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The cosmic rays of highest energy are protons that have kinetic energy on the order of $10^{13}$ MeV. (a) From the point of view of the proton, how many kilometers across is the galaxy? (b) How long would it take a proton of this energy to travel across the Milky Way galaxy, having a diameter $\sim 10^{5}$ light-years, as measured in the proton's frame?

a) $10^{11} m$

b) 333$s$

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Well, relativistic factor is equal to one place kinetic energy divided by rest moss and Hershey. And putting the values one place, uh, 10 to the power through Dean divided by 9 38 And this is equal to 1.7 Multiply by 10 to the power down. Okay. And also al is equal to LP divided by gamma. Here, proper Landis stand to the qualify in government is running 0.7 multiplied. We tend to be powered 10. This is equal to 10 to the power minus fine line tears and which is approximately equal to, uh, 10 to the parlor. 11 meters. All right, No, let's do the second part. Well, speed is equal to, uh, squared Rudolph one minus one, divided by gonna square times. See and plug in the values Well scared route off one minus one, divided by down to the color. Done. And then we have square motive lively, see, and this is approximately equal to see and using. Delta T is equal to divided by B Well, l is down to the fallen 11 meters and he is approximately equal to see on and therefore Delta T is equal to uh 10 to the power 11 meters, divided by speed of light, made a poor second and therefore Time Musical 2333 seconds.