Find the kinetic energy of an electron moving at (a) \(0.0010 c\) (b) \(0.60 c,\) and \((\mathrm{c}) 0.99 c .\) Use suitable approximations where possible.

When we talk about the non-relativistic kinetic energy of an electron, we refer to the energy that it possesses due to its motion at speeds much lower than the speed of light. The formula to calculate this energy is derived from classical mechanics and is given by the equation:
\[ KE = \frac{1}{2} m v^2 \]
where \( KE \) is the kinetic energy, \( m \) is the mass of the electron, and \( v \) is its velocity. For an electron with mass \(9.11 \times 10^{-31} \) kilograms, moving at speed \( v = 0.0010 c \) – where \( c \) is the speed of light – the non-relativistic kinetic energy can be a useful approximation. This calculation assumes that the effects of Einstein's theories of relativity can be ignored, which is generally acceptable for speeds that are much less than the speed of light.
Keep in mind that this approximation becomes less accurate as the electron's speed approaches a significant fraction of the speed of light. Hence, for higher speeds, such as \( 0.60 c \) and \( 0.99 c \), we consider a different approach for high precision in calculations.

As an electron approaches speeds comparable to that of light, its kinetic energy can no longer be accurately represented by the classical equation. Instead, we must turn to Einstein's theory of relativity. The relativistic kinetic energy accounts for the fact that as objects move faster, their mass effectively increases and thus requires more energy to accelerate. The relativistic kinetic energy formula is:
\[ K = mc^2\left(\frac{1}{\sqrt{1 - \left(\frac{v^2}{c^2}\right)}} - 1\right) \]
In this equation, \( K \) is the relativistic kinetic energy, \( m \) is the rest mass of the electron, \( v \) is its velocity, and \( c \) is the speed of light. For electrons moving at significant fractions of the speed of light, such as \( 0.60 c \) or \( 0.99 c \), using this relativistic modification is crucial to derive an accurate measure of kinetic energy.
This formula shows that as the speed \( v \) of the electron approaches the speed of light \( c \), the denominator approaches zero, making the kinetic energy infinitely large. This reflects the principle that no particle with mass can reach or exceed the speed of light.

Approximations are a fundamental tool in physics calculations, allowing us to simplify complex equations into more manageable forms when certain conditions are met. When dealing with speeds that are significantly lower than the speed of light – as is the case with an electron moving at \( 0.0010 c \) – we can approximate the value of \( \sqrt{1 - (v^2/c^2)} \) to be nearly equal to 1.
This results in the relativistic kinetic energy equation simplifying to:
\[ K \approx mc^2\left(\frac{1}{1} - 1\right) = 0 \]
This simplification is handy when dealing with lower speeds since it negates the need to compute the square root. However, this approximation becomes increasingly faulty as the velocity of the electron gets closer to the speed of light. In physics, the art of approximation requires a critical understanding of when and to what extent an approximation remains valid—the difference between a useful simplification and an erroneous result hinges on this judgment.