Problem 45 - Essential Calculus Early Transcendentals

Question

In the theory of relativity, the mass of a particle is $$m=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}}$$ where \(m_{0}\) is the rest mass of the particle, \(m\) is the mass when the particle moves with speed \(v\) relative to the observer, and \(c\) is the speed of light. Sketch the graph of \(m\) as a function of \(v .\)

Step-by-Step Solution

Answer

The graph of \( m \) vs. \( v \) starts at \( m_0 \) and increases steeply to infinity as \( v \) approaches \( c \).

Step 1: Understand the Expression

The equation for the relativistic mass is \( m = \frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}} \). This equation shows that as the velocity \( v \) approaches the speed of light \( c \), the denominator approaches zero, implying that \( m \) increases.

Step 2: Identify Important Values of v

We need to consider some critical points: when \( v = 0 \), \( v = c/2 \), \( v = \frac{3c}{4} \), and as \( v \) approaches \( c \) from the left. These values will help determine how the function behaves.

Step 3: Calculate Mass at Key Points

- At \( v = 0 \), \( m = m_{0}/\sqrt{1-0} = m_{0} \).- At \( v = c/2 \), \( m = \frac{m_{0}}{\sqrt{1-(1/4)}} = \frac{m_{0}}{\sqrt{3/4}} = \frac{m_{0}}{\sqrt{0.75}} \approx 1.155 m_{0} \).- As \( v \) approaches \( c \), the expression for \( m \) shows the mass grows towards infinity because the denominator approaches zero.

Step 4: Sketch the Function

Plot \( m \) against \( v \) on a graph with \( v \) ranging from 0 to \( c \). The curve starts at \( m = m_{0} \) when \( v = 0 \) and increases to infinity as \( v \) gets closer to \( c \). The curve is steepest as it nears \( v = c \), showing a rapid increase in mass.

Step 5: Understand Domain and Range

The domain of \( m(v) \) is \( v \in [0, c) \) because the velocity \( v \) cannot reach \( c \) or exceed it. The range is \( m \in [m_{0}, \infty) \), reflecting that the mass is always greater than or equal to the rest mass \( m_{0} \).

Key Concepts

Theory of Relativity

The theory of relativity, developed by Albert Einstein, transformed our understanding of space, time, and gravity. There are two parts to it: special relativity and general relativity. Special relativity, relevant to our discussion here, addresses the physics of objects moving at significant fractions of the speed of light. It suggests that time and space are not absolute; instead, they vary with speed.
Einstein's famous equation, $E=mc^2$, suggests a fundamental link between mass and energy. In the context of special relativity, this means as the speed of an object increases, its relativistic mass also changes, which has implications for how we understand motion at high speeds.

• Special Relativity: Deals with objects moving in a straight line at constant speeds close to the speed of light.
• Mass-Energy Equivalence: Indicates mass can be converted to energy and vice versa.

These concepts are crucial for explaining why particles behave differently near the speed of light compared to everyday speeds.

Speed of Light

The speed of light, denoted by the symbol $c$, is a constant in the vacuum of space and is approximately equal to $299,792,458$ meters per second. This speed is central to the theory of relativity and sets an upper limit on how fast information and matter can travel.
Light speed's importance in relativity cannot be understated. It acts as a cosmic speed limit. No object with mass can reach or exceed this speed in vacuum due to the effects predicted by relativity, such as the increase in relativistic mass.

• Constant Nature: Light's speed does not change regardless of the observer's movement.
• Upper Limit: Sets the maximum speed at which all energy, information, and matter can travel.

The speed of light ensures that as particles approach this speed, their physical properties, like mass, start behaving in extraordinary ways.

Rest Mass

Rest mass, often denoted as $m_0$, is the mass of a particle when it is at rest relative to the observer. This concept is critical for understanding motion from the perspective of relativity.
In various physical reactions and phenomena, this mass remains invariant and is used as a baseline for understanding changes in an object's mass as it moves at high speeds. Under normal conditions, the rest mass of an object is what we measure in everyday life and does not change.

• Invariant Property: Remains unchanged regardless of the object's velocity relative to the observer.
• Baseline Measurement: Used as the initial reference point for comparing relativistic changes in mass.

A solid understanding of rest mass helps grasp how it's used in measurements and calculations within the domain of relativistic physics.

Mass-Velocity Relationship

The mass-velocity relationship explores how a particle's mass changes as its speed increases. According to special relativity, as a particle's speed approaches the speed of light, its relativistic mass increases infinitely. This is due to the equation:\[m = \frac{m_0}{\sqrt{1-v^2/c^2}}\]
This equation highlights several key insights:

• Mass Increase: As velocity \(v\) increases towards \(c\), the denominator approaches zero, causing mass \(m\) to grow significantly.
• Never Reaching Light Speed: No object with mass can reach or exceed \(c\), as its mass would become infinite.
• Graphical Representation: A graph of \(m\) vs. \(v\) shows a steep curve, beginning at \(m_0\) when \(v=0\) and trending upwards as \(v\) nears \(c\).

This relationship illustrates the profound effects of high velocities on an object's physical properties, making it a cornerstone concept in relativistic mechanics.