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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

Four identical balls are lined in a straight grove made on a horizontal frictionless surface as shown. Two similar balls each moving with a velocity v collide elastically with the row of 4 balls from left. What will happen (A) One ball from the right rolls out with a speed \(2 \mathrm{v}\) and the remaining balls will remain at rest. (B) Two balls from the right roll out speed \(\mathrm{v}\) each and the remaining balls will remain stationary. (C) All the four balls in the row will roll out with speed \(\mathrm{v}(\mathrm{v} / 4)\) each and the two colliding balls will come to rest. (D) The colliding balls will come to rest and no ball rolls out from right.

Two bodies of masses \(\mathrm{m}\) and \(3 \mathrm{~m}\) have same momentum. their respective kinetic energies \(E_{1}\) and \(E_{2}\) are in the ratio..... (A) \(1: 3\) (B) \(3: 1\) (C) \(1: 3\) (D) \(1: 6\)

A body of mass \(6 \mathrm{~kg}\) is under a force, which causes a displacement in it given by $\mathrm{S}=\left[\left(2 \mathrm{t}^{3}\right) / 3\right](\mathrm{in} \mathrm{m}) .$ Find the work done by the force in first one seconds. (A) \(2 \mathrm{~J}\) (B) \(3.8 \mathrm{~J}\) (C) \(5.2 \mathrm{~J}\) (D) \(24 \mathrm{~J}\)

A bomb of mass \(3.0 \mathrm{~kg}\) explodes in air into two pieces of masses \(2.0 \mathrm{~kg}\) and \(1.0 \mathrm{~kg}\). The smaller mass goes at a speed of \(80 \mathrm{~m} / \mathrm{s}\). The total energy imparted to the two fragments is (A) \(1.07 \mathrm{KJ}\) (B) \(2.14 \mathrm{KJ}\) (C) \(2.4 \mathrm{KJ}\) (D) \(4.8 \mathrm{KJ}\)

A bullet of mass \(\mathrm{m}\) moving with velocity \(\mathrm{v}\) strikes a block of mass \(\mathrm{M}\) at rest and gets embedded into it. The kinetic energy of the composite block will be (A) \((1 / 2) \mathrm{mv}^{2} \times[\mathrm{M} /(\mathrm{m}+\mathrm{m})]\) (B) \((1 / 2) \mathrm{mv}^{2} \times[(\mathrm{m}+\mathrm{m}) / \mathrm{M}]\) (C) \((1 / 2) \mathrm{MV}^{2} \times[\mathrm{m} /(\mathrm{m}+\mathrm{M})]\) (D) \((1 / 2) \mathrm{mv}^{2} \times[\mathrm{m} /(\mathrm{m}+\mathrm{M})]\)
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