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Chapter 9: Problem 22

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

Short Answer

Expert verified
The internal energy of the system increases to 8K while the center-of-mass energy remains the same (K) after the explosion.

Step by step solution

01

Determine the initial kinetic energy (K)

The initial kinetic energy (K) of the object is given in the problem, though no specific value has been provided. As there is no reference value for K, we will refer to it as K throughout the solution.
02

Calculate the kinetic energy of the exploded object

Once the object explodes, it separates into two pieces, each moving with twice the speed of the original object.The kinetic energy of a moving object is given by \(\frac{1}{2} m v^2\). Here, since the speed is doubled, the kinetic energy is quadrupled (since the kinetic energy is proportional to the square of the speed). Thus, the kinetic energy of each piece will be \(4 \times K\). However, since there are two of these pieces, the total kinetic energy becomes \(2 \times 4K = 8K\).
03

Comparing the center-of-mass energy and internal energy

The center-of-mass energy of a system is solely dependent on the motion of the center of mass (which, in this case, does not change as the object explodes symmetrically — this is taken as zero) and is separate from the internal energy (which is the kinetic energy of the individual components, measured from the center of mass frame). Thus, after the explosion, the internal energy is \(8K\), while the center-of-mass energy remains \(K\).

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Most popular questions from this chapter

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