Question

Alpha decay is nuclear decay in which a helium nucleus is emitted. If the helium nucleus has a mass of $6.80\times 10^{-27}\textrm{ kg}$ and is given 5.00 MeV of kinetic energy, what is its velocity?

Final Answer

$1.53\times 10^{7}\textrm{ m/s}$

### Solution video

# OpenStax College Physics, Chapter 28, Problem 54 (Problems & Exercises)

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Video Transcript

This is College Physics Answers with Shaun Dychko. During

*λ*-decay, a helium nucleus is emitted and it has a kinetic energy of 5.00 megaelectron volts and given its rest mass of 6.80 times 10 to the minus 27 kilograms, what must its speed be? We know kinetic energy is*γ*minus 1 times the rest mass times speed of light squared and we can solve this for*γ*and then we have a formula for*γ*in terms of speed and so we'll solve this for*v*afterwards. So first we divide both sides by*mc squared*and then switch the sides around so we have*γ*minus 1 equals kinetic energy divided by rest mass times speed of light squared then we add 1 to both sides and now we have solved for*γ*. So we plug in numbers: that's 5.00 megaelectron volts converted into joules by multiplying by 1.602 times 10 to the minus 13 joules per megaelectron volt and we divide that by the rest mass of the helium particle and then times that by the speed of light squared and then add 1 and we get 1.0013106 so we have to turn this number into a speed now. So*γ*is the Lorentz factor and that equals 1 over the square root of 1 minus the speed squared divided by speed of light squared. We can square both sides to get rid of square root on the bottom and then we'll multiply both sides by this denominator 1 minus*v squared*over*c squared*then divide both sides by*γ squared*then we will end up with 1 minus*v squared*over*c squared*on the left and 1 over*γ squared*on the right and then add*v squared*over*c squared*to both sides and subtract 1 over*γ squared*from both sides and then switch the sides around and you get*v squared*over*c squared*is 1 minus 1 over*γ squared*then multiply both sides by*c squared*and you get*v squared*equals*c squared*times 1 minus 1 over*γ squared*, square root both sides and*v*then is*c*times square root 1 minus 1 over*γ squared*. So that's 2.998 times 10 to the 8 meters per second times square root of 1 minus 1 over 1.0013106 squared and that's 1.53 times 10 to the 7 meters per second.