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

Physicists use binomial expansion for small velocities to approximate the relativistic mass equation.

Step 1: Understand the Problem

We need to approximate the relativistic mass formula \( m = \frac{m_0}{\sqrt{1-v^2/c^2}} \) for velocities much less than the speed of light (\( v << c \)). This approximation is used to simplify calculations under classical mechanics conditions.

Step 2: Apply Binomial Expansion

For \( x << 1 \), a common approximation is \((1-x)^{-1/2} \approx 1 + \frac{1}{2}x \). Here, we will use \( x = \frac{v^2}{c^2} \) in our formula. This applies to the expression \( m = \frac{m_0}{\sqrt{1-v^2/c^2}} \).

Step 3: Substitute and Simplify

Substitute \( x = \frac{v^2}{c^2} \) into the binomial expansion to get:\[ (1 - \frac{v^2}{c^2})^{-1/2} \approx 1 + \frac{1}{2} \frac{v^2}{c^2} \].Substitute this back into the mass formula to get:\[ m \approx m_0 (1 + \frac{1}{2} \frac{v^2}{c^2}) \].

Step 4: Expand and Conclude

Expand the final expression:\[ m \approx m_0 + \frac{m_0}{2} \left(\frac{v^2}{c^2}\right) \].Therefore, for small velocities compared to the speed of light, the mass can be approximated as being close to the rest mass plus an additional term based on the velocity.

Key Concepts

Theory of Relativity

The Theory of Relativity, introduced by Albert Einstein, revolutionized our understanding of physics. It comprises two theories: Special Relativity and General Relativity. Special Relativity focuses on objects moving at constant speeds, particularly those close to the speed of light. A key aspect is the concept that the laws of physics are the same for all non-accelerating observers. It also introduces the idea that time and space are intertwined into a single continuum known as spacetime. This leads to intriguing phenomena such as time dilation and length contraction, where time can slow down, and lengths can contract based on an object's relative speed.
In the context of mass, Special Relativity introduces the concept of relativistic mass, where the mass of an object increases with its velocity. The formula \[ m = \frac{m_0}{\sqrt{1-v^2/c^2}} \]gives the relativistic mass \( m \), where \( m_0 \) is the object's rest mass, \( v \) its velocity, and \( c \) the speed of light. If an object's velocity is significantly less than the speed of light, the effect on mass is negligible, aligning relativity with classical mechanics. This insight allows physicists to make approximations in calculations, simplifying complex relativistic equations for practical use when speeds are much less than \( c \).
Understanding these concepts is crucial for appreciating why relativistic effects matter when dealing with high-speed objects. It provides a framework for how matter and energy behave at these extreme conditions.

Binomial Expansion

Binomial Expansion transforms complex mathematical expressions into simpler ones by approximating them for small values. It's particularly useful when dealing with equations where an exact solution is cumbersome or not required, especially in physics. For functions involving terms like \((1-x)^{-1/2}\), a binomial expansion can approximate expressions when \( x \) is much smaller than 1. The generic form used here is \[(1-x)^{-1/2} \approx 1 + \frac{1}{2}x \]. This is derived from the formula for the binomial series, which provides a way to expand expressions involving powers.
In the solution, this expansion method helps simplify the relativistic mass formula to a more manageable form by setting \( x = \frac{v^2}{c^2} \). This approximation reduces the relativistic mass equation without requiring a detailed or complex computation, making it accessible within the realms of classical mechanics, where the velocity \( v \) is much less than the speed of light \( c \).
Understanding binomial expansion is not only fundamental in algebra but also crucial for practical applications in physics and engineering, where it aids in making quick, reliable estimations that ease computational intensity.

Classical Mechanics

Classical Mechanics is a branch of physics that deals with the motion of objects and the forces acting on them under everyday conditions. Established long before the theories of relativity, classical mechanics relies on Newton's laws of motion to describe the effects and interactions of forces. It assumes that time and space are absolute and independent, which holds true for scenarios involving slow-moving objects relative to the speed of light.

• Objects follow deterministic paths determined by known initial conditions and forces.
• Mass is a constant property of matter, independent of velocity.

When dealing with velocities that are a tiny fraction of the speed of light, Relativistic effects are negligible, thus classical mechanics can effectively approximate reality. It simplifies computations of motion, collisions, and dynamics, providing a straightforward framework for understanding most day-to-day physical phenomena.
The approximated formula \( m \approx m_0 + \frac{m_0}{2}\left(\frac{v^2}{c^2}\right) \) derived from the relativistic mass equation, aligns with the principles of classical mechanics. This means that under conditions of low velocities, classical and relativistic mechanics coincide closely, reinforcing the applicability and robustness of classical physics under such circumstances.
Appreciating classical mechanics allows you to grasp the limitations and conditions where more advanced theories like relativity become necessary, bridging the gap between everyday physics and the universe's more extreme conditions.