physics / relativity.htmlon commit Merge branch 'master' of ssh://charles/tank/andrew/school/notes (74f3723)
   1<!DOCTYPE html>
   2<html>
   3  <head>
   4      <meta charset="utf-8" />
   5      <title>relativity</title>
   6<script type="text/x-mathjax-config">
   7  MathJax.Hub.Config({
   8    jax: ["input/TeX","output/HTML-CSS"],
   9    extensions: [],
  10    TeX: {
  11      extensions: ["AMSmath.js","AMSsymbols.js","noErrors.js","noUndefined.js"]
  12    },
  13    showMathMenu: false
  14  });
  15</script>
  16<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js">
  17</script>
  18      <style>
  19
  20      html {
  21          font-family: "Helvetica", sans-serif;
  22          letter-spacing: -0.01em;
  23          font-size: 0.7em;
  24      }
  25
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 556  </head>
 557  <body class='markdown-preview'><h1>Special Relativity</h1>
 558<h2>Postulates of special relativity</h2>
 559<ol>
 560<li>
 561<p>Laws of physics are constant in all inertial reference frames</p>
 562</li>
 563<li>
 564<p>Speed of light <span class="math"><script type="math/tex">c</script></span> is the same to all observers (proved by Michelson-Morley)</p>
 565</li>
 566</ol>
 567<p><span class="math"><script type="math/tex">\therefore</script></span>, <span class="math"><script type="math/tex">t</script></span> must dilate as speed changes.</p>
 568<p>Time and space - four dimensional relationship</p>
 569<p><strong>Inertial reference frame</strong> - not accelerating</p>
 570<p><strong>Proper time / length</strong> - measured by observer in same frame as events</p>
 571<h2>Lorentz factor</h2>
 572<span class="math"><script type="math/tex; mode=display">\gamma = {1 \over \sqrt{1-{v^2 \over c^2}}}
 573</script></span>
 574<p>Proper (time|length) denoted by (<span class="math"><script type="math/tex">t|l_0</script></span>) is time observed from moving frame.</p>
 575<p>Applied:</p>
 576<p><span class="math"><script type="math/tex">t = t_0 \gamma</script></span> (time - longer in moving frame)<br>
 577<span class="math"><script type="math/tex">l={l_0 \over \gamma}</script></span> (length - only contracts in direction of motion - shorter in moving frame)<br>
 578<span class="math"><script type="math/tex">m=m_0 \gamma</script></span> (mass dilation)</p>
 579<span class="math"><script type="math/tex; mode=display">v=c \sqrt{1-{1 \over \gamma ^2}}
 580</script></span>
 581<p>(<span class="math"><script type="math/tex">c=3 \times 10^8</script></span> m / s)</p>
 582<h2>Mass-energy equivalence</h2>
 583<p><span class="math"><script type="math/tex">E_0=mc^2</script></span> (rest energy)</p>
 584<p><span class="math"><script type="math/tex">E=\gamma mc^2</script></span> (total energy)</p>
 585<ul>
 586<li>mass is a dense form of energy</li>
 587</ul>
 588<h3>Relativistic momentum</h3>
 589<span class="math"><script type="math/tex; mode=display">\rho = {mv \over {\sqrt{1-{v^2 \over c^2}}}} = \gamma mv=\gamma \rho_0
 590</script></span>
 591<p>Low speeds - little difference between relativistic &amp; non-relativistic momentum<br>
 592<span class="math"><script type="math/tex">\rho</script></span> approaches <span class="math"><script type="math/tex">\infty</script></span> as speed approaches <span class="math"><script type="math/tex">c</script></span><br>
 593Impossible to reach speed of light (speed <span class="math"><script type="math/tex">c</script></span> requires <span class="math"><script type="math/tex">\infty</script></span> energy)<br>
 594Photon has no mass - zero space or time</p>
 595<h3>Total energy of an object</h3>
 596<span class="math"><script type="math/tex; mode=display">E_{total}=E_k+E_{rest}=\rho mc^2
 597</script></span>
 598<h3>Kinetic energy</h3>
 599<span class="math"><script type="math/tex; mode=display">E_k=(\gamma - 1)mc^2
 600</script></span>
 601<p>(takes relativity into account - use for fast objects)</p>
 602<h3>Rest energy</h3>
 603<span class="math"><script type="math/tex; mode=display">E_{rest}=mc^2
 604</script></span>
 605<p>(where <span class="math"><script type="math/tex">v<0.1c</script></span>) - this is the innate mass energy (proper energy)</p>
 606<h3>Work</h3>
 607<span class="math"><script type="math/tex; mode=display">W = \Delta E = \Delta mc^2
 608</script></span>
 609<h3>Velocity</h3>
 610<span class="math"><script type="math/tex; mode=display">v={\rho \over {m \sqrt{1+{p^2 \over {m^2c^2}}}}}
 611</script></span>
 612<h3>Fusion / Fission equations</h3>
 613<p>eV to J: multiply by <span class="math"><script type="math/tex">1.6\times 10^{-19}</script></span> (charge of an electron)</p>
 614<h2>High altitude muons</h2>
 615<ul>
 616<li>Time dilation - more muons reach Earth than expected</li>
 617<li>Normal half-life is <span class="math"><script type="math/tex">2.2 \mu s</script></span> in stationary frame of reference</li>
 618<li>At close to <span class="math"><script type="math/tex">c</script></span>, muons observed from Earth have lifetime <span class="math"><script type="math/tex">> 2.2 \mu s</script></span></li>
 619<li>Slower time - more time to travel, so more muons reach surface</li>
 620</ul></body>
 621</html>