starts moving at a speed of $20 \mathrm{m} {\mathrm{s}}^{-1}$ at an angle $\theta$ (degree) with respect to the $X$-axis. The configuration of pieces after the collision is shown in the figure. The value of $\theta$(in degrees) to the nearest integer is _________.