Q127 / 150 · MHT CET 2021 (22 Sep Shift 1) · Physics

A nucleus breaks into two nuclear parts, which have their velocity ratio $2: 1$. The ratio of their nuclear radii will be

  • A
    $\sqrt{2}$
  • B
    $\frac{1}{2}$
  • Correct answer
    C
    $\frac{1}{2^{1 / 3}}$
  • D
    $\frac{1}{\sqrt{2}}$

Solution

By law of conservation of momentum, the two parts will have equal and opposite momentum, assuming the nucleus was initially at rest. $$ \begin{aligned} & \therefore \mathrm{m}_1 \mathrm{v}_1=\mathrm{m}_2 \mathrm{v}_2 \text { or } \frac{\mathrm{m}_1}{\mathrm{~m}_2}=\frac{\mathrm{v}_2}{\mathrm{v}_1}=\frac{1}{2} \\ & \frac{\mathrm{m}_1}{\mathrm{~m}_2}=\left(\frac{\mathrm{r}_1}{\mathrm{r}_2}\right)^3=\frac{1}{2} \\ & \therefore \frac{\mathrm{r}_1}{\mathrm{r}_2}=\frac{1}{2^{1 / 3}} \end{aligned} $$

Hence, the correct answer is option: 'C'.

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