One important aspect of Einstein's theory of relativity is...

One important aspect of Einstein's theory of relativity is that mass is not constant. For a person with mass $m_{0}$ at rest, the mass will equal $m=m_{0} / \sqrt{1-v^{2} / c^{2}}$ at velocity $v$ (where $c$ is the speed of light). Thinking of $m$ as a function of $v$, find the linear approximation of $m(v)$ at $v=0$. Use the linear approximation to show that mass is essentially constant for small velocities.

Key Concepts

Special Relativity

Special relativity is a fundamental theory in physics developed by Albert Einstein, which reshapes our understanding of space and time. It introduces the idea that the laws of physics are the same for all inertial observers and that the speed of light in a vacuum is constant, regardless of the observer's motion. This framework leads to non-intuitive phenomena such as time dilation, length contraction, and the velocity-dependent behavior of mass.

Relativistic Mass

Relativistic mass refers to the concept that the mass of an object increases with its velocity relative to an observer. According to special relativity, the mass of an object is given by the formula m = m0/?(1-v²/c²), where m0 is the rest mass, v is the velocity, and c is the speed of light. This relationship implies that as an object's velocity approaches the speed of light, its mass tends toward infinity, highlighting the prohibitive energy requirements to reach light speed.

Linear Approximation

Linear approximation, often based on the Taylor series expansion, is a mathematical method used to approximate complex functions by a linear function near a specific point. This technique simplifies the analysis by considering only the first-order terms in the expansion, making it easier to understand the behavior of functions like the velocity-dependent mass function near v = 0.

Small Velocity Limit

The small velocity limit considers situations where the velocity of an object is much smaller compared to the speed of light (v << c). In this regime, the relativistic effects become negligible, and the mass function can be approximated linearly. The linear approximation shows that the mass remains essentially constant, closely equal to the rest mass, which reaffirms why classical mechanics is sufficient for describing everyday experiences at low speeds.

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