{"id":266,"date":"2021-02-03T23:36:36","date_gmt":"2021-02-03T23:36:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=266"},"modified":"2026-02-06T21:10:42","modified_gmt":"2026-02-06T21:10:42","slug":"putting-it-together-limits","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/putting-it-together-limits\/","title":{"raw":"Putting It Together: Limits","rendered":"Putting It Together: Limits"},"content":{"raw":"
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Einstein's Equation<\/h3>\r\n

At the beginning of the module, we mentioned briefly how Albert Einstein showed that a limit exists to how fast any object can travel. Given Einstein\u2019s equation for the mass of a moving object, what is the value of this bound?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nOur starting point is Einstein\u2019s equation for the mass of a moving object,\r\n

[latex]m=\\dfrac{m_0}{\\sqrt{1-\\frac{v^2}{c^2}}}[\/latex]<\/p>\r\nwhere [latex]m_0[\/latex]\u00a0is the object\u2019s mass at rest, [latex]v[\/latex]\u00a0is its speed, and\u00a0[latex]c[\/latex]\u00a0is the speed of light. To see how the mass changes at high speeds, we can graph the ratio of masses [latex]\\frac{m}{m_0}[\/latex]\u00a0as a function of the ratio of speeds,\u00a0[latex]\\frac{v}{c}[\/latex].\r\n\r\n[caption id=\"attachment_5013\" align=\"aligncenter\" width=\"325\"]\"A Figure 2.23 This graph shows the ratio of masses as a function of the ratio of speeds in Einstein\u2019s equation for the mass of a moving object.[\/caption]\r\n\r\nWe can see that as the ratio of speeds approaches 1\u2014that is, as the speed of the object approaches the speed of light\u2014the ratio of masses increases without bound. In other words, the function has a vertical asymptote at [latex]\\frac{v}{c}=1[\/latex]. We can try a few values of this ratio to test this idea.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]\\dfrac{v}{c}[\/latex]<\/td>\r\n[latex]\\sqrt{1-\\frac{v^2}{c^2}}[\/latex]<\/td>\r\n[latex]\\dfrac{m}{m_0}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]0.99[\/latex]<\/td>\r\n[latex]0.1411[\/latex]<\/td>\r\n[latex]7.089[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]0.999[\/latex]<\/td>\r\n[latex]0.0447[\/latex]<\/td>\r\n[latex]22.37[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]0.9999[\/latex]<\/td>\r\n[latex]0.0141[\/latex]<\/td>\r\n[latex]70.71[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo what does this mean?\r\n\r\nAccording to Table 1, if an object with mass [latex]100[\/latex] kg is traveling at [latex]0.9999c[\/latex], its mass becomes [latex]7071[\/latex] kg. Since no object can have an infinite mass, we conclude that no object can travel at or more than the speed of light.","rendered":"
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Einstein’s Equation<\/h3>\n

At the beginning of the module, we mentioned briefly how Albert Einstein showed that a limit exists to how fast any object can travel. Given Einstein\u2019s equation for the mass of a moving object, what is the value of this bound?<\/p>\n<\/div>\n<\/div>\n

Our starting point is Einstein\u2019s equation for the mass of a moving object,<\/p>\n

[latex]m=\\dfrac{m_0}{\\sqrt{1-\\frac{v^2}{c^2}}}[\/latex]<\/p>\n

where [latex]m_0[\/latex]\u00a0is the object\u2019s mass at rest, [latex]v[\/latex]\u00a0is its speed, and\u00a0[latex]c[\/latex]\u00a0is the speed of light. To see how the mass changes at high speeds, we can graph the ratio of masses [latex]\\frac{m}{m_0}[\/latex]\u00a0as a function of the ratio of speeds,\u00a0[latex]\\frac{v}{c}[\/latex].<\/p>\n

\"A<\/p>\n

Figure 2.23 This graph shows the ratio of masses as a function of the ratio of speeds in Einstein\u2019s equation for the mass of a moving object.<\/p>\n<\/div>\n

We can see that as the ratio of speeds approaches 1\u2014that is, as the speed of the object approaches the speed of light\u2014the ratio of masses increases without bound. In other words, the function has a vertical asymptote at [latex]\\frac{v}{c}=1[\/latex]. We can try a few values of this ratio to test this idea.<\/p>\n\n\n\n\n\n\n
[latex]\\dfrac{v}{c}[\/latex]<\/td>\n[latex]\\sqrt{1-\\frac{v^2}{c^2}}[\/latex]<\/td>\n[latex]\\dfrac{m}{m_0}[\/latex]<\/td>\n<\/tr>\n
[latex]0.99[\/latex]<\/td>\n[latex]0.1411[\/latex]<\/td>\n[latex]7.089[\/latex]<\/td>\n<\/tr>\n
[latex]0.999[\/latex]<\/td>\n[latex]0.0447[\/latex]<\/td>\n[latex]22.37[\/latex]<\/td>\n<\/tr>\n
[latex]0.9999[\/latex]<\/td>\n[latex]0.0141[\/latex]<\/td>\n[latex]70.71[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

So what does this mean?<\/p>\n

According to Table 1, if an object with mass [latex]100[\/latex] kg is traveling at [latex]0.9999c[\/latex], its mass becomes [latex]7071[\/latex] kg. Since no object can have an infinite mass, we conclude that no object can travel at or more than the speed of light.<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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