A nucleus disintegrates into two nuclear parts which have their velocities in the ratio 2 : 1. The r

Q: Why is the initial momentum taken as zero?

The problem states the nucleus 'disintegrates', which implies it was initially a single entity at rest. Therefore, its initial velocity and momentum are zero.

Q: Why do we assume nuclear density is constant?

It is a fundamental property of nuclear matter. Nucleons (protons and neutrons) are closely packed, and the volume of a nucleus is directly proportional to its mass number, making the density roughly the same for all nuclei.

Q: Does the lighter fragment always move faster?

Yes. To conserve momentum ($m_1 v_1 = m_2 v_2$), the fragment with the smaller mass must have a proportionally larger velocity.

Q: What if the nucleus split into three parts instead of two?

Momentum conservation would still apply, but it would require the vector addition of all three momenta to be zero ($p_1 + p_2 + p_3 = 0$). You couldn't simply equate two magnitudes.

Q: How does this relate to the kinetic energy of the fragments?

Since kinetic energy is $K = p^2 / 2m$ and both fragments have the same momentum magnitude $p$, the lighter fragment will carry away more kinetic energy.

Question

A nucleus disintegrates into two nuclear parts which have their velocities in the ratio $2 : 1$. The ratio of their nuclear sizes will be

Options:

(A) $2^{1/3} : 1$

(B) $1 : 3^{1/2}$

(C) $3^{1/2} : 1$

(D) $1 : 2^{1/3}$

Sigma Percentile · Accepted Answer

The phenomenon of nuclear disintegration is a fascinating display of fundamental physics principles in action. When a heavy parent nucleus at rest suddenly splits into two fragments, it sets the stage for a perfect application of the **Law of Conservation of Linear Momentum**.

### Analyzing the Setup
Imagine a parent nucleus sitting perfectly still. Its initial velocity is zero, which means its initial momentum is also zero. Suddenly, it disintegrates into two parts. Because there are no external forces acting on the system, the total momentum must remain conserved.

This implies that the two new fragments must fly apart in exactly opposite directions, and their momenta must be equal in magnitude. Mathematically, we express this as:
$$m_1 v_1 = m_2 v_2$$

### The Velocity-Mass Relationship
The problem states that the velocities of the two fragments are in the ratio $2 : 1$. Let's call the faster fragment part 1 and the slower one part 2. So, we have:
$$\frac{v_1}{v_2} = \frac{2}{1}$$

Using our momentum conservation equation, we can rearrange it to find the ratio of their masses:
$$\frac{m_1}{m_2} = \frac{v_2}{v_1}$$

Substituting the given velocity ratio, we get:
$$\frac{m_1}{m_2} = \frac{1}{2}$$

This tells us a crucial physical truth: the fragment that moves twice as fast must be half as massive. The lighter fragment is always the faster one!

### Connecting Mass to Nuclear Size
Now, we need to find the ratio of their nuclear sizes, which means the ratio of their radii. How do we connect mass to radius?

The key lies in the properties of nuclear matter. The density of nuclear matter, denoted by $ho$, is incredibly high but, more importantly, it is **constant** for almost all nuclei. Since density is mass per unit volume, we can write the mass of a spherical nucleus as:
$$m = 	ext{Volume} 	imes 	ext{Density} = \frac{4}{3}\pi r^3 ho$$

Because $\frac{4}{3}\pi$ and $ho$ are constants, we can conclude that the mass of a nucleus is directly proportional to the cube of its radius:
$$m \propto r^3$$

### Final Calculation
We can now substitute this proportionality into our mass ratio equation:
$$\frac{m_1}{m_2} = \frac{r_1^3}{r_2^3}$$

Since we already know that $\frac{m_1}{m_2} = \frac{1}{2}$, we can equate the two:
$$\frac{r_1^3}{r_2^3} = \frac{1}{2}$$

To find the ratio of the radii, we simply take the cube root of both sides:
$$\frac{r_1}{r_2} = \left(\frac{1}{2}ight)^{1/3} = \frac{1}{2^{1/3}}$$

Therefore, the ratio of their nuclear sizes is:
$$r_1 : r_2 = 1 : 2^{1/3}$$

This elegant result perfectly matches option (d). It's a beautiful reminder of how macroscopic conservation laws and microscopic nuclear properties intertwine to give us precise predictions about the subatomic world.

Animated Solution for Physics - Atoms and Nuclei: A nucleus disintegrates into two nuclear parts which have their velocities in the ratio $2 : 1$ . The ratio of their nuclear sizes will be

Select Answer:

Visualized Solution

\text{Nuclear Disintegration}

\text{Conservation of Momentum}

\text{Mass Ratio}

\text{Mass and Radius Relation}

\text{Radius Ratio}

\text{Final Calculation}

\text{Kinetic Energy Distribution}

The Sigma Insight: Nucleus and Nuclear Reaction

Analyzing the Setup Imagine a parent nucleus sitting perfectly still

The Velocity-Mass Relationship The problem states that the velocities of the two fragments are in the ratio $2 : 1$

Connecting Mass to Nuclear Size Now, we need to find the ratio of their nuclear sizes, which means the ratio of their radii

Final Calculation

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Select Answer:

Visualized Solution

\text{Nuclear Disintegration}

\text{Conservation of Momentum}

\text{Mass Ratio}

\text{Mass and Radius Relation}

\text{Radius Ratio}

\text{Final Calculation}

\text{Kinetic Energy Distribution}

The Sigma Insight: Nucleus and Nuclear Reaction

Analyzing the Setup Imagine a parent nucleus sitting perfectly still

The Velocity-Mass Relationship The problem states that the velocities of the two fragments are in the ratio .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2:1

Connecting Mass to Nuclear Size Now, we need to find the ratio of their nuclear sizes, which means the ratio of their radii

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