Chapter 1.6, Problem 72E

When an object of mass m moves with a velocity v that is small compared to the velocity of light, c, its energy is given approximately by

$E\approx \frac{1}{2}m{v}^2$.

If v is comparable in size to c, then the energy must be computed by the exact formula

$E=m{c}^2\left(\frac{1}{\sqrt{1-v^2/c^2}}-1\right)$.

1. (a) Plot a graph of both functions for E against v for 0 ≤ v ≤ 5⋅ 10⁸ and 0 ≤ E ≤ 5⋅ 10¹⁷. Take m = 1 kg and c = 3⋅ 10⁸ m/sec. Explain how you can predict from the exact formula the position of the vertical asymptote.
2. (b) What do the graphs tell you about the approximation? For what values of v does the first formula give a good approximation to E?