{"id":2814,"date":"2019-03-28T18:13:03","date_gmt":"2019-03-28T18:13:03","guid":{"rendered":"https:\/\/staging.opencurve.info\/wp\/?p=2814"},"modified":"2020-07-02T00:43:00","modified_gmt":"2020-07-01T22:43:00","slug":"simple-problems-on-relativistic-energy-and-momentum-2","status":"publish","type":"post","link":"https:\/\/staging.opencurve.info\/wp\/simple-problems-on-relativistic-energy-and-momentum-2\/","title":{"rendered":"Simple problems on relativistic energy and momentum"},"content":{"rendered":"
<\/div>\n
\n\n\n\n

We will focus on a few simple problems where we will manipulate the equations for relativistic energy and momentum.<\/h3>\n\n\n\n

This could be seen as a second-year university-level post.<\/em><\/p>\n\n\n\n\n\n\n\n

\n\n\n\n\t\t<\/span> <\/span> \n\t\t<\/span><\/span>\n\n\n\n

Einstein had shown that the Lorentz transformations were the correct way to switch between the coordinate systems of different frames of reference [1]. He also taught us that Newton\u2019s laws weren\u2019t at all proper relativistic laws. For instance, Newtonian momentum $ \\mathbf{p} = m \\mathbf{v} $, and energy $ E = mv^2 \/ 2 $ were not at all accurate at speeds approaching that of light.<\/p>\n\n\n\n

Instead, we have all come to learn that the relativistic momentum is written as<\/p>\n\n\n\n

\\begin{equation} \\label{eq:relativistic momentum}\n\\mathbf{p} = \\frac{m \\mathbf{v}}{\\sqrt{1 – \\dfrac{v^2}{c^2}}}.\n\\end{equation}<\/p>\n\n\n\n

And that the correct relativistic expression for total energy is<\/p>\n\n\n\n

\\begin{equation} \\label{eq:relativistic energy}\nE_{\\text{tot}} = \\frac{mc^2}{\\sqrt{1 – \\dfrac{v^2}{c^2}}}.\n\\end{equation}<\/p>\n\n\n\n

We will solve the following problem set:<\/p>\n\n\n\n

  1. Prove, for a particle travelling at $ c $, that the magnitude of the relativistic energy is given by $ E = pc $.<\/li>
  2. Show that the energy-momentum relation for a particle with any<\/em> mass $ m $ travelling at any<\/em> speed $ v $ is correct and do mind it is not<\/em> the famous $ E = mc^2 $ we are referring to. Use the correct one, if you please.<\/li>
  3. Given that the mass of a proton is $ m_p $, calculate its exact speed when its relativistic  translational kinetic energy (which is the relativistic total energy minus its relativistic mass energy) is four times its relativistic mass energy.<\/li><\/ol>\n\n\n\n

    Problem I<\/strong><\/h3>\n\n\n\n

    Since $ E $ is expressed in terms of $ p $, we need to rewrite Eq. $ \\eqref{eq:relativistic momentum} $ by solving for $ m $:<\/p>\n\n\n\n

    \\[ m = \\frac{p \\sqrt{1 – \\dfrac{v^2}{c^2}}}{v}. \\]\n\n\n\n

    Note, we do not use the vector quantities, just the magnitudes. We can now proceed to substitute this into Eq. $ \\eqref{eq:relativistic energy} $:<\/p>\n\n\n\n

    \\[ E_{\\text{tot}} = \\frac{\\left(\\dfrac{p \\sqrt{1 – \\dfrac{v^2}{c^2}}}{v}\\right)c^2}{\\sqrt{1-\\dfrac{v^2}{c^2}}}. \\]\n\n\n\n

    This reduces to<\/p>\n\n\n\n

    \\begin{align}
    E_{\\text{tot}} &= \\frac{pc^2 \\sqrt{1 – \\dfrac{v^2}{c^2}}}{v \\sqrt{1 – \\dfrac{v^2}{c^2}}}, \\\\
    \\therefore E_{\\text{tot}} &= \\frac{pc^2}{v}. \\label{eq:E=pc^2\/v}
    \\end{align}<\/p>\n\n\n\n

    As we are dealing with a particle travelling at speed $ c $, we know $ v = c $, rendering Eq. $ \\eqref{eq:E=pc^2\/v} $ to<\/p>\n\n\n\n

