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Step-by-Step Solution
Step 1: Identify the Physical Principle
When the heavy nucleus breaks apart, the total initial momentum (zero, since it is at rest) must equal the total final momentum of the two fragments. This is the law of conservation of momentum.
Step 2: Write the Conservation of Momentum Equation
Since the nucleus is initially at rest, we have
$ m_1 v_1 + m_2 v_2 = 0 $
but because they move in opposite directions, a convenient form is
$ m_1 v_1 = m_2 v_2 $.
This implies
$ \frac{v_1}{v_2} = \frac{m_2}{m_1} $.
Step 3: Insert the Given Velocity Ratio
We are given that
$ v_1 : v_2 = 8 : 27 $.
Therefore,
$ \frac{m_2}{m_1} = \frac{v_1}{v_2} = \frac{8}{27}. $
Step 4: Relate Masses to Radii of the Spherical Nuclei
The mass of each nucleus can be approximated by
$ m = \rho \frac{4}{3} \pi r^3 $
where
$ \rho $
is the density and
$ r $
is the nucleus radius. Thus,
$ \frac{m_2}{m_1}
= \frac{\rho \frac{4}{3} \pi r_2^3}{\rho \frac{4}{3} \pi r_1^3}
= \left(\frac{r_2}{r_1}\right)^3. $
We already have
$ \frac{m_2}{m_1} = \frac{8}{27}. $
Thus,
$ \left(\frac{r_2}{r_1}\right)^3 = \frac{8}{27}. $
Step 5: Solve for the Radius Ratio
Taking the cube root on both sides,
$ \frac{r_2}{r_1} = \frac{2}{3}. $
Hence,
$ r_1 : r_2 = 3 : 2. $
Answer
The ratio of the radii of the two nuclei is 3 : 2.
