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How fast (in $\frac{\mathrm{m}}{\mathrm{s}}$) are typical neutrons before and after slowing down in a nuclear reactor (with U-235 and $H_2O$ as moderator)? Do you have any reference for the values?

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Water cooled nuclear reactors slow neutrons down to "thermal" speed to increase the probability of their interactions with other nuclei. Thermal neutrons have a kinetic energy of about 0.025 eV (electron volt) or about 4.0 × 10−21 J, or a speed of 2.2 km/s. Neutrons released from a fission reaction have a mean energy of 2 MeV, or 20,000 km/s. Their speed can vary widely though, as their energy can be anywhere from 0.1 to 15 MeV, so their speeds range up to 54,000 km/s.

See this Wikipedia article for more details.

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