The ultra particle

6559 /www/nrich/html/content/id/6559/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/6504">Warm-up</a></li> <li><a href="http://nrich.maths.org/6560">Try this next</a></li> <li><a href="http://nrich.maths.org/6569">Think higher</a></li> <li><a href="http://nrich.maths.org/6631">Read: mathematics</a></li> <li><a href="http://plus.maths.org/content/whats-so-special-about-special-relativity">Read: science</a></li> <li><a href="http://plus.maths.org/content/secret-symmetry-and-higgs-boson-part-i">Explore further</a></li> </ul> <div> </div> Note: Assume standard Newtonian mechanics for the first parts of this problem. Required data are provided at the foot of the problem. On October 15, 1991 an ultra-high energy proton came to earth, near Utah, from space with an energy of $3.2\pm 0.9\times 10^{20}$ electron volts. This REALLY is a lot of energy for a proton to possess, so much so it was dubbed the 'Oh-my-god' particle! To see why, explore the energy it contains by comparing this figure with the kinetic energy for the motion of more every-day objects - find an everyday situation which really gives an intuitive sense of the amount of energy of this ultra-high energy proton. If you used the Newtonian expression $KE=\frac{1}{2}mv^2$ for the energy, how fast would the proton be travelling? How does this compare with the speed of light? What does this tell you? Imagine that a small, hand-sized, ball of meteoric iron with the same kinetic energy per kilogram as this ultra-high energy proton struck earth. Analyse the possible effect this would have. Extension: Clearly, the energies referred to in the question push the furthest reaches of Einstein's theory of special relativity. Use this situation to determine the actual velocity of the proton relative to Earth using the formula involving the rest mass $m_0$ of a stationary object and the speed of light $c$ $$E= \frac{m_0c}{\sqrt{1-\frac{v^2}{c^2}}}$$ Data for the problem 1 electron volt (eV) = $1.602 176 46 \times 10^{-19}$J Rest mass of the proton is $9.3828\times 10^8$ eV Mass of proton $1.672 621 58\times 10^{-27}$kg Speed of light $2.99 792 458\times 10^8$ ms$^{-1}$ <div class="framework"> NOTES AND BACKGROUND Cosmic rays are simply particles of matter moving through space; typically high energy protons, helium nuclei or electrons. Modern models of physics currently under investigation by theoretical physicists predict bounds on the energies of cosmic rays or the appearence of more exotic particles in cosmic rays. <a href="http://en.wikipedia.org/wiki/Ultra-high-energy_cosmic_ray">Ultra-high energy particles</a> are occasionally found in cosmic rays. These defy well understood bounds, confound theoretical physicists and help to drive forward the bounds of our understanding of the universe. An interesting article on the particle referred to in this problem is found <a href="http://www.fourmilab.ch/documents/OhMyGodParticle/">here</a> .</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> $E = 3.2\times 10^{20} \times 1.602\times 10^{-19} = 51.264 J$ If we equate this to a hammer head of mass $0.5$kg, being flung at some speed, we find that $v = 14.32m/s = 52 km/h$ This is an absolutely vast amount of energy for a single, fundamental particle to have! If we rearrange $E = \frac{1}{2}mv^2$, using the rest mass of the particle for m, we find that $v = 2.5\times 10^{14}m/s$, which is about a million times faster than light. We can therefore assume that the mass of the particle is greater than the rest mass. The energy per kilo of the photon (in terms of rest mass) is $51.264/(1.67\times 10^{-27}) = 3.07 \times 10^{29}J/kg$ Given that the mass of the earth is only $6\times 10^{24}$kg, the ball of iron would contain enough energy to propel the earth to a velocity of about $320m/s$, or $1151$km/h, using the formula for kinetic energy. Perhaps it maybe might just pass straight through the Earth, vaporising everything it touched, leaving the bulk a little shaken, but intact. Even one proton possessing this energy is extremely rare though, thankfully If you rearrange the given formula to $v = c\sqrt{1 - (\frac{m_0c}{E})^2} \approx 3\times 10^{8}m/s$, i.e. as far as my calculator is concerned, almost the speed of light. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> The final speeds relative to the earth are in the range (299792233.8, 299792387.4 ) m/s </mdoxml> 2 3 0 0 0 0 1 The ultra particle Explore the energy of this incredibly energetic particle Mechanics Energy Mechanics Relativity, black holes and quantum mechanics Applications physics Applications physics Admin Individual Collections Fundamental Particles