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Chapter 9: Problem 24

An object with kinetic energy \(K\) explodes into two pieces, each of which moves with twice the speed of the original object. Find the ratio of the internal kinetic energy to the center-of-mass energy after the explosion.

Short Answer

Expert verified
The ratio of the internal kinetic energy to the center-of-mass energy after the explosion is 2.

Step by step solution

01

Understanding the Problem

Analyze the information given in the problem. Understand that the initial object splits into two pieces that move with twice the speed of the original object.
02

Setting up the Kinetic Energy Equation

Start by writing the equation for kinetic energy as \(K = 1/2 mv^2\), where \(m\) is the mass and \(v\) is the velocity of the object. For this exercise, realize that the kinetic energy of each piece after the explosion will be given by \(K1 = 1/2 m(2v)^2\) and \(K2 = 1/2 m(2v)^2\), as the velocity doubles due to the explosion.
03

Computing the Internal Kinetic Energy

The internal kinetic energy after the explosion will be the sum of the kinetic energies of pieces, \(K_{internal} = K1 + K2 = m(2v)^2 = 2K\) where \(K\) is the kinetic energy of the original object. Thus, the internal kinetic energy after the explosion is twice the initial kinetic energy of the object.
04

Finding the Center-of-Mass Energy

The center-of-mass energy after the explosion should remain the same as it was before the explosion. This is because the center of mass of a system of particles is not dependent on any internal interactions between the particles. Thus the center-of-mass energy would still be \(K\) as before the explosion.
05

Calculating the Required Ratio

The required ratio of the internal kinetic energy to the center-of-mass energy after the explosion is \(K_{internal} / K_{cm} = 2K/K = 2\).

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