Problem 54 - Calculus Early Transcendentals: Pearson New International Edition

Question

If an object of rest mass \(m_{0}\) has velocity \(v\), then (according to the theory of relativity) its mass \(m\) is given by \(m=\) \(m_{0} / \sqrt{1-v^{2} / c^{2}}\), where \(c\) is the velocity of light. Explain how physicists get the approximation $$ m \approx m_{0}+\frac{m_{0}}{2}\left(\frac{v}{c}\right)^{2} $$

Step-by-Step Solution

Answer

Use the binomial expansion on the relativistic equation for small \(v\) values.

Step 1: Understand the Equation

The problem provides the relativistic mass equation for an object: \[ m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} \]where \(m_0\) is the rest mass, \(v\) is the velocity of the object, and \(c\) is the speed of light.

Step 2: Identify the Series Expansion

To approximate for small velocities \(v\), we can use a binomial expansion of the term \[ \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]The binomial expansion formula is \[(1-x)^n \approx 1 + nx + \frac{n(n-1)}{2}x^2 + \cdots\]We apply this expansion to \[ \sqrt{1 - \frac{v^2}{c^2}} \] by setting \(x = \frac{v^2}{c^2}\) and \(n = -\frac{1}{2}\).

Step 3: Apply the Binomial Expansion

Expanding \[ \sqrt{1-\frac{v^2}{c^2}} \] using the binomial theorem:\[ \left(1 - \frac{v^2}{c^2}\right)^{-\frac{1}{2}} \approx 1 + \frac{1}{2} \cdot \frac{v^2}{c^2} \] We neglect higher-order terms because they become significantly small when \(v\) is much smaller than \(c\).

Step 4: Substitute the Expansion

After approximating, substitute back into the original formula:\[ m = m_0 \cdot \left(1 + \frac{1}{2} \cdot \frac{v^2}{c^2}\right) \]Simplifying this expression gives:\[ m \approx m_0 + \frac{m_0}{2} \left(\frac{v}{c}\right)^2 \] This matches the provided approximation formula.

Key Concepts

Theory of Relativity

The Theory of Relativity, proposed by Albert Einstein, changed our understanding of physics. It includes the Special Theory of Relativity, which describes how objects behave at high speeds close to the speed of light and how mass changes with velocity. The key formula derived from this theory to calculate the relativistic mass is: \[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]Where:- \(m\) is the relativistic mass.- \(m_0\) is the rest mass, which is the mass of the object when it is not moving relative to the observer.- \(v\) is the velocity of the object.- \(c\) is the velocity of light.This formula shows that as an object's velocity approaches the speed of light, \(v\) gets closer to \(c\), the denominator of the equation gets closer to zero, and consequently, the relativistic mass \(m\) increases.

Binomial Expansion

The Binomial Expansion is a mathematical series expansion that helps approximate expressions raised to a power, especially useful when exact computation is complex. In simpler terms, when you have an expression like \[(1-x)^n \]you can expand it as: \[1 + nx + \frac{n(n-1)}{2}x^2 + \cdots\]To solve the given problem with relativistic mass, we applied binomial expansion to the expression: \[\sqrt{1 - \frac{v^2}{c^2}}\]By substituting \(x = \frac{v^2}{c^2}\) and \(n = -\frac{1}{2}\) in the expansion formula, we could approximate that for small values of \(v\) compared to \(c\), higher-order terms are negligible. The simplified expansion becomes: \[1 + \frac{1}{2}x\]This binomial approximation is vital for simplifying the calculation of relativistic effects at velocities much less than the speed of light.

Rest Mass

The concept of rest mass \(m_0\) is central to the theory of relativity. It refers to the intrinsic mass of an object when it is stationary relative to the observer.Unlike relativistic mass, the rest mass does not change regardless of the object's velocity.In the relativistic mass formula, rest mass acts as a reference point from which changes in mass can be measured as velocity changes.This makes the rest mass a useful and constant quantity, providing a stable measure, especially significant when evaluating the mass-energy equivalence expressed in Einstein's famous formula: \[E=mc^2\] which describes how energy \(E\) is related to mass \(m\), emphasizing that the inherent energy content of an object is attributable to its rest mass.

Velocity of Light

The velocity of light, represented as \(c\), is a fundamental constant in physics. It is approximately equal to \(299,792,458 \text{ meters per second} \) and is crucial for understanding the limits imposed by the universe on how fast information and matter can travel.In the context of relativistic physics, the velocity of light serves as the maximum speed at which massive particles and information can travel.When calculating relativistic mass, as an object's velocity approaches the velocity of light, relativistic effects become pronounced.That's why, in the relativistic mass formula: \[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]the factor \(\frac{v^2}{c^2}\) becomes critical as \(v\) nears \(c\), leading to the dramatic increase in mass observed as objects move at relativistic speeds.