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Protons in an accelerator at the Fermi National Laboratory near Chicago are accelerated to a total energy that is 400 times their rest energy. (a) What is the speed of these protons in terms of $c$ ? (b) What is their kinetic energy in MeV?

a) 0.999997$c$

b) $3.74 \times 10^{5} \mathrm{MeV}$

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speed of protons is given by weight is equal to, uh, square root off. Uh, one divided by one minus Gonna square with mommy's little lad Mystic factor times the speed off light Uh, we have gamma and gamma is equal to 400 and we know the speed of light, which is three. Multiply by 10 to the power seat meter per second. Let's plug in the values Speed of protons is equal to, uh, squared route off one divided by one minus 400 square Multiply by speed off light. Uh, see and solving. This weekend we is recalled to zero point are mine Mine mine 99 times seven s Sorry. 0.999997 times the sweet off light Okay. And no, let's calculate Oh, the kinetic energy We know that kinetic energy ke equals, uh, Garma minus one times the rest energy. And we have got Mom, which is 400 minus one and rest energy. Um, well, uh, arrest energy is 930 feet minder and 38 point tree mega electron volts mega LeCrone balls. And our 404 100 minus one is but treat 119 mine into 9 38.3 mega electron balls and multiplying the two numbers We get three point 74 3.74 multiply by 10 to the power five mega electron volts.