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) \(8: 1\)

Short answer

The ratio of the velocities of the two parts after the nucleus splits is 8:1. This is determined using the conservation of momentum and the given ratio of radii and density. The ratio v1/v2 is calculated as \(8r^3/r^3\), resulting in a ratio of 8:1. So, the correct answer is (D) $8:1$.

Step-by-step solution

Identify the given information and the unknown variables

We are given that the nucleus splits into two parts of radii in the ratio 1:2. These parts have the same density, and we need to find the ratio of their velocities. Let the radii of the two parts be r and 2r, their densities be ρ, and their velocities be v1 and v2 respectively. Our aim is to find the ratio v1/v2.

Calculate the masses of the parts using density

The volume of a sphere is given by the formula: V = \(\frac{4}{3}\pi{r^3}\) For parts 1 and 2, we have: V1 = \(\frac{4}{3}\pi{r^3}\) and V2 = \(\frac{4}{3}\pi{(2r)^3}\) The mass of an object can be calculated using the formula: mass = density × volume Therefore, mass1 (m1) = ρ × V1 = ρ × \(\frac{4}{3}\pi{r^3}\) mass2 (m2) = ρ × V2 = ρ × \(\frac{4}{3}\pi{(2r)^3}\)

Apply the conservation of momentum

As the nucleus was initially at rest, the total initial momentum is zero. So, the total momentum of the two parts should also be equal to zero. This can be written mathematically as: m1 × v1 = m2 × v2 Substitute the masses of the parts (m1 and m2) calculated in step 2: (ρ × \(\frac{4}{3}\pi{r^3}\)) × v1 = (ρ × \(\frac{4}{3}\pi{(2r)^3}\)) × v2

Solve for the ratio of velocities v1/v2

We can cancel out the ρ and \(\frac{4}{3}\pi\) terms from both sides of the equation: \(r^3\) × v1 = \((2r)^3\) × v2 Now, group the terms of v1 and v2: \(\frac{v1}{v2}\) = \(\frac{(2r)^3}{r^3}\) \(\frac{v1}{v2}\) = \(\frac{8r^3}{r^3}\) \(\frac{v1}{v2}\) = \(8\) The ratio of velocities of the two parts is 8:1, therefore the correct answer is option (D).

Key concepts

Nuclear Physics

Nuclear physics delves into the core of matter, examining the nucleus of atoms and its components. When a nucleus splits, it's called nuclear fission. This event releases a large amount of energy and causes changes in the momentum of the resulting parts.
In the exercise, a nucleus divides into two parts with different sizes, suggesting a fission event. Since the nucleus was initially at rest, the concept of conservation of momentum applies. Momentum is conserved in a closed system, meaning the total momentum before and after the event remains the same. This fundamental principle allows us to predict and calculate outcomes in nuclear reactions.
Understanding the behavior of subatomic particles helps scientists develop nuclear energy and manage nuclear reactions safely. By studying nuclear physics, you grasp the forces that hold a nucleus together and the energy released when these bonds are broken or formed.

Density and Volume

Density is a measure of how much mass is contained in a given volume. In this problem, the nuclei parts have the same density, which means the mass is distributed uniformly.
The volume of the parts is linked to their radii, given through the formula for the volume of a sphere: \[ V = \frac{4}{3}\pi{r^3} \]
For two nuclei parts with radii in ratio 1:2, if the small part has a radius \( r \), the larger part has a radius \( 2r \). Hence, their volumes will be proportional to \( r^3 \) and \( (2r)^3 \), respectively.
This proportionality indicates that the volume—and subsequently the mass—of the larger part will be eight times that of the smaller part, assuming identical density. Understanding these relationships is crucial in calculating the masses when analyzing physical scenarios like nuclear fission.

Ratio of Velocities

When the nucleus divides, it splits into two parts that shoot apart. Their velocities are tied to the concept of momentum conservation, meaning the momentum (product of mass and velocity) before and after must be equal. If a system starts at rest, such as our splitting nucleus, the total initial momentum is zero.
This gives the equation: \[ m_1 \times v_1 = m_2 \times v_2 \]
As calculated, if the radii—and thus the masses—of the parts are in a certain ratio, their velocities will adjust to maintain balanced momentum. The larger the radii, the larger the mass; in return, this changes how fast or slow each part moves.
Following through the steps given, the ratio of velocities is derived as \( 8:1 \). This directly results from balancing the mass differences with changes in velocity, ensuring both parts together maintain the total zero initial momentum of the initial system.