Question

A nucleus at rest splits into two nuclear parts having same density and radii in the ratio $1:2$. Their velocities are in the ratio

• A. $2:1$
• B. $4:1$
• C. $6:1$
• D. $8:1$

Answer 1

The correct option is D

$8:1$

Let, the mass of a part of the nucleus be $$\begin{gathered} m=\rho v \\ =\rho x\left(\frac{4}{3}\right)\pi r^3 \end{gathered}$$ [Where, $m$ is the mass, $\rho$ is density, $v$ is the volume of nucleus, $r$ is the radius of the nucleus.]

Applying the law of conservation of momentum,

$m_1{v}_1=m_2{v}_2$ [$m_1,m_2$ are the masses of the split part of the nucleus, $v_1,v_2$ are the volumes of the spit part of the nucleus.]

$$\begin{gathered} [\rho x(4/3\left)\pi {r_1}^3\right]v_1=[\rho x(4/3\left)\pi {r_2}^3\right]v_2 \\ \Rightarrow \frac{v_1}{v_2}={\left(\frac{r_2}{r_1}\right)}^3 \\ \Rightarrow \frac{v_1}{v_2}={\left(\frac{2}{1}\right)}^3 \\ \Rightarrow \frac{v_1}{v_2}=\left(\frac{8}{1}\right) \end{gathered}$$[Where, the ratio of the split part of the nucleus $\frac{r_1}{r_2}=\frac{1}{2}$]

Their velocities are in the ratio $8:1$.

Hence, option D is correct.