(a) Find the kinetic energy of a 78.0-kg spacecraft launched...

(a) Find the kinetic energy of a 78.0-kg spacecraft launched out of the solar system with speed 106 $\mathrm{km} / \mathrm{s}$ by using the classical equation $K=\frac{1}{2} m u^{2} .$ (b) What If? Calculate its kinetic energy using the relativistic equation. (c) Explain the result of comparing the answers of parts (a) and (b).

Key Concepts

Lorentz Factor

The Lorentz factor, denoted as ?, is defined by the equation ? = 1/?(1 - u²/c²). It quantifies the extent to which time, length, and relativistic mass change for objects in motion relative to an observer. This factor is fundamental in converting classical to relativistic expressions of kinetic energy and is key to understanding how energy and momentum behave at high speeds.

Comparison Between Classical and Relativistic Calculations

Comparing the classical and relativistic calculations of kinetic energy highlights how, at lower speeds (when u is much less than c), the relativistic kinetic energy converges to the classical value. This result underlines the validity of classical mechanics under typical conditions, while demonstrating the necessity of relativistic corrections as speeds increase. It provides insight into transitioning between regimes where either classical or relativistic physics is applicable.

Classical Kinetic Energy

This concept involves the calculation of kinetic energy using the formula K = ½ m u², where m is the mass of an object and u is its velocity. It assumes that the speeds involved are much lower than the speed of light, so that relativistic effects can be neglected. This method is central to classical mechanics and gives accurate results under everyday conditions.

Relativistic Kinetic Energy

In contrast to the classical approach, the relativistic kinetic energy is calculated using the formula K = (? - 1) m c². Here, c is the speed of light and ? (the Lorentz factor) accounts for the relativistic effects that arise at speeds approaching c. This adjustment is crucial when objects move at a significant fraction of the speed of light, ensuring that energy calculations remain valid under the framework of Einstein's theory of special relativity.

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