    \\begin{align}
    E_{\\text{tot}} &= \\frac{pc^2}{c}, \\\\
    \\therefore E_{\\text{tot}} &= pc.
    \\end{align}<\/p>\n\n\n\n

    Problem II<\/strong><\/h3>\n\n\n\n

    The energy-momentum relation is<\/p>\n\n\n\n

    \\[ E^2_{\\text{tot}} = p^2c^2 + m^2c^4. \\]\n\n\n\n

    Substituting Eqs. $ \\eqref{eq:relativistic momentum} $ and $ \\eqref{eq:relativistic energy} $, yields<\/p>\n\n\n\n

    \\[ \\left(\\frac{mc^2}{\\sqrt{1 – \\dfrac{v^2}{c^2}}}\\right)^2 = \\left(\\frac{m \\mathbf{v}}{\\sqrt{1 – \\dfrac{v^2}{c^2}}}\\right)^2c^2 + m^2c^4, \\]\n\n\n\n

    which we can continue to work out as follows:<\/p>\n\n\n\n

    \\begin{align*}\\left(\\frac{mc^2}{\\sqrt{1 – \\dfrac{v^2}{c^2}}}\\right)^2 – \\left(\\frac{m \\mathbf{v}}{\\sqrt{1 – \\dfrac{v^2}{c^2}}}\\right)^2c^2 – m^2c^4 &= 0, \\\\
    \\frac{m^2c^4}{1 – \\dfrac{v^2}{c^2}} – \\frac{m^2v^2c^2}{1 – \\dfrac{v^2}{c^2}} – m^2c^4 &= 0, \\\\
    \\left(1-\\dfrac{v^2}{c^2}\\right)\\left(\\frac{m^2c^4}{1-\\dfrac{v^2}{c^2}}\\right) \\qquad &\\qquad \\\\ – \\left(1-\\dfrac{v^2}{c^2}\\right)\\left(\\frac{m^2v^2c^2}{1-\\dfrac{v^2}{c^2}}\\right) &\\qquad \\\\ – \\left(1-\\dfrac{v^2}{c^2}\\right)m^2c^4 &= 0, \\\\
    m^2c^4 – m^2v^2c^2 – m^2c^4 + \\frac{m^2v^2c^4}{c^2} &= 0, \\\\
    m^2c^4 – m^2c^4 – m^2v^2c^2 + m^2v^2c^2 &= 0, \\\\
    0 – 0 &= 0.
    \\end{align*}<\/p>\n\n\n\n

    Hence, for every value of $ m $, $ p $, and thus $ v $, the relation holds.<\/p>\n\n\n\n

    Problem III<\/strong><\/h3>\n\n\n\n
    \"\"
    Hydrogen bubble chamber Fermilab<\/figcaption><\/figure><\/div>\n\n\n\n

    The relativistic (total) energy is<\/p>\n\n\n\n

    \\[ E_{\\text{tot}} = E_{\\text{trans}} + E_{\\text{mass}}. \\]\n\n\n\n

    If the relativistic translational kinetic energy is four times the relativistic mass energy, then we can write<\/p>\n\n\n\n

    \\[ E_{\\text{trans}} = 4E_{\\text{mass}}. \\]\n\n\n\n

    In our case, this then yields for the relativistic (total) energy:<\/p>\n\n\n\n

    \\[ E_{\\text{tot}} = 4E_{\\text{mass}} + E_{\\text{mass}} = 5E_{text{mass}}. \\]\n\n\n\n

    To calculate the proton\u2019s speed, we then write<\/p>\n\n\n\n

    \\begin{align*}
    \\frac{m_pc^2}{\\sqrt{1 – \\dfrac{v^2}{c^2}}} &= 5E_{\\text{mass}} = 5m_pc^2, \\\\
    \\frac{1}{\\sqrt{1 – \\dfrac{v^2}{c^2}}} &= 5, \\\\
    \\sqrt{1-v^2\/c^2} &= \\frac{1}{5}, \\\\
    1-\\frac{v^2}{c^2} &= \\frac{1}{25}, \\\\
    \\frac{v^2}{c^2} &= \\frac{24}{25}, \\\\
    v^2 &= \\frac{24c^2}{25}, \\\\
    \\therefore v &= \\sqrt{\\frac{24c}{25}} = \\frac{2\\sqrt{6}c}{5},
    \\end{align*}<\/p>\n\n\n\n

    which is about $ 0.98c $ rounded to two decimals, which means that the proton zips at about 98% of the speed of light through the fabric of the cosmos.<\/p>\n\n\n\n

