Putting It Together: Limits

Einstein’s Equation

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’s equation for the mass of a moving object, what is the value of this bound?

Our starting point is Einstein’s equation for the mass of a moving object,

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

where [latex]m_0[/latex] is the object’s mass at rest, [latex]v[/latex] is its speed, and [latex]c[/latex] is 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] as a function of the ratio of speeds, [latex]\frac{v}{c}[/latex].

A graph showing the ratio of masses as a function of the ratio of speed in Einstein’s equation for the mass of a moving object. The x axis is the ratio of the speeds, v/c. The y axis is the ratio of the masses, m/m0. The equation of the function is m = m0 / sqrt(1 – v2 / c2 ). The graph is only in quadrant 1. It starts at (0,1) and curves up gently until about 0.8, where it increases seemingly exponentially; there is a vertical asymptote at v/c (or x) = 1.

Figure 2.23 This graph shows the ratio of masses as a function of the ratio of speeds in Einstein’s equation for the mass of a moving object.

We can see that as the ratio of speeds approaches 1—that is, as the speed of the object approaches the speed of light—the 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.

[latex]\dfrac{v}{c}[/latex] [latex]\sqrt{1-\frac{v^2}{c^2}}[/latex] [latex]\dfrac{m}{m_0}[/latex]
[latex]0.99[/latex] [latex]0.1411[/latex] [latex]7.089[/latex]
[latex]0.999[/latex] [latex]0.0447[/latex] [latex]22.37[/latex]
[latex]0.9999[/latex] [latex]0.0141[/latex] [latex]70.71[/latex]

So what does this mean?

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.