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Chapter 33: Problem 54

Among the most energetic cosmic rays ever detected are protons with energies around \(10^{20}\) eV. Find the momentum of such a proton, and compare with that of a 25 -mg insect crawling at \(2 \mathrm{mm} / \mathrm{s}.\)

Short Answer

Expert verified
After calculating the momentum of both the proton and the insect, one can see that despite the vastly different sizes of the two entities, the protons' momentum can be of the same order of magnitude as that of the insect due to its high velocity. This comparison illustrates the incredible energy carried by high-energy cosmic rays despite their microscopic size.

Step by step solution

01

Determine the velocity of the proton

The energy of the proton is given in electron volts (eV). To derive the velocity from this, the kinetic energy equation \(KE = \frac{1}{2} mv^2\) can be used, where \(KE\) is the kinetic energy, \(m\) is the mass, and \(v\) is the velocity. The mass of the proton is known to be about \(1.67 \times 10^{-27}\) kg. Substituting this value and the given energy into the equation, \(v = \sqrt{\frac{2KE}{m}}\), the result will be in m/s.
02

Calculate the momentum of the proton

The relativistic momentum equation can then be applied, giving \(p = \frac{mv}{\sqrt{1 - (\frac{v^2}{c^2})}}\), where \(c\) is the speed of light (\(3 \times 10^8\) m/s). This will give the momentum of the proton.
03

Calculate the momentum of the insect

For the insect, the mass and velocity are directly given, so the non-relativistic momentum equation \(p = mv\) can be used. The mass must be converted from mg to kg, and the velocity from mm/s to m/s, beforehand.
04

Compare the momentum

The final step is to compare the momentum of the proton and that of the insect. This will give a clear picture of how much larger or smaller the impact of the proton is compared to that of the insect, despite its tiny mass.

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