    Image Hydrogen bubble chamber Fermilab: Proton with 300 GeV energy producing 26 charged particles in the 30 inch hydrogen bubble chamber at Fermilab. Source: <\/em>Wikimedia Commons<\/em><\/a>
    <\/p>\n\n\n\n

    [1] Einstein, A. (1905) \u2018Zur Elektrodynamik bewegter K\u00f6rper\u2019, Annalen der Physik<\/em>, 322(10), pp. 891\u2013921. doi: 10.1002\/andp.19053221004<\/a>.<\/p>\n\n\n\n

    \n\n\n\n

    This is a repost. Slight errors in the parsing of LaTeX in the original article of 24 December 2018 have been corrected.<\/p>\n","protected":false},"excerpt":{"rendered":"

    We will focus on a few simple problems where we will manipulate Einstein’s equations for relativistic energy and momentum.<\/p>\n","protected":false},"author":1,"featured_media":2846,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[23,7,4],"tags":[89,170,172,176,174],"class_list":["post-2814","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-level-2-undergraduate","category-physics-tutorial","category-tutorial","tag-einstein","tag-energy","tag-momentum","tag-problem-set-en","tag-relativity"],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/staging.opencurve.info\/wp\/wp-content\/uploads\/2019\/03\/flare-666138-e1588487585282.jpg?fit=1920%2C768&ssl=1","uagb_featured_image_src":{"full":["https:\/\/i0.wp.com\/staging.opencurve.info\/wp\/wp-content\/uploads\/2019\/03\/flare-666138-e1588487585282.jpg?fit=1920%2C768&ssl=1",1920,768,false],"thumbnail":["https:\/\/i0.wp.com\/staging.opencurve.info\/wp\/wp-content\/uploads\/2019\/03\/flare-666138-e1588487585282.jpg?resize=150%2C150&ssl=1",150,150,true],"medium":["https:\/\/i0.wp.com\/staging.opencurve.info\/wp\/wp-content\/uploads\/2019\/03\/flare-666138-e1588487585282.jpg?fit=300%2C120&ssl=1",300,120,true],"medium_large":["https:\/\/i0.wp.com\/staging.opencurve.info\/wp\/wp-content\/uploads\/2019\/03\/flare-666138-e1588487585282.jpg?fit=768%2C307&ssl=1",768,307,true],"large":["https:\/\/i0.wp.com\/staging.opencurve.info\/wp\/wp-content\/uploads\/2019\/03\/flare-666138-e1588487585282.jpg?fit=1200%2C480&ssl=1",1200,480,true],"1536x1536":["https:\/\/i0.wp.com\/staging.opencurve.info\/wp\/wp-content\/uploads\/2019\/03\/flare-666138-e1588487585282.jpg?fit=1536%2C614&ssl=1",1536,614,true],"2048x2048":["https:\/\/i0.wp.com\/staging.opencurve.info\/wp\/wp-content\/uploads\/2019\/03\/flare-666138-e1588487585282.jpg?fit=1920%2C768&ssl=1",1920,768,true],"crp_thumbnail":["https:\/\/i0.wp.com\/staging.opencurve.info\/wp\/wp-content\/uploads\/2019\/03\/flare-666138-e1588487585282.jpg?resize=150%2C150&ssl=1",150,150,true]},"uagb_author_info":{"display_name":"KJ Runia","author_link":"https:\/\/staging.opencurve.info\/wp\/author\/kjrunia\/"},"uagb_comment_info":0,"uagb_excerpt":"We will focus on a few simple problems where we will manipulate Einstein's equations for relativistic energy and momentum.","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/posts\/2814","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/comments?post=2814"}],"version-history":[{"count":42,"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/posts\/2814\/revisions"}],"predecessor-version":[{"id":6836,"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/posts\/2814\/revisions\/6836"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/media\/2846"}],"wp:attachment":[{"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/media?parent=2814"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/categories?post=2814"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/staging.opencurve.info\/wp\/wp-json\/wp\/v2\/tags?post=2814"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}