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Chapter 4: Problem 488

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

Expert verified
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

01

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.
02

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 × \(\frac{4}{3}\pi{(2r)^3}\)
03

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
04

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).

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Most popular questions from this chapter